7005:
14742:
7964:
6521:
8476:
6301:
12747:
7536:
25980:
7000:{\displaystyle {\begin{aligned}K(x_{f},t_{f};x_{i},t_{i})&=Qe^{\frac {iS_{\text{c}}}{\hbar }}\prod _{j=1}^{\infty }{\frac {j\pi }{\sqrt {2}}}\int da_{j}\exp {\left({\frac {i}{2\hbar }}a_{j}^{2}{\frac {m}{2}}\left({\frac {(j\pi )^{2}}{t_{f}-t_{i}}}-\omega ^{2}(t_{f}-t_{i})\right)\right)}\\&=e^{\frac {iS_{\text{c}}}{\hbar }}Q\prod _{j=1}^{\infty }\left(1-\left({\frac {\omega (t_{f}-t_{i})}{j\pi }}\right)^{2}\right)^{-{\frac {1}{2}}}\end{aligned}}}
8040:
5911:
23511:, the "sum over histories" interpretation, the path integral is taken to be fundamental, and reality is viewed as a single indistinguishable "class" of paths that all share the same events. For this interpretation, it is crucial to understand what exactly an event is. The sum-over-histories method gives identical results to canonical quantum mechanics, and Sinha and Sorkin claim the interpretation explains the
12261:
7959:{\displaystyle {\begin{aligned}K(x_{f},t_{f};x_{i},t_{i})&=\left({\frac {m\omega }{2\pi i\hbar \sin \omega T}}\right)^{\frac {1}{2}}\exp {\left({\frac {i}{\hbar }}{\tfrac {1}{2}}m\omega {\frac {(x_{i}^{2}+x_{f}^{2})\cos \omega T-2x_{i}x_{f}}{\sin \omega T}}\right)}\\&=\sum _{n=0}^{\infty }\exp {\left(-{\frac {iE_{n}T}{\hbar }}\right)}\psi _{n}(x_{f})\psi _{n}(x_{i})^{*}~.\end{aligned}}}
19261:
7505:
3789:
8471:{\displaystyle K(x_{f},t_{f};x_{i},t_{i})=\left({\frac {m\omega }{\pi \hbar }}\right)^{\frac {1}{2}}e^{\frac {-i\omega T}{2}}\left(1-e^{-2i\omega T}\right)^{-{\frac {1}{2}}}\exp {\left(-{\frac {m\omega }{2\hbar }}\left(\left(x_{i}^{2}+x_{f}^{2}\right){\frac {1+e^{-2i\omega T}}{1-e^{-2i\omega T}}}-{\frac {4x_{i}x_{f}e^{-i\omega T}}{1-e^{-2i\omega T}}}\right)\right)}.}
20419:
15955:, this would not have been such a big problem – it would just have been a bad choice of coordinates. But the lack of symmetry means that the infinite quantities must be cut off, and the bad coordinates make it nearly impossible to cut off the theory without spoiling the symmetry. This makes it difficult to extract the physical predictions, which require a
9298:
20968:
2859:
6296:{\displaystyle {\begin{aligned}S_{\text{c}}&=\int _{t_{i}}^{t_{f}}{\mathcal {L}}\,dt=\int _{t_{i}}^{t_{f}}\left({\tfrac {1}{2}}m{\dot {x}}^{2}-{\tfrac {1}{2}}m\omega ^{2}x^{2}\right)\,dt\\&={\frac {1}{2}}m\omega \left({\frac {(x_{i}^{2}+x_{f}^{2})\cos \omega (t_{f}-t_{i})-2x_{i}x_{f}}{\sin \omega (t_{f}-t_{i})}}\right)~.\end{aligned}}}
10408:
13278:
12742:{\displaystyle \langle x_{f}|e^{-{\frac {i}{\hbar }}{\hat {H}}(t-t_{1})}F_{1}({\hat {x}})e^{-{\frac {i}{\hbar }}{\hat {H}}(t_{1}-t_{2})}F_{2}({\hat {x}})e^{-{\frac {i}{\hbar }}{\hat {H}}(t_{2})}|x_{i}\rangle =\int _{x(0)=x_{i}}^{x(t)=x_{f}}{\mathcal {D}}F_{1}(x(t_{1}))F_{2}(x(t_{2}))e^{{\frac {i}{\hbar }}\int dtL(x(t),{\dot {x}}(t))}}
10201:
4296:
21458:, but in the sense that the transformed value of the field at any given point under an infinitesimal transformation would only depend on the field configuration over an arbitrarily small neighborhood of the point in question). Let's also assume that the action is local in the sense that it is the integral over spacetime of a
19002:
6510:
7223:
5900:
14180:
Much of the study of quantum field theories from the path-integral perspective, in both the mathematics and physics literatures, is done in the
Euclidean setting, that is, after a Wick rotation. In particular, there are various results showing that if a Euclidean field theory with suitable properties
20693:
15962:
The problem of lost symmetry also appears in classical mechanics, where the
Hamiltonian formulation also superficially singles out time. The Lagrangian formulation makes the relativistic invariance apparent. In the same way, the path integral is manifestly relativistic. It reproduces the Schrödinger
5011:
in the numerator, the time slicing produces convolved
Gaussians, just as before. Now, however, the convolution product is marginally singular, since it requires careful limits to evaluate the oscillating integrals. To make the factors well defined, the easiest way is to add a small imaginary part to
4850:
The proportionality constant is not really determined by the time-slicing approach, only the ratio of values for different endpoint choices is determined. The proportionality constant should be chosen to ensure that between each two time slices the time evolution is quantum-mechanically unitary, but
2284:
possible paths of the system in between the initial and final states, including those that are absurd by classical standards. In calculating the probability amplitude for a single particle to go from one space-time coordinate to another, it is correct to include paths in which the particle describes
20073:
19633:
17158:
The integral above is not trivial to interpret because of the square root. Fortunately, there is a heuristic trick. The sum is over the relativistic arc length of the path of an oscillating quantity, and like the nonrelativistic path integral should be interpreted as slightly rotated into imaginary
16843:
It is also possible to reexpress the nonrelativistic time evolution in terms of propagators going toward the past, since the Schrödinger equation is time-reversible. The past propagator is the same as the future propagator except for the obvious difference that it vanishes in the future, and in the
1139:
The
Hamiltonian is a function of the position and momentum at one time, and it determines the position and momentum a little later. The Lagrangian is a function of the position now and the position a little later (or, equivalently for infinitesimal time separations, it is a function of the position
18627:
Unlike the nonrelativistic case, it is impossible to produce a relativistic theory of local particle propagation without including antiparticles. All local differential operators have inverses that are nonzero outside the light cone, meaning that it is impossible to keep a particle from travelling
15970:
Further, different choices of canonical variables lead to very different-seeming formulations of the same theory. The transformations between the variables can be very complicated, but the path integral makes them into reasonably straightforward changes of integration variables. For these reasons,
2227:
Feynman showed that Dirac's quantum action was, for most cases of interest, simply equal to the classical action, appropriately discretized. This means that the classical action is the phase acquired by quantum evolution between two fixed endpoints. He proposed to recover all of quantum mechanics
16992:
For a nonrelativistic theory, the time as measured along the path of a moving particle and the time as measured by an outside observer are the same. In relativity, this is no longer true. For a relativistic theory the propagator should be defined as the sum over all paths that travel between two
13674:
3282:
20231:
18306:
The second term has a nonrelativistic limit also, but this limit is concentrated on frequencies that are negative. The second pole is dominated by contributions from paths where the proper time and the coordinate time are ticking in an opposite sense, which means that the second term is to be
4854:
The result has a probability interpretation. The sum over all paths of the exponential factor can be seen as the sum over each path of the probability of selecting that path. The probability is the product over each segment of the probability of selecting that segment, so that each segment is
1077:
15974:
The price of a path integral representation is that the unitarity of a theory is no longer self-evident, but it can be proven by changing variables to some canonical representation. The path integral itself also deals with larger mathematical spaces than is usual, which requires more careful
18506:
12057:
14369:
9045:
22494:
18905:
17153:
12899:
4086:
20712:
8661:
8839:
2661:
9689:
and this is the Schrödinger equation. The normalization of the path integral needs to be fixed in exactly the same way as in the free particle case. An arbitrary continuous potential does not affect the normalization, although singular potentials require careful treatment.
11190:
9577:
3539:
18145:
3792:
10216:
17852:
12249:
14011:
3796:
3795:
3791:
3790:
5396:
14181:
can be constructed, one can then undo the Wick rotation to recover the physical, Lorentzian theory. On the other hand, it is much more difficult to give a meaning to path integrals (even
Euclidean path integrals) in quantum field theory than in quantum mechanics.
13082:
15975:
mathematics, not all of which has been fully worked out. The path integral historically was not immediately accepted, partly because it took many years to incorporate fermions properly. This required physicists to invent an entirely new mathematical object – the
3797:
19293:
Since this formulation of quantum mechanics is analogous to classical action principle, one might expect that identities concerning the action in classical mechanics would have quantum counterparts derivable from a functional integral. This is often the case.
22224:
20220:
17695:
10034:
18619:
So in the relativistic case, the
Feynman path-integral representation of the propagator includes paths going backwards in time, which describe antiparticles. The paths that contribute to the relativistic propagator go forward and backwards in time, and the
4097:
9684:
19256:{\displaystyle \operatorname {P} (B\mid A)={\frac {\sum _{F\subset A\cap B}\left|\int {\mathcal {D}}\varphi O_{\text{in}}e^{i{\mathcal {S}}}F\right|^{2}}{\sum _{F\subset A}\left|\int {\mathcal {D}}\varphi O_{\text{in}}e^{i{\mathcal {S}}}F\right|^{2}}},}
11387:
just set it equal to 1. In physics, such a quantity is "equal to 1 as an operator identity". In mathematics, it "weakly converges to 1". In either case, it is 1 in any expectation value, or when averaged over any interval, or for all practical purpose.
10019:
10517:
9033:
2357:
15986:
The integration variables in the path integral are subtly non-commuting. The value of the product of two field operators at what looks like the same point depends on how the two points are ordered in space and time. This makes some naive identities
23684:, the path integral formation can be used to determine an effective action and pre-exponential model to see the effect of dissipation on tunnelling. From this model, tunneling rates of macroscopic systems (at finite temperatures) can be predicted.
21074:
19885:
7212:
16383:
14485:, i.e., make time imaginary, and integrate over all possible beginning-ending configurations. The Wick-rotated path integral—described in the previous subsection, with the ordinary action replaced by its "Euclidean" counterpart—now resembles the
21846:
17306:
19727:
23038:
and the path integral formulation, an ordering prescription is required to resolve the ambiguity in the correspondence between non-commutative operators and the commutative functions that appear in path integrands. For example, the operator
17439:
11063:
16003:
The nonrelativistic formulation is traditionally given in terms of particle paths, not fields. There, the path integral in the usual variables, with fixed boundary conditions, gives the probability amplitude for a particle to go from point
23649:
16972:
5617:
3794:
12914:
from real to imaginary times. In the setting of quantum field theory, the Wick rotation changes the geometry of space-time from
Lorentzian to Euclidean; as a result, Wick-rotated path integrals are often called Euclidean path integrals.
11854:
16812:
5494:
17558:
7500:{\displaystyle K(x_{f},t_{f};x_{i},t_{i})=Qe^{\frac {iS_{\text{c}}}{\hbar }}{\sqrt {\frac {\omega (t_{f}-t_{i})}{\sin \omega (t_{f}-t_{i})}}}=e^{\frac {iS_{c}}{\hbar }}{\sqrt {\frac {m\omega }{2\pi i\hbar \sin \omega (t_{f}-t_{i})}}}.}
13518:
6316:
2004:
representation it is reinterpreted as a quantity associated to the path. In the limit that one takes a large power of this operator, one reconstructs the full quantum evolution between two states, the early one with a fixed value of
15999:
In relativistic theories, there is both a particle and field representation for every theory. The field representation is a sum over all field configurations, and the particle representation is a sum over different particle paths.
22820:
18294:
16168:
11291:
5686:
15841:
Both the Schrödinger and
Heisenberg approaches to quantum mechanics single out time and are not in the spirit of relativity. For example, the Heisenberg approach requires that scalar field operators obey the commutation relation
1254:
20526:
14142:
10947:
19896:
19520:
4440:
2525:
4654:
21209:
9404:
4561:
20414:{\displaystyle \langle F\rangle _{J}={\frac {\int {\mathcal {D}}\varphi Fe^{i\left({\mathcal {S}}+\langle J,\varphi \rangle \right)}}{\int {\mathcal {D}}\varphi e^{i\left({\mathcal {S}}+\langle J,\varphi \rangle \right)}}}.}
18920:
here is a concise way to represent the infinite-dimensional integral over all possible field configurations on all of space-time. As stated above, the unadorned path integral in the denominator ensures proper normalization.
13537:
14631:
10748:
3135:
113:
18339:
17315:
of the exponential of minus the length. This can be given a probability interpretation. The sum over all paths is a probability average over a path constructed step by step. The total number of steps is proportional to
14547:
Clearly, such a deep analogy between quantum mechanics and statistical mechanics cannot be dependent on the formulation. In the canonical formulation, one sees that the unitary evolution operator of a state is given by
23497:
23121:
9819:
9315:. In this case, to lowest order the potential energy is constant, and only the kinetic energy contribution is nontrivial. (This separation of the kinetic and potential energy terms in the exponent is essentially the
9293:{\displaystyle \psi (y;t+\varepsilon )=\int _{-\infty }^{\infty }\psi (x;t)\int _{x(t)=x}^{x(t+\varepsilon )=y}e^{i\int _{t}^{t+\varepsilon }{\bigl (}{\frac {1}{2}}{\dot {x}}^{2}-V(x){\bigr )}dt}Dx(t)\,dx\qquad (1)}
3124:
1093:
is the generator of time translations. This means that the state at a slightly later time differs from the state at the current time by the result of acting with the
Hamiltonian operator (multiplied by the negative
14195:
3783:
20963:{\displaystyle F\leftG=(-i)^{n}{\frac {\partial ^{k_{1}}}{\partial x_{1}^{k_{1}}}}{\frac {\delta }{\delta J(x_{1})}}\cdots {\frac {\partial ^{k_{n}}}{\partial x_{n}^{k_{n}}}}{\frac {\delta }{\delta J(x_{n})}}G.}
10825:
1421:
22335:
18780:
16999:
12759:
8915:
3919:
19358:
16490:, the propagator can be evaluated either explicitly from the path integral or by noting that the Schrödinger equation is a diffusion equation in imaginary time, and the solution must be a normalized Gaussian:
10533:, at the same moment in time, the manipulations above are only heuristic, because the multiplication rules for these quantities is just as noncommuting in the path integral as it is in the operator formalism.
7092:
3901:
17968:
5197:
2643:
17570:
is a weight factor, the relative importance of paths of different proper time. By the translation symmetry in proper time, this weight can only be an exponential factor and can be absorbed into the constant
9038:
The path integral reproduces the Schrödinger equation for the initial and final state even when a potential is present. This is easiest to see by taking a path-integral over infinitesimally separated times.
8504:
5002:
2854:{\displaystyle \int \limits _{-\infty }^{+\infty }\cdots \int \limits _{-\infty }^{+\infty }\exp \left({\frac {i}{\hbar }}\int _{t_{a}}^{t_{b}}L{\big (}x(t),v(t){\big )}\,dt\right)\,dx_{0}\,\cdots \,dx_{n},}
11890:
8695:
4845:
21450:
for the classical case? Does it have a quantum analog as well? Yes, but with a caveat. The functional measure would have to be invariant under the one parameter group of symmetry transformation as well.
21400:
18597:
16585:
10649:
1482:
1315:
7541:
6526:
5916:
5112:
4742:
10403:{\displaystyle \int \psi _{0}(x)\int _{u(0)=x}-\left(\int \left({\frac {d}{dt}}{\frac {\partial S}{\partial {\dot {u}}}}-{\frac {\partial S}{\partial u}}\right)\varepsilon (t)\,dt\right)e^{iS}\,Du\,}
11078:
9434:
3384:
21743:
17983:
11495:
25013:
Discusses the definition of Path
Integrals for systems whose kinematical variables are the generators of a real separable, connected Lie group with irreducible, square integrable representations.
23534:
Whereas in quantum mechanics the path integral formulation is fully equivalent to other formulations, it may be that it can be extended to quantum gravity, which would make it different from the
22324:
17734:
13868:
13273:{\displaystyle \psi (x,t)={\frac {1}{Z}}\int _{\mathbf {x} (0)=x}e^{-S_{\mathrm {Euclidean} }(\mathbf {x} ,{\dot {\mathbf {x} }})/\hbar }\psi _{0}(\mathbf {x} (t))\,{\mathcal {D}}\mathbf {x} \,}
22097:
5212:
18624:
of this is that the amplitude for a free particle to travel between two points includes amplitudes for the particle to fluctuate into an antiparticle, travel back in time, then forward again.
16817:
which is the reciprocal of the operator that annihilates the wavefunction in the Schrödinger equation, which wouldn't have come out right if the proportionality factor weren't constant in the
11723:
11307:
is a rapidly fluctuating statistical quantity, whose average value is 1, i.e. a normalized "Gaussian process". The fluctuations of such a quantity can be described by a statistical Lagrangian
21646:
19275:
is the superposition of all incoming states that could lead to the states we are interested in. In particular, this could be a state corresponding to the state of the Universe just after the
11638:
in Schrödinger quantum mechanics. One may, however, solve this problem by transforming the time-sliced flat-space path integral to curved space using a multivalued coordinate transformation (
22881:
18602:
which when factored, produces opposite-sign infinitesimal terms in each factor. This is the mathematically precise form of the relativistic particle propagator, free of any ambiguities. The
16679:
12065:
9706:
variables, but it is instructive to see this directly. The direct approach shows that the expectation values calculated from the path integral reproduce the usual ones of quantum mechanics.
9012:
the singularity is removed and a time-sliced approximation exists, which is exactly integrable, since it can be made harmonic by a simple coordinate transformation, as discovered in 1979 by
21289:
15963:
equation, the Heisenberg equations of motion, and the canonical commutation relations and shows that they are compatible with relativity. It extends the Heisenberg-type operator algebra to
11370:
21537:
10196:{\displaystyle \int \psi _{0}(x)\int _{u(0)=x}\left(\int {\frac {\partial S}{\partial u}}\varepsilon +{\frac {\partial S}{\partial {\dot {u}}}}{\dot {\varepsilon }}\,dt\right)e^{iS}\,Du\,}
2121:
will vary periodically with a very high frequency about the value zero, as a result of which its integral will be practically zero. The only important part in the domain of integration of
23003:
22108:
20084:
17581:
22020:
16481:
14705:
14422:
4291:{\displaystyle K(x,y;T)=\int _{x(0)=x}^{x(T)=y}\prod _{t}\exp \left(-{\tfrac {1}{2}}\left({\frac {x(t+\varepsilon )-x(t)}{\varepsilon }}\right)^{2}\varepsilon \right)\,{\mathcal {D}}x,}
3571:
9007:
1814:
14463:
13335:
13020:
1878:
23203:
23162:
13070:
9592:
23413:
23372:
23308:
23267:
19444:
18691:
Much of the formal study of QFT is devoted to the properties of the resulting functional integral, and much effort (not yet entirely successful) has been made toward making these
4920:
21911:
9894:
1977:
15934:
10430:
13853:
3793:
3630:
3370:
Thus, in contrast to classical mechanics, not only does the stationary path contribute, but actually all virtual paths between the initial and the final point also contribute.
20982:
19788:
13708:
10573:
do not commute. In the path integral, these are just integration variables and they have no obvious ordering. Feynman discovered that the non-commutativity is still present.
7107:
18665:
is itself a function of space and time, and the square brackets are a reminder that the action depends on all the field's values everywhere, not just some particular value.
16201:
21754:
1645:
17184:
14710:
which is precisely the partition function of statistical mechanics for the same system at the temperature quoted earlier. One aspect of this equivalence was also known to
13816:
3593:
2150:. We can apply this argument to each of the variables of integration ... and obtain the result that the only important part in the domain of integration is that for which
1703:
19652:
9020:. The combination of a path-dependent time transformation and a coordinate transformation is an important tool to solve many path integrals and is called generically the
17334:
14175:
14050:
11635:
1347:
18640:
application to quantum field theory, in which the "paths" or histories being considered are not the motions of a single particle, but the possible time evolutions of a
10953:
13794:
13739:
23795:
18726:) makes the functional integral even more like a statistical partition function and also tames some of the mathematical difficulties of working with these integrals.
11580:
1898:
factor contains all the dynamical information – it pushes the state forward in time. The first part and the last part are just Fourier transforms to change to a pure
964:
is obscure in the formulation. The path-integral approach has proven to be equivalent to the other formalisms of quantum mechanics and quantum field theory. Thus, by
23575:
19489:
19279:, although for actual calculation this can be simplified using heuristic methods. Since this expression is a quotient of path integrals, it is naturally normalised.
16861:
5520:
11737:
16709:
11634:
For a particle in curved space the kinetic term depends on the position, and the above time slicing cannot be applied, this being a manifestation of the notorious
6505:{\displaystyle S=S_{\text{c}}+\sum _{n=1}^{\infty }{\tfrac {1}{2}}a_{n}^{2}{\frac {m}{2}}\left({\frac {(n\pi )^{2}}{t_{f}-t_{i}}}-\omega ^{2}(t_{f}-t_{i})\right).}
5418:
17458:
12967:
10549:, which is now reinterpreted as the condition for constructive interference. This can be shown using the method of stationary phase applied to the propagator. As
23881:
23331:
23226:
13343:
11624:
13767:
12941:
10576:
To see this, consider the simplest path integral, the brownian walk. This is not yet quantum mechanics, so in the path-integral the action is not multiplied by
5895:{\displaystyle x_{\text{c}}(t)=x_{i}{\frac {\sin \omega (t_{f}-t)}{\sin \omega (t_{f}-t_{i})}}+x_{f}{\frac {\sin \omega (t-t_{i})}{\sin \omega (t_{f}-t_{i})}}.}
3670:
3650:
2220:
Dirac's work did not provide a precise prescription to calculate the sum over paths, and he did not show that one could recover the Schrödinger equation or the
18757:
11639:
20688:{\displaystyle F={\frac {\partial ^{k_{1}}}{\partial x_{1}^{k_{1}}}}\varphi (x_{1})\cdots {\frac {\partial ^{k_{n}}}{\partial x_{n}^{k_{n}}}}\varphi (x_{n}),}
8925:
Feynman's time-sliced approximation does not, however, exist for the most important quantum-mechanical path integrals of atoms, due to the singularity of the
26224:
22505:
20068:{\displaystyle {\frac {\delta ^{n}Z}{\delta J(x_{1})\cdots \delta J(x_{n})}}=i^{n}\,Z\,\left\langle \varphi (x_{1})\cdots \varphi (x_{n})\right\rangle _{J},}
19628:{\displaystyle \left\langle {\frac {\delta F}{\delta \varphi }}\right\rangle =-i\left\langle F{\frac {\delta {\mathcal {S}}}{\delta \varphi }}\right\rangle }
19367:; the equation means that the action is stationary under small changes in the field configuration). The quantum analogues of these equations are called the
18186:
16989:
in Green's function can either be the energy if the paths are going toward the future, or the negative of the energy if the paths are going toward the past.
16022:
11201:
3913:
in the exponential, so that large deviations are suppressed by small numbers, not by cancelling oscillatory contributions. The amplitude (or Kernel) reads:
1165:
19375:
14058:
10843:
9586:
is diffusive from the free particle propagation, with an extra infinitesimal rotation in phase that slowly varies from point to point from the potential:
5117:
which, with the same normalization as before (not the sum-squares normalization – this function has a divergent norm), obeys a free Schrödinger equation:
4356:
2407:
14465:, the path of minimum action dominates the integral, because the phase of any path away from this fluctuates rapidly and different contributions cancel.
13669:{\displaystyle {\frac {1}{Z}}\int _{\mathbf {x} (0)=x}f(\mathbf {x} )e^{-{\frac {m}{2}}\int |{\dot {\mathbf {x} }}|^{2}dt}\,{\mathcal {D}}\mathbf {x} \,}
13528:
is from the context of quantum field theory, where the change from real to imaginary time changes the space-time geometry from Lorentzian to Euclidean.)
4576:
21085:
13710:
includes all the remaining dependence of the integrand on the path. This integral has a rigorous mathematical interpretation as integration against the
9325:
4473:
3277:{\displaystyle \exp \left({\frac {i}{\hbar }}\varepsilon \sum _{j=1}^{n+1}L\left({\tilde {x}}_{j},{\frac {x_{j}-x_{j-1}}{\varepsilon }},j\right)\right)}
25224:
19764:
24461:
This course, designed for mathematicians, is a rigorous introduction to perturbative quantum field theory, using the language of functional integrals.
1155:
In quantum mechanics, the Legendre transform is hard to interpret, because the motion is not over a definite trajectory. In classical mechanics, with
26285:
21429:
region, for example inside a double light-cone. This gives a more mathematically precise and physically rigorous definition of quantum field theory.
18501:{\displaystyle K(p)={\frac {i}{p_{0}-{\sqrt {{\vec {p}}^{2}+m^{2}}}+i\varepsilon }}+{\frac {i}{p_{0}-{\sqrt {{\vec {p}}^{2}+m^{2}}}-i\varepsilon }}.}
14554:
10671:
1136:. So the Hamiltonian is different in different frames, and this type of symmetry is not apparent in the original formulation of quantum mechanics.
43:
23526:
have attempted to make more rigorous the notion of extracting a classical-like "coarse-grained" history from the space of all possible histories.
25777:
14364:{\displaystyle Z=\int e^{\frac {i{\mathcal {S}}}{\hbar }}\,{\mathcal {D}}\mathbf {x} \quad {\text{where }}{\mathcal {S}}=\int _{0}^{t_{f}}L\,dt}
968:
either approach from the other, problems associated with one or the other approach (as exemplified by Lorentz covariance or unitarity) go away.
23418:
23042:
21320:
15951:
covariant, but the symmetry is not apparent in intermediate stages. If naive field-theory calculations did not produce infinite answers in the
15828:
9715:
16855:. In this case, the interpretation is that these are the quantities to convolve the final wavefunction so as to get the initial wavefunction:
3013:
1929:
to be expressed as a simple function along each path. This function is the quantum analog of the classical action. This observation is due to
25425:
23035:
18628:
faster than light. Such a particle cannot have a Green's function that is only nonzero in the future in a relativistically invariant theory.
379:
22489:{\displaystyle \langle q(x)\rangle +i\langle Fq(x)\rangle =\langle q(x)\rangle +i\left\langle F\partial _{\mu }j^{\mu }(x)\right\rangle =0.}
18900:{\displaystyle \langle F\rangle ={\frac {\int {\mathcal {D}}\varphi Fe^{i{\mathcal {S}}}}{\int {\mathcal {D}}\varphi e^{i{\mathcal {S}}}}}.}
17148:{\displaystyle K(x-y,\mathrm {T} )=\int _{x(0)=x}^{x(\mathrm {T} )=y}e^{i\int _{0}^{\mathrm {T} }{\sqrt {{\dot {x}}^{2}}}-\alpha \,d\tau }.}
12894:{\displaystyle \langle F\rangle ={\frac {\int {\mathcal {D}}F(\phi )e^{{\frac {i}{\hbar }}S}}{\int {\mathcal {D}}e^{{\frac {i}{\hbar }}S}}}}
4081:{\displaystyle K(x-y;T)=\int _{x(0)=x}^{x(T)=y}\exp \left(-\int _{0}^{T}{\frac {{\dot {x}}^{2}}{2}}\,\mathrm {d} t\right)\,{\mathcal {D}}x.}
3678:
14148:
Although the Wick-rotated Schrödinger equation does not have a direct physical meaning, interesting properties of the Schrödinger operator
5505:
10770:
10553:
decreases, the exponential in the integral oscillates rapidly in the complex domain for any change in the action. Thus, in the limit that
1370:
26158:
23694:
14911:
8850:
8656:{\displaystyle K(x_{f},t_{f};x_{i},t_{i})=\left({\frac {m\omega }{\pi \hbar }}\right)^{\frac {1}{2}}e^{\frac {-i\omega T}{2}}\cdot R(T).}
880:
22233:
and its first partial derivatives! More general Lagrangians would require a modification to this definition!). We're not insisting that
21557:
If we don't assume any special boundary conditions, this would not be a "true" symmetry in the true sense of the term in general unless
19307:
10545:
by many orders of magnitude, we typically have destructive interference other than in the vicinity of those trajectories satisfying the
9698:
Since the states obey the Schrödinger equation, the path integral must reproduce the Heisenberg equations of motion for the averages of
7016:
3842:
26274:
26180:
18699:
17879:
14486:
12052:{\displaystyle \langle x_{f}|e^{-{\frac {i}{\hbar }}{\hat {H}}(t-t')}F({\hat {x}})e^{-{\frac {i}{\hbar }}{\hat {H}}(t')}|x_{i}\rangle }
8834:{\displaystyle e^{\frac {-i\omega T}{2}}e^{-in\omega T}=e^{-i\omega T\left({\frac {1}{2}}+n\right)}\quad {\text{for }}n=0,1,2,\ldots .}
5123:
2573:
2316:
when the Hamiltonian is at most quadratic in the momentum. An amplitude computed according to Feynman's principles will also obey the
25948:
24455:
16688:-space the proportionality factor here is constant in time, as will be verified in a moment. The Fourier transform in time, extending
4931:
2968:(more precisely, since one must work with a continuous spectrum, the respective densities) to find the quantum mechanical particle at
2183:, which is just the action function, which classical mechanics requires to be stationary for small variations in all the intermediate
26016:
23561:
can be modeled by using the path integral formation to determine the action of the trajectory through a potential barrier. Using the
18711:
14951:
18307:
interpreted as the antiparticle. The nonrelativistic analysis shows that with this form the antiparticle still has positive energy.
16993:
points in a fixed proper time, as measured along the path (these paths describe the trajectory of a particle in space and in time):
25274:
18167:
space just means multiplying by the propagator, the second term is suppressed and the first term is enhanced. For frequencies near
4757:
2362:
One common approach to deriving the path integral formula is to divide the time interval into small pieces. Once this is done, the
131:
19:
This article is about a formulation of quantum mechanics. For integrals along a path, also known as line or contour integrals, see
23851:
21346:
18526:
18150:
For states where one nonrelativistic particle is present, the initial wavefunction has a frequency distribution concentrated near
16496:
10830:
This shows that the random walk is not differentiable, since the ratio that defines the derivative diverges with probability one.
10586:
1432:
1265:
960:(this is related to conservation of probability; the probabilities of all physically possible outcomes must add up to one) of the
26126:
25960:
23512:
18621:
5026:
4665:
4467:. The result is easy to evaluate by taking the Fourier transform of both sides, so that the convolutions become multiplications:
1080:
These are five of the infinitely many paths available for a particle to move from point A at time t to point B at time t’(>t).
928:(time and space components of quantities enter equations in the same way) is easier to achieve than in the operator formalism of
587:
11185:{\displaystyle -={\frac {{\big (}x(t+\varepsilon )-x(t){\big )}^{2}}{\varepsilon }}\approx {\frac {\varepsilon }{\varepsilon }}}
9572:{\displaystyle \psi (y;t+\varepsilon )\approx \int \psi (x;t)e^{-i\varepsilon V(x)}e^{\frac {i(x-y)^{2}}{2\varepsilon }}\,dx\,.}
3534:{\displaystyle \psi (x,t)={\frac {1}{Z}}\int _{\mathbf {x} (0)=x}{\mathcal {D}}\mathbf {x} \,e^{iS}\psi _{0}(\mathbf {x} (t))\,}
25644:
25217:
18140:{\displaystyle 2p_{0}K(p)={\frac {i}{p_{0}-{\sqrt {{\vec {p}}^{2}+m^{2}}}}}+{\frac {i}{p_{0}+{\sqrt {{\vec {p}}^{2}+m^{2}}}}}.}
10424:. The conclusion is that this first order variation is zero for an arbitrary initial state and at any arbitrary point in time:
9835:
at each separate time is a separate integration variable. So it is legitimate to change variables in the integral by shifting:
360:
19514:(after a Wick rotation, followed by a Wick rotation back) to get the following Schwinger–Dyson equations for the expectation:
14647:. If one makes a Wick rotation here, and finds the amplitude to go from any state, back to the same state in (imaginary) time
26169:
25578:
25141:
25118:
25093:
25006:
24940:
24918:
24898:
24875:
24847:
24813:
24789:
24767:
24748:
24729:
24702:
24652:
24633:
24588:
24558:
24496:
24321:
18511:
Without these terms, the pole contribution could not be unambiguously evaluated when taking the inverse Fourier transform of
17847:{\displaystyle K(p)=\int _{0}^{\infty }e^{-\mathrm {T} p^{2}-\mathrm {T} \alpha }\,d\mathrm {T} ={\frac {1}{p^{2}+\alpha }},}
16824:
The infinitesimal term in the denominator is a small positive number, which guarantees that the inverse Fourier transform in
9420:
locally by an amount proportional to the potential energy. The second term is the free particle propagator, corresponding to
940:
descriptions of the same quantum system. Another advantage is that it is in practice easier to guess the correct form of the
543:
23802:
21670:
14006:{\displaystyle \psi (x,t)=\int e^{-\int V(\mathbf {x} (t))\,dt/\hbar }\,\psi _{0}(\mathbf {x} (t))\,d\mu _{x}(\mathbf {x} )}
11401:
4851:
a more illuminating way to fix the normalization is to consider the path integral as a description of a stochastic process.
25501:
5391:{\displaystyle \psi _{t}(y)=\int \psi _{0}(x)K(x-y;t)\,dx=\int \psi _{0}(x)\int _{x(0)=x}^{x(t)=y}e^{iS}\,{\mathcal {D}}x,}
466:
22262:
15686:
26049:
23714:
23508:
18310:
The proper way to express this mathematically is that, adding a small suppression factor in proper time, the limit where
1062:
22031:
12244:{\displaystyle \int _{x(0)=x_{i}}^{x(t)=x_{f}}{\mathcal {D}}F(x(t'))e^{{\frac {i}{\hbar }}\int dtL(x(t),{\dot {x}}(t))}}
8844:
Comparison to the above eigenstate expansion yields the standard energy spectrum for the simple harmonic oscillator,
26213:
25163:
24610:
15821:
11660:
1030:
also declaring it to be the "fundamental mathematical construct of modern quantum mechanics and quantum field theory".
21593:
11654:
Sometimes (e.g. a particle moving in curved space) we also have measure-theoretic factors in the functional integral:
11585:
and in quantum mechanics, the extra imaginary unit in the action converts this to the canonical commutation relation,
25247:
25210:
24212:
23724:
22841:
16828:
will be nonzero only for future times. For past times, the inverse Fourier transform contour closes toward values of
16608:
10413:
But this was just a shift of integration variables, which doesn't change the value of the integral for any choice of
6306:
Next, expand the deviation from the classical path as a Fourier series, and calculate the contribution to the action
126:
22219:{\displaystyle j^{\mu }(x)=f^{\mu }(x)-{\frac {\partial }{\partial (\partial _{\mu }\varphi )}}{\mathcal {L}}(x)Q\,}
21421:
The path integrals are usually thought of as being the sum of all paths through an infinite space–time. However, in
21220:
20215:{\displaystyle Z^{,i_{1}\cdots i_{n}}=i^{n}Z\left\langle \varphi ^{i_{1}}\cdots \varphi ^{i_{n}}\right\rangle _{J},}
17690:{\displaystyle K(x-y)=\int _{0}^{\infty }e^{-{\frac {(x-y)^{2}}{\mathrm {T} }}-\alpha \mathrm {T} }\,d\mathrm {T} .}
11313:
3801:
The diagram shows the contribution to the path integral of a free particle for a set of paths, eventually drawing a
25357:
21468:
15456:
9021:
2000:
representation the quantity that is being summed over the intermediate states is an obscure matrix element, in the
215:
24566:
The historical reference written by the inventor of the path integral formulation himself and one of his students.
26370:
26298:
26136:
25915:
25624:
25619:
25342:
22892:
18333:
slightly, so that the inverse Fourier transform will pick up a small decay factor in one of the time directions:
14754:
4862:
The probability interpretation gives a natural normalization choice. The path integral should be defined so that
4855:
probabilistically independently chosen. The fact that the answer is a Gaussian spreading linearly in time is the
2321:
2306:
1090:
1070:
953:
873:
322:
302:
170:
21922:
16425:
14657:
14398:
13524:
Note the sign change between this and the normal action, where the potential energy term is negative. (The term
3547:
26355:
25927:
25599:
24258:
23543:
9679:{\displaystyle {\frac {\partial \psi }{\partial t}}=i\cdot \left({\tfrac {1}{2}}\nabla ^{2}-V(x)\right)\psi \,}
8960:
2373:
paths, which in one dimension is a product of ordinary integrals. For the motion of the particle from position
2221:
1765:
562:
292:
21294:
If the functional measure is not translationally invariant, it might be possible to express it as the product
14427:
13287:
12972:
1832:
26375:
26365:
26147:
26009:
25905:
25682:
25604:
25387:
25257:
24951:
23167:
23126:
21569:
15814:
15019:
14868:
14828:
13025:
9303:
Since the time separation is infinitesimal and the cancelling oscillations become severe for large values of
984:
602:
340:
240:
10014:{\displaystyle \int \psi _{0}(x)\int _{u(0)=x}e^{iS(u+\varepsilon ,{\dot {u}}+{\dot {\varepsilon }})}\,Du\,}
26350:
26059:
25639:
25573:
25568:
25539:
25252:
19368:
19288:
17724:
contribution is a Gaussian, gives whose Fourier transform is another Gaussian with reciprocal width. So in
15556:
15486:
14858:
10512:{\displaystyle \left\langle \psi _{0}\left|{\frac {\delta S}{\delta x}}(t)\right|\psi _{0}\right\rangle =0}
2031:
Crucially, Dirac identified the effect of the classical limit on the quantum form of the action principle:
538:
533:
504:
355:
136:
23377:
23336:
23272:
23231:
21454:
Let's just assume for simplicity here that the symmetry in question is local (not local in the sense of a
19403:
17701:
4868:
3378:
In terms of the wave function in the position representation, the path integral formula reads as follows:
25707:
25614:
21864:
21438:
21069:{\displaystyle \left\langle {\frac {\delta {\mathcal {S}}}{\delta \varphi (x)}}+J(x)\right\rangle _{J}=0}
19880:{\displaystyle Z=\int {\mathcal {D}}\varphi e^{i\left({\mathcal {S}}+\langle J,\varphi \rangle \right)}.}
19298:
15014:
14956:
10546:
1946:
1145:
1053:, whose 1933 paper gave birth to path integral formulation. The complete method was developed in 1948 by
909:. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or
902:
572:
317:
307:
21319:
is a translationally invariant measure. This is true, for example, for nonlinear sigma models where the
18688:. In principle, one integrates Feynman's amplitude over the class of all possible field configurations.
15848:
7207:{\displaystyle \prod _{j=1}^{\infty }\left(1-{\frac {x^{2}}{j^{2}}}\right)={\frac {\sin \pi x}{\pi x}},}
1708:
and if this is also interpreted as a matrix multiplication, the sum over all states integrates over all
26360:
26094:
25983:
25745:
25553:
25524:
25085:
21422:
17973:
This expression can be interpreted in the nonrelativistic limit, where it is convenient to split it by
16378:{\displaystyle \psi _{T}(y)=\int _{x}\psi _{0}(x)K(x,y;T)\,dx=\int ^{x(T)=y}\psi _{0}(x(0))e^{iS}\,Dx.}
15964:
14863:
14189:
The path integral is just the generalization of the integral above to all quantum mechanical problems—
13821:
3598:
866:
518:
489:
22256:
is. And we also assume the even stronger assumption that the functional measure is locally invariant:
21841:{\displaystyle \langle Q\rangle +i\left\langle F\int _{\partial V}f^{\mu }\,ds_{\mu }\right\rangle =0}
20436:
is viewed as a functional distribution (this shouldn't be taken too literally as an interpretation of
15666:
1526:. So consider two states separated in time and act with the operator corresponding to the Lagrangian:
944:
of a theory, which naturally enters the path integrals (for interactions of a certain type, these are
26191:
26079:
26054:
25767:
25634:
25558:
25519:
25476:
25407:
25300:
21459:
17301:{\displaystyle K(x-y,\mathrm {T} )=e^{-\alpha \mathrm {T} }\int _{x(0)=x}^{x(\mathrm {T} )=y}e^{-L}.}
15004:
13746:
13682:
1117:
1066:
1061:. The original motivation stemmed from the desire to obtain a quantum-mechanical formulation for the
1046:
941:
523:
484:
437:
335:
195:
19722:{\displaystyle \left\langle F_{,i}\right\rangle =-i\left\langle F{\mathcal {S}}_{,i}\right\rangle .}
2208:– the classical limit – the path integral is dominated by solutions that are in the neighborhood of
26099:
26002:
25834:
25814:
25804:
25794:
25750:
25325:
25017:
23019:
21333:
is some topologically nontrivial space, the concept of a translation does not even make any sense.
19768:
18753:
18743:
18645:
17434:{\displaystyle K(x-y,\mathrm {T} )=e^{-\alpha \mathrm {T} }e^{-{\frac {(x-y)^{2}}{\mathrm {T} }}}.}
15491:
15121:
14966:
14931:
14541:
2335:) as a weighted sum of all possible histories of the system from the initial to the final state. A
2289:, curves in which the particle shoots off into outer space and flies back again, and so forth. The
1532:
980:
607:
24331:
17444:
The usual definition of the relativistic propagator only asks for the amplitude is to travel from
15481:
14718:
after Wick rotation. Note, however, that the Euclidean path integral is actually in the form of a
13799:
11643:
11058:{\displaystyle =x{\frac {dx}{dt}}=x(t+\varepsilon ){\frac {x(t+\varepsilon )-x(t)}{\varepsilon }}}
3576:
1660:
26324:
26069:
26044:
25529:
25455:
25151:
23719:
21655:. This property is called the invariance of the measure, and this does not hold in general. (See
18942:
15980:
15766:
15236:
15044:
15039:
14901:
13859:
10557:
goes to zero, only points where the classical action does not vary contribute to the propagator.
2934:, which, apart from a nonessential factor, is directly the product of the probability amplitudes
2366:
tells us that the noncommutativity of the kinetic and potential energy operators can be ignored.
1141:
1124:. The Hamiltonian indicates how to march forward in time, but the time is different in different
1110:
776:
494:
402:
25943:
25491:
24128:
18324:
limit of the second term must vanish. In the Fourier transform, this means shifting the pole in
16840:. The infinitesimal term can be interpreted as an infinitesimal rotation toward imaginary time.
14151:
14020:
4925:
This condition normalizes the Gaussian and produces a kernel that obeys the diffusion equation:
2317:
1323:
988:
452:
350:
116:
26064:
25854:
25629:
25609:
25534:
25402:
24410:
24402:
23644:{\displaystyle \Gamma =A_{\mathrm {o} }\exp \left(-{\frac {S_{\mathrm {eff} }}{\hbar }}\right)}
20500:
19499:
19387:
16967:{\displaystyle G_{\text{B}}(p,E)={\frac {-i}{-E-{\frac {i{\vec {p}}^{2}}{2m}}+i\varepsilon }}.}
15636:
15616:
15116:
15049:
14961:
14926:
10565:
The formulation of the path integral does not make it clear at first sight that the quantities
9013:
5612:{\displaystyle {\mathcal {L}}={\tfrac {1}{2}}m{\dot {x}}^{2}-{\tfrac {1}{2}}m\omega ^{2}x^{2}.}
2313:
2305:. Contributions from paths wildly different from the classical trajectory may be suppressed by
1058:
1023:
929:
781:
499:
327:
297:
260:
25445:
13772:
13717:
11849:{\displaystyle S=\int \left({\frac {m}{2}}g_{ij}{\dot {x}}^{i}{\dot {x}}^{j}-V(x)\right)\,dt,}
9034:
Relation between Schrödinger's equation and the path integral formulation of quantum mechanics
4859:, which can be interpreted as the first historical evaluation of a statistical path integral.
2358:
Relation between Schrödinger's equation and the path integral formulation of quantum mechanics
2276:
In order to find the overall probability amplitude for a given process, then, one adds up, or
407:
26202:
26089:
25879:
25392:
25372:
24226:
23023:
22835:, we can forget about all the boundary conditions and locality assumptions. We'd simply have
20441:
19772:
19391:
19364:
18719:
18703:
17321:
16807:{\displaystyle G_{\text{F}}(p,E)={\frac {-i}{E-{\frac {{\vec {p}}^{2}}{2m}}+i\varepsilon }},}
15526:
15386:
15082:
14946:
14808:
14469:
11517:
5489:{\displaystyle i{\frac {\partial }{\partial t}}\psi _{t}=-{\frac {\nabla ^{2}}{2}}\psi _{t}.}
4856:
2244:
1499:
996:
937:
914:
250:
235:
24309:
19455:
19390:(we'll assume this for the rest of this article, although this does not hold for, let's say
17553:{\displaystyle K(x-y)=\int _{0}^{\infty }K(x-y,\mathrm {T} )W(\mathrm {T} )\,d\mathrm {T} ,}
26303:
26104:
26084:
25910:
25839:
25784:
25514:
25337:
25295:
25033:
24966:
24682:
24525:
24510:
24476:
24425:
24363:
24343:
24280:
24238:
24099:
23699:
23516:
23026:. In fact, renormalization is the major obstruction to making path integrals well-defined.
21447:
20477:
20437:
19511:
18735:
15771:
15551:
15087:
14941:
14733:
13513:{\displaystyle S_{\mathrm {Euclidean} }(\mathbf {x} ,{\dot {\mathbf {x} }})=\int \left\,dt}
3004:
2332:
2328:
2269:
1057:. Some preliminaries were worked out earlier in his doctoral work under the supervision of
1019:
976:
975:
processes, and this provided the basis for the grand synthesis of the 1970s, which unified
567:
479:
205:
162:
21851:
where the integral is over the boundary. This is the quantum analog of Noether's theorem.
16703:
to be zero for negative times, gives Green's function, or the frequency-space propagator:
14853:
14711:
14472:
follows. Considering only paths that begin and end in the same configuration, perform the
811:
8:
25895:
25864:
25809:
25789:
25697:
25654:
25509:
25435:
25362:
25352:
25264:
23034:
Regardless of whether one works in configuration space or phase space, when equating the
20974:
18692:
18300:
15781:
15111:
12946:
11505:
9316:
2931:
2363:
1106:
921:
910:
906:
666:
474:
392:
220:
200:
152:
25185:
25037:
24994:
24970:
24686:
24529:
24480:
24429:
24347:
24284:
24242:
24103:
23745:
23313:
23208:
15496:
15226:
11591:
1109:
between frequency and energy, and the general relation is consistent with that plus the
26380:
26279:
25955:
25824:
25722:
25430:
25377:
25269:
25056:
25021:
24982:
24887:
24824:
24805:
Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets
24708:
24663:
24622:
24547:
24367:
24296:
24270:
23939:
23830:
23673:
23558:
22815:{\displaystyle q(x)\leftZ+J(x)Q\leftZ=\partial _{\mu }j^{\mu }(x)\leftZ+J(x)Q\leftZ=0.}
18289:{\displaystyle 2mK_{\text{NR}}(p)={\frac {i}{(p_{0}-m)-{\frac {{\vec {p}}^{2}}{2m}}}}.}
16163:{\displaystyle K(x,y;T)=\langle y;T\mid x;0\rangle =\int _{x(0)=x}^{x(T)=y}e^{iS}\,Dx.}
15976:
15791:
15716:
15646:
15531:
15516:
15446:
15411:
15401:
15361:
15276:
15166:
15126:
14916:
14878:
14833:
14821:
14798:
14793:
14715:
14490:
13752:
13076:.) If we repeat the derivation of the path-integral formula in this setting, we obtain
12926:
11286:{\displaystyle f(t)={\frac {{\big (}x(t+\varepsilon )-x(t){\big )}^{2}}{\varepsilon }}}
3655:
3635:
1125:
1121:
992:
925:
387:
312:
245:
157:
1249:{\displaystyle \varepsilon H=p(t){\big (}q(t+\varepsilon )-q(t){\big )}-\varepsilon L}
26254:
26109:
25965:
25874:
25844:
25772:
25735:
25730:
25712:
25677:
25667:
25382:
25347:
25330:
25233:
25159:
25137:
25114:
25089:
25078:
25061:
25022:"The correspondence principle in the statistical interpretation of quantum mechanics"
25002:
24986:
24936:
24914:
24894:
24871:
24843:
24809:
24785:
24763:
24744:
24725:
24712:
24698:
24648:
24629:
24606:
24584:
24554:
24492:
24437:
24371:
24317:
24300:
24250:
24208:
23882:"Finally We May Have a Path to the Fundamental Theory of Physics… and It's Beautiful"
23677:
23562:
23538:
model. Feynman had some success in this direction, and his work has been extended by
21657:
21577:
20485:
17974:
15988:
15947:, and this is not a relativistically invariant concept. The results of a calculation
15746:
15676:
15561:
15406:
15341:
15311:
15281:
15141:
15136:
14981:
14906:
14896:
14788:
14137:{\displaystyle \hbar {\frac {\partial }{\partial t}}\psi (x,t)=-{\hat {H}}\psi (x,t)}
10942:{\displaystyle =x{\frac {dx}{dt}}=x(t){\frac {x(t+\varepsilon )-x(t)}{\varepsilon }}}
10665:
is fluctuating, and the derivative is defined as the limit of a discrete difference.
8926:
5506:
Propagator § Basic examples: propagator of free particle and harmonic oscillator
1720:
898:
821:
796:
736:
731:
631:
597:
577:
175:
34:
25180:
24383:
23983:
15979:– which also allowed changes of variables to be done naturally, as well as allowing
15466:
4435:{\displaystyle K(x-y;T)=G_{\varepsilon }*G_{\varepsilon }*\cdots *G_{\varepsilon },}
2520:{\displaystyle t_{a}=t_{0}<t_{1}<\cdots <t_{n-1}<t_{n}<t_{n+1}=t_{b}}
2103:, as varying continuously while the other ones are fixed. Owing to the smallness of
1909:
Another way of saying this is that since the Hamiltonian is naturally a function of
25692:
25687:
25544:
25440:
25129:
25051:
25041:
24974:
24690:
24598:
24533:
24484:
24433:
24351:
24288:
24246:
24107:
23704:
18739:
18616:, which in the Minkowski version is a small exponential suppression of long paths.
16836:
propagates the particle into the future and is the reason for the subscript "F" on
15786:
15701:
15681:
15651:
15601:
15426:
15421:
15416:
15301:
15286:
15216:
14921:
14848:
14838:
14773:
14375:
4649:{\displaystyle {\tilde {G}}_{\varepsilon }(p)=e^{-{\frac {\varepsilon p^{2}}{2}}},}
2261:
2209:
2187:
s. This shows the way in which equation (11) goes over into classical results when
1149:
826:
816:
806:
706:
686:
671:
641:
509:
397:
21204:{\displaystyle {\frac {\delta {\mathcal {S}}}{\delta \varphi (x)}}\leftZ+J(x)Z=0,}
11068:
In elementary calculus, the two are only different by an amount that goes to 0 as
9399:{\displaystyle e^{-i\varepsilon V(x)}e^{i{\frac {{\dot {x}}^{2}}{2}}\varepsilon }}
4556:{\displaystyle {\tilde {K}}(p;T)={\tilde {G}}_{\varepsilon }(p)^{T/\varepsilon }.}
4305:
is interpreted as a finite collection of integrations at each integer multiple of
2247:
is given by adding together the contributions of all paths in configuration space.
26039:
26025:
25922:
25849:
25829:
25799:
25762:
25757:
25662:
25486:
25107:
25073:
24777:
24576:
24504:
The 1942 thesis. Also includes Dirac's 1933 paper and Feynman's 1948 publication.
24466:
24359:
24129:"The sum over all possibilities: The path integral formulation of quantum theory"
23877:
23856:
23681:
23539:
22249:
21330:
19643:
18641:
17324:, the result of many independent steps is a Gaussian of variance proportional to
15956:
15952:
15731:
15726:
15696:
15691:
15661:
15621:
15591:
15571:
15476:
15371:
15366:
15331:
15316:
15291:
15271:
15251:
15181:
15171:
15092:
15054:
14971:
14936:
14768:
14746:
14741:
2336:
2237:
2202:
1054:
1042:
1038:
1027:
1007:
851:
721:
701:
447:
287:
24905:
Highly readable textbook; introduction to relativistic QFT for particle physics.
24743:. Proceedings Of The 9Th International Conference. World Scientific Publishing.
24261:; DeWitt-Morette, Cécile (1995). "A new perspective on Functional Integration".
19736:
EL equations. The time ordering is taken before the time derivatives inside the
18714:, and the two are essentially mathematically identical except for the factor of
13531:
Now, the contribution of the kinetic energy to the path integral is as follows:
25900:
25869:
25859:
25481:
25471:
25305:
25026:
Proceedings of the National Academy of Sciences of the United States of America
24799:
24601:(1993). "Most of the Good Stuff". In Brown, Laurie M.; Rigden, John S. (eds.).
24406:
21573:
21455:
15756:
15706:
15626:
15611:
15596:
15576:
15546:
15541:
15536:
15521:
15506:
15471:
15461:
15391:
15326:
15296:
15231:
15196:
15186:
15161:
15032:
14999:
14873:
14626:{\displaystyle |\alpha ;t\rangle =e^{-{\frac {iHt}{\hbar }}}|\alpha ;0\rangle }
13742:
13711:
10743:{\displaystyle {\frac {dx}{dt}}={\frac {x(t+\varepsilon )-x(t)}{\varepsilon }}}
9017:
3129:
and the abovementioned "zigzagging" corresponds to the appearance of the terms
2312:
Feynman showed that this formulation of quantum mechanics is equivalent to the
2302:
1156:
1129:
1095:
1034:
786:
746:
726:
696:
676:
626:
592:
442:
432:
225:
25196:
25190:
24803:
24694:
24537:
24314:
Path Integrals in Physics Volume 1: Stochastic Process & Quantum Mechanics
24112:
24087:
18941:
The answer to this is a number between 0 and 1, which can be interpreted as a
15511:
10024:
The change in the integral from the shift is, to first infinitesimal order in
3822:
as an integral over all paths. For a free-particle action (for simplicity let
3814:
The path integral representation gives the quantum amplitude to go from point
913:, over an infinity of quantum-mechanically possible trajectories to compute a
26344:
26292:
25819:
25672:
25563:
25397:
25367:
25320:
24662:
Grosche, Christian (1992). "An Introduction into the Feynman Path Integral".
24451:
24222:
23535:
21581:
21406:
20445:
19395:
18749:
of field configurations (which only depends locally on the fields), then the
18723:
15761:
15751:
15741:
15736:
15656:
15641:
15441:
15436:
15266:
15261:
15221:
15191:
15156:
15146:
15077:
15072:
14473:
14378:
of the classical problem in which one investigates the path starting at time
13073:
12911:
7098:
5509:
5020:. Then the same convolution argument as before gives the propagation kernel:
5017:
2298:
2265:
1936:
Dirac further noted that one could square the time-evolution operator in the
1011:
846:
841:
771:
741:
711:
582:
528:
255:
230:
108:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }
20:
23676:, in which the systems and surroundings must be modeled together. Using the
2023:. The result is a sum over paths with a phase, which is the quantum action.
26245:
25702:
25315:
25310:
25065:
24572:
24191:
19756:
18750:
17866:
to be imaginary gives the usual relativistic propagator, up to a factor of
15796:
15721:
15606:
15581:
15566:
15501:
15381:
15376:
15256:
15246:
15241:
15206:
15201:
15176:
15131:
15009:
14976:
14803:
14783:
13858:
We then have a rigorous version of the Feynman path integral, known as the
3802:
2369:
For a particle in a smooth potential, the path integral is approximated by
2340:
2212:
of the action. The classical path arises naturally in the classical limit.
1144:, and the condition that determines the classical equations of motion (the
836:
831:
766:
751:
716:
210:
25046:
24334:(1972). "Feynman's path integral: Definition without limiting procedure".
11501:
2655:
An approximation for the path integral can be computed as proportional to
25740:
25102:
24928:
24859:
Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral
24829:
24275:
23523:
23492:{\displaystyle {\frac {1}{2}}({\hat {q}}{\hat {p}}+{\hat {p}}{\hat {q}})}
23116:{\displaystyle {\frac {1}{2}}({\hat {q}}{\hat {p}}+{\hat {p}}{\hat {q}})}
16416:
15776:
15711:
15671:
15631:
15586:
15451:
15356:
15351:
15336:
15321:
15306:
9814:{\displaystyle \int \psi _{0}(x)\int _{x(0)=x}e^{iS(x,{\dot {x}})}\,Dx\,}
4340:
3288:
2233:
1133:
1000:
933:
932:. Unlike previous methods, the path integral allows one to easily change
801:
756:
691:
646:
15971:
the Feynman path integral has made earlier formalisms largely obsolete.
14424:
denotes the integration measure over all paths. In the classical limit,
3119:{\displaystyle L(x,{\dot {x}})=T-V={\frac {1}{2}}m|{\dot {x}}|^{2}-V(x)}
26313:
24978:
24668:
24379:
24355:
23709:
19760:
19503:
16174:
15431:
15346:
15211:
15151:
14843:
14778:
2879:
is the Lagrangian of the one-dimensional system with position variable
1930:
1050:
972:
791:
761:
681:
656:
651:
636:
25125:
A great introduction to Path Integrals (Chapter 1) and QFT in general.
18929:
Strictly speaking, the only question that can be asked in physics is:
18636:
However, the path integral formulation is also extremely important in
16419:, the final state is the initial state convolved with the propagator:
11508:, and the (euclideanized) canonical commutation relations in physics.
11072:
goes to 0. But in this case, the difference between the two is not 0:
3778:{\displaystyle S=\int dt\,L(\mathbf {x} (t),{\dot {\mathbf {x} }}(t))}
1033:
The basic idea of the path integral formulation can be traced back to
25594:
25290:
24823:
MacKenzie, Richard (2000). "Path Integral Methods and Applications".
24292:
23744:
For a simplified, step-by-step derivation of the above relation, see
23502:
18681:
9709:
Start by considering the path integral with some fixed initial state
2327:
The path integral formulation of quantum field theory represents the
2286:
1650:
If the multiplications implicit in this formula are reinterpreted as
1015:
957:
920:
This formulation has proven crucial to the subsequent development of
282:
25202:
24470:
23018:
Path integrals as they are defined here require the introduction of
10820:{\displaystyle x(t+\varepsilon )-x(t)\approx {\sqrt {\varepsilon }}}
1416:{\displaystyle \varepsilon L=\varepsilon p{\dot {q}}-\varepsilon H,}
23835:
23547:
23522:
Some advocates of interpretations of quantum mechanics emphasizing
21444:
19733:
19276:
15396:
11870:. This measure cannot be expressed as a functional multiplying the
11859:
then it means that each spatial slice is multiplied by the measure
8910:{\displaystyle E_{n}=\left(n+{\tfrac {1}{2}}\right)\hbar \omega ~.}
7530:. One may write this propagator in terms of energy eigenstates as
3291:
approximating the time integral, which are finally integrated over
2277:
1006:
The path integral has impacted a wide array of sciences, including
961:
661:
25994:
23944:
21572:
that generates the one parameter group in question. We could have
19353:{\displaystyle {\frac {\delta {\mathcal {S}}}{\delta \varphi }}=0}
17857:
which is the Euclidean propagator for a scalar particle. Rotating
17452:, after summing over all the possible proper times it could take:
14540:. Strictly speaking, though, this is the partition function for a
7087:{\displaystyle Q={\sqrt {\frac {m}{2\pi i\hbar (t_{f}-t_{i})}}}~.}
3896:{\displaystyle S=\int {\frac {{\dot {x}}^{2}}{2}}\,\mathrm {d} t,}
1894:
This is not very different from just ordinary time evolution: the
1026:. It has been called the "most powerful formula in physics", with
995:
with an imaginary diffusion constant, and the path integral is an
25191:
A mathematically rigorous approach to perturbative path integrals
24488:
17963:{\displaystyle K(p)={\frac {i}{p_{0}^{2}-{\vec {p}}^{2}-m^{2}}}.}
15967:, which are new relations difficult to see in the old formalism.
5192:{\displaystyle {\frac {d}{dt}}K(x;T)=i{\frac {\nabla ^{2}}{2}}K.}
2638:{\displaystyle \varepsilon =\Delta t={\frac {t_{b}-t_{a}}{n+1}}.}
1105:). For states with a definite energy, this is a statement of the
17178:
can be evaluated when the sum is over paths in Euclidean space:
14052:
satisfies the Wick-rotated version of the Schrödinger equation,
11511:
For a general statistical action, a similar argument shows that
4997:{\displaystyle {\frac {d}{dt}}K(x;T)={\frac {\nabla ^{2}}{2}}K.}
1076:
24720:
Inomata, Akira; Kuratsuji, Hiroshi; Gerry, Christopher (1992).
14714:
who remarked that the equation named after him looked like the
2916:
th time step, if the time integral is approximated by a sum of
2370:
1522:
is only definite on states that are indefinite with respect to
23829:
Hari Dass, N. D. (2020-03-28). "Dirac and the Path Integral".
23757:
For a brief account of the origins of these difficulties, see
2201:
That is, in the limit of action that is large compared to the
24202:
4840:{\displaystyle K(x-y;T)\propto e^{-{\frac {(x-y)^{2}}{2T}}}.}
26192:
Feynman's Lost Lecture: The Motion of Planets Around the Sun
25156:
Quantum Mechanics Formalism, Methodologies, and Applications
21395:{\displaystyle {\hat {\mathcal {S}}}={\mathcal {S}}-i\ln M.}
18644:
over all space. The action is referred to technically as a
18592:{\displaystyle K(p)={\frac {i}{p^{2}-m^{2}+i\varepsilon }},}
16580:{\displaystyle K(x,y;T)\propto e^{\frac {im(x-y)^{2}}{2T}}.}
12918:
10644:{\displaystyle S=\int \left({\frac {dx}{dt}}\right)^{2}\,dt}
4309:. Each factor in the product is a Gaussian as a function of
1477:{\displaystyle {\dot {q}}={\frac {\partial H}{\partial p}},}
1310:{\displaystyle p={\frac {\partial L}{\partial {\dot {q}}}},}
24762:. Oxford Mathematical Monographs. Oxford University Press.
24511:"Space-Time Approach to Non-Relativistic Quantum Mechanics"
22825:
The above two equations are the Ward–Takahashi identities.
14184:
11879:
measure because they belong to entirely different classes.
5206:
s will also obey the same equation, by linearity. Defining
5107:{\displaystyle K(x-y;T)\propto e^{\frac {i(x-y)^{2}}{2T}},}
4737:{\displaystyle {\tilde {K}}(p;T)=e^{-{\frac {Tp^{2}}{2}}}.}
1518:
can be interpreted as noncommuting operators. The operator
24840:
Mathematical Feynman path integrals and their applications
24681:. Graduate Texts in Mathematics. Vol. 267. Springer.
24647:. Springer Tracts in Modern Physics 145. Springer-Verlag.
24203:
Albeverio, S.; Hoegh-Krohn, R. & Mazzucchi, S (2008).
23013:
2128:
is thus that for which a comparatively large variation in
2092:
tends to zero. Let us now picture one of the intermediate
23499:
for the same respective choice of ordering prescription.
21413:= 0, we get the entire set of Schwinger–Dyson equations.
19297:
In the language of functional analysis, we can write the
14493:
with inverse temperature proportional to imaginary time,
11882:
10753:
The distance that a random walk moves is proportional to
3909:
To do this, it is convenient to start without the factor
2035:...we see that the integrand in (11) must be of the form
1120:, which is a more fundamental quantity in the context of
1116:
The Hamiltonian in classical mechanics is derived from a
25186:
Path Integrals in Quantum Theories: A Pedagogic 1st Step
25084:, The Quantum Theory of Fields, vol. 1, Cambridge:
23852:"How Our Reality May Be a Sum of All Possible Realities"
23746:
Path Integrals in Quantum Theories: A Pedagogic 1st Step
17320:, and each step is less likely the longer it is. By the
1982:
and this gives the time-evolution operator between time
24760:
The Feynman Integral and Feynman's Operational Calculus
24722:
Path Integrals and Coherent States of SU(2) and SU(1,1)
18698:
Such a functional integral is extremely similar to the
16177:. Superposing different values of the initial position
4751:, and it is a Gaussian again with reciprocal variance:
3330:
is an arbitrary value of the interval corresponding to
2280:, the amplitude of the 3rd postulate over the space of
2240:
of a complex number called the "probability amplitude".
2154:
is stationary for small variations in all intermediate
1917:, exponentiating this quantity and changing basis from
24952:"A Sum-over-histories Account of an EPR(B) Experiment"
24229:(1983). "Quantum tunnelling in a dissipative system".
21738:{\displaystyle \int {\mathcal {D}}\varphi \,Q\left=0,}
16977:
Given the nearly identical only change is the sign of
11490:{\displaystyle =x{\frac {dx}{dt}}-{\frac {dx}{dt}}x=1}
9631:
8879:
7690:
6361:
6054:
6017:
5572:
5535:
4198:
2139:. This part is the neighbourhood of a point for which
26225:
Perfectly Reasonable Deviations from the Beaten Track
24719:
24257:
24207:. Lecture Notes in Mathematics 523. Springer-Verlag.
23578:
23421:
23380:
23339:
23316:
23275:
23234:
23211:
23170:
23129:
23045:
22895:
22844:
22508:
22338:
22265:
22111:
22034:
21925:
21867:
21757:
21673:
21596:
21471:
21349:
21223:
21088:
20985:
20715:
20529:
20234:
20087:
19899:
19791:
19655:
19523:
19458:
19406:
19310:
19005:
18783:
18722:
the integral to an imaginary time variable (called a
18529:
18342:
18189:
17986:
17882:
17737:
17584:
17461:
17337:
17187:
17002:
16864:
16712:
16611:
16499:
16428:
16204:
16025:
15851:
14660:
14557:
14430:
14401:
14198:
14154:
14061:
14023:
13871:
13824:
13802:
13775:
13755:
13720:
13685:
13540:
13346:
13290:
13085:
13028:
12975:
12949:
12929:
12762:
12264:
12068:
11893:
11740:
11663:
11594:
11520:
11404:
11316:
11204:
11081:
10956:
10846:
10773:
10674:
10589:
10433:
10219:
10037:
9897:
9718:
9595:
9437:
9328:
9048:
8963:
8853:
8698:
8683:: All terms in this expansion get multiplied by the
8507:
8043:
7539:
7226:
7110:
7019:
6524:
6319:
5914:
5689:
5523:
5514:
The Lagrangian for the simple harmonic oscillator is
5421:
5215:
5126:
5029:
4934:
4871:
4760:
4668:
4579:
4476:
4359:
4100:
3922:
3845:
3681:
3658:
3638:
3601:
3579:
3550:
3387:
3138:
3016:
2664:
2576:
2410:
2346:
1949:
1835:
1768:
1663:
1535:
1435:
1373:
1326:
1268:
1168:
1089:
In quantum mechanics, as in classical mechanics, the
46:
24624:
Quantum Physics: A Functional Integral Point of View
23542:
and others. Approaches that use this method include
22319:{\displaystyle \int {\mathcal {D}}\varphi \,q(x)=0.}
21425:
we would restrict everything to lie within a finite
16832:
where there is no singularity. This guarantees that
12255:
This generalizes to multiple operators, for example
5633:
as the classical trajectory plus some perturbation,
1140:
and velocity). The relation between the two is by a
24583:. Mineola, NY: Dover Publications. pp. 29–31.
24472:
Feynman's Thesis — A New Approach to Quantum Theory
23952:
23022:. Changing the scale of the regulator leads to the
19763:of the field configurations (which has at least an
1826:Finally, the last factor in this interpretation is
1132:, while the Hamiltonian is the time component of a
25106:
25077:
24886:
24621:
24546:
23643:
23503:Path integral in quantum-mechanical interpretation
23491:
23407:
23366:
23325:
23302:
23261:
23220:
23197:
23156:
23115:
22997:
22875:
22814:
22488:
22318:
22218:
22092:{\displaystyle q(x)=\partial _{\mu }j^{\mu }(x)\,}
22091:
22014:
21905:
21840:
21737:
21640:
21531:
21394:
21283:
21203:
21068:
20962:
20687:
20413:
20214:
20067:
19879:
19721:
19627:
19483:
19438:
19352:
19255:
18899:
18591:
18500:
18288:
18139:
17962:
17846:
17728:-space, the propagator can be reexpressed simply:
17689:
17552:
17433:
17300:
17147:
16966:
16806:
16673:
16579:
16475:
16377:
16162:
15928:
14699:
14625:
14457:
14416:
14363:
14169:
14136:
14044:
14005:
13847:
13810:
13788:
13761:
13733:
13702:
13668:
13512:
13329:
13272:
13064:
13014:
12961:
12935:
12893:
12741:
12243:
12051:
11848:
11717:
11618:
11574:
11489:
11364:
11285:
11184:
11057:
10941:
10819:
10742:
10643:
10560:
10511:
10402:
10195:
10013:
9813:
9678:
9571:
9398:
9292:
9001:
8909:
8833:
8689:factor in the front, yielding terms of the form
8655:
8470:
7958:
7499:
7206:
7086:
6999:
6504:
6295:
5894:
5611:
5488:
5390:
5191:
5106:
4996:
4914:
4839:
4736:
4648:
4555:
4434:
4290:
4080:
3895:
3777:
3664:
3644:
3624:
3587:
3565:
3533:
3276:
3118:
2853:
2637:
2519:
2143:is stationary with respect to small variations in
1971:
1872:
1808:
1697:
1639:
1487:and the partial derivative now is with respect to
1476:
1415:
1341:
1309:
1248:
956:. Possible downsides of the approach include that
107:
24911:Techniques & Applications of Path Integration
24570:
24307:
24073:
23938:Bernstein, Jeremy (2010-04-20). "Another Dirac".
23904:
22229:(this is assuming the Lagrangian only depends on
17873:and an ambiguity, which will be clarified below:
12910:It is very common in path integrals to perform a
11718:{\displaystyle \int \mu e^{iS}\,{\mathcal {D}}x.}
10536:
9428:they are additive; in any case one has with (1):
7097:Using the infinite-product representation of the
26342:
24643:Grosche, Christian & Steiner, Frank (1998).
21641:{\displaystyle \int {\mathcal {D}}\varphi \,Q=0}
18163:. When convolving with the propagator, which in
13747:Einstein's mathematical model of Brownian motion
8954:by another path-dependent pseudo-time parameter
25113:(Second ed.). Princeton University Press.
24758:Johnson, Gerald W.; Lapidus, Michel L. (2002).
24221:
24174:
22876:{\displaystyle \left\langle Q\right\rangle =0.}
20512:is defined to be the operator that substitutes
18299:This is the expression for the nonrelativistic
17704:. Taking a Fourier transform over the variable
16674:{\displaystyle K(p;T)=e^{\frac {iTp^{2}}{2m}},}
26286:Quantum Man: Richard Feynman's Life in Science
25199:| PBS Space Time. July 7, 2017. (Video, 15:48)
24868:Path Integrals Methods in Quantum Field Theory
24856:
24757:
24642:
24411:"Solution of the path integral for the H-atom"
24330:
21284:{\displaystyle {\mathcal {S}}_{,i}Z+J_{i}Z=0.}
21079:we get the "master" Schwinger–Dyson equation:
18606:term introduces a small imaginary part to the
17311:This describes a sum over all paths of length
11365:{\displaystyle {\mathcal {L}}=(f(t)-1)^{2}\,,}
9424:times a diffusion process. To lowest order in
2250:The contribution of a path is proportional to
26010:
25218:
24808:(4th ed.). Singapore: World Scientific.
24205:Mathematical Theory of Feynman Path Integrals
24067:
23310:, or Weyl ordering prescription; conversely,
21532:{\displaystyle Q=\partial _{\mu }f^{\mu }(x)}
18931:What fraction of states satisfying condition
15822:
11266:
11225:
11152:
11111:
9245:
9194:
8950:at the origin. Only after replacing the time
5680:. The classical trajectory can be written as
5499:
5007:For oscillatory path integrals, ones with an
2799:
2765:
1627:
1596:
1556:
1546:
1320:where the partial derivative with respect to
1232:
1192:
874:
24861:(2nd ed.). Singapore: World Scientific.
24738:
24544:
24401:
24003:
22429:
22405:
22399:
22372:
22363:
22339:
22244:is the generator of a symmetry (i.e. we are
21773:
21758:
21432:
20395:
20383:
20326:
20314:
20242:
20235:
19864:
19852:
19282:
18790:
18784:
16077:
16053:
14620:
14572:
12769:
12763:
12517:
12265:
12046:
11894:
11728:This factor is needed to restore unitarity.
11649:
5905:This trajectory yields the classical action
5408:obeys the free Schrödinger equation just as
4570:is another Gaussian of reciprocal variance:
1821:evolve an infinitesimal time into the future
1745:. That is the action on the Hilbert space –
102:
76:
26159:QED: The Strange Theory of Light and Matter
25128:
24949:
24782:A Modern Approach to Functional Integration
24741:Path Integrals--New Trends And Perspectives
24152:
24085:
23695:Static forces and virtual-particle exchange
22998:{\displaystyle \int d^{d}x\,J(x)Q\leftZ=0.}
21336:In that case, we would have to replace the
12905:
11629:
10525:If the action contains terms that multiply
10522:this is the Heisenberg equation of motion.
9027:
2215:
2115:varying extremely rapidly. This means that
1084:
1045:. This idea was extended to the use of the
971:The path integral also relates quantum and
26017:
26003:
25225:
25211:
24950:Sinha, Sukanya; Sorkin, Rafael D. (1991).
24933:Functional Integration and Quantum Physics
24619:
24061:
22015:{\displaystyle q(x)=\delta ^{(d)}(X-y)Q\,}
18718:in the exponent in Feynman's postulate 3.
16476:{\displaystyle \psi _{T}=\psi _{0}*K(;T).}
16388:For a spatially homogeneous system, where
15829:
15815:
14740:
14700:{\displaystyle Z=\operatorname {Tr} \left}
14417:{\displaystyle {\mathcal {D}}\mathbf {x} }
10837:is ambiguous, with two possible meanings:
9888:, since the endpoints are not integrated:
3906:the integral can be evaluated explicitly.
3566:{\displaystyle {\mathcal {D}}\mathbf {x} }
3007:of the one-dimensional system considered,
2351:
2343:contribution to the transition amplitude.
2224:from this rule. This was done by Feynman.
1925:at each step allows the matrix element of
881:
867:
26181:What Do You Care What Other People Think?
25055:
25045:
25016:
24837:
24828:
24822:
24667:
24597:
24391:Physikalische Zeitschrift der Sowjetunion
24274:
24163:
24111:
23943:
23937:
23926:
23834:
23828:
22912:
22279:
22215:
22088:
22011:
21890:
21813:
21687:
21610:
20003:
19990:
18317:of the first term must vanish, while the
17804:
17675:
17538:
17133:
16365:
16280:
16150:
14354:
14243:
13978:
13947:
13930:
13665:
13652:
13503:
13269:
13256:
12919:Wick rotation and the Feynman–Kac formula
11836:
11701:
11358:
10634:
10399:
10392:
10367:
10192:
10185:
10160:
10010:
10003:
9810:
9803:
9675:
9565:
9558:
9273:
9002:{\displaystyle s=\int {\frac {dt}{r(t)}}}
8481:One may absorb all terms after the first
6093:
5972:
5374:
5284:
4899:
4350:with copies of itself at adjacent times:
4274:
4091:Splitting the integral into time slices:
4064:
4050:
3881:
3725:
3530:
3458:
2834:
2830:
2816:
2804:
1809:{\displaystyle e^{-i\varepsilon H(p,q)},}
1364:fixed. The inverse Legendre transform is
25150:
25072:
24908:
24798:
24196:Mathematical Integrals in Quantum Nature
23781:
23672:. This form is specifically useful in a
23029:
21651:for any polynomially-bounded functional
19638:for any polynomially-bounded functional
18631:
14458:{\displaystyle {\mathcal {S}}\gg \hbar }
14185:Path integral and the partition function
13330:{\displaystyle S_{\mathrm {Euclidean} }}
13015:{\displaystyle e^{-it{\hat {H}}/\hbar }}
9307:, the path integral has most weight for
4339:. The multiple integrals are a repeated
3787:
2135:produces only a very small variation in
2012:and the later one with a fixed value of
1873:{\displaystyle e^{ipq(t+\varepsilon )},}
1159:in time, the Legendre transform becomes
1075:
999:of a method for summing up all possible
24776:
24739:Janke, W.; Pelster, Axel, eds. (2008).
24661:
24508:
24465:
24450:
24308:Chaichian, M.; Demichev, A. P. (2001).
24015:
23958:
23876:
23793:
23198:{\displaystyle qp+{\frac {i\hbar }{2}}}
23157:{\displaystyle qp-{\frac {i\hbar }{2}}}
23014:Need for regulators and renormalization
21554:(and possibly the spacetime position).
21340:in this equation by another functional
18180:, the dominant first term has the form
15939:for two simultaneous spatial positions
14725:
13065:{\displaystyle e^{-t{\hat {H}}/\hbar }}
10541:If the variation in the action exceeds
2314:canonical approach to quantum mechanics
26343:
24865:
24336:Communications in Mathematical Physics
24126:
23569:) can be determined to be of the form
19771:for the functional measure), then the
19732:These equations are the analog of the
18957:. In terms of path integration, since
17720:separately, and because each separate
14489:of statistical mechanics defined in a
11883:Expectation values and matrix elements
9873:is a different shift at each time but
9693:
4566:The Fourier transform of the Gaussian
1041:for solving problems in diffusion and
25998:
25232:
25206:
24927:
24884:
24857:Müller-Kirsten, Harald J. W. (2012).
24545:Feynman, R. P.; Hibbs, A. R. (1965).
24384:"The Lagrangian in Quantum Mechanics"
24378:
24316:. Taylor & Francis. p. 1ff.
24190:
24050:
23970:
23915:
23824:
23822:
23228:depending on whether one chooses the
18729:
13741:. The Wiener measure, constructed by
12753:and to the general expectation value
5202:This means that any superposition of
1654:multiplications, the first factor is
1498:In quantum mechanics, the state is a
1018:. In physics, it is a foundation for
26127:There's Plenty of Room at the Bottom
24993:
24676:
24581:Quantum Mechanics and Path Integrals
24549:Quantum Mechanics and Path Integrals
24038:
24026:
23849:
23758:
23553:
23408:{\displaystyle {\hat {p}}{\hat {q}}}
23367:{\displaystyle {\hat {q}}{\hat {p}}}
23303:{\displaystyle {\hat {p}}{\hat {q}}}
23262:{\displaystyle {\hat {q}}{\hat {p}}}
21854:Now, let's assume even further that
19439:{\displaystyle e^{i{\mathcal {S}}},}
11379:derived from extremizing the action
9409:The first term rotates the phase of
9319:.) The exponential of the action is
8920:
4915:{\displaystyle \int K(x-y;T)\,dy=1.}
4445:where the number of convolutions is
26024:
25134:Path Integrals in Quantum Mechanics
25101:
24913:. New York: John Wiley & Sons.
24456:"Geometry and Quantum Field Theory"
23509:interpretation of quantum mechanics
21906:{\displaystyle Q=\int d^{d}x\,q(x)}
18924:
3573:denotes integration over all paths
2912:corresponds to the position at the
2339:is a graphical representation of a
2324:corresponding to the given action.
1972:{\displaystyle e^{i\varepsilon S},}
13:
26214:The Pleasure of Finding Things Out
26170:Surely You're Joking, Mr. Feynman!
25109:Quantum Field Theory in a Nutshell
24645:Handbook of Feynman Path Integrals
24620:Glimm, J. & Jaffe, A. (1981).
23819:
23625:
23622:
23619:
23591:
23579:
23529:
22656:
22447:
22271:
22189:
22169:
22162:
22158:
22060:
21795:
21679:
21602:
21501:
21480:
21369:
21354:
21250:
21227:
21097:
21000:
20886:
20869:
20804:
20787:
20635:
20618:
20565:
20548:
20366:
20343:
20297:
20262:
19835:
19812:
19697:
19595:
19417:
19319:
19208:
19171:
19103:
19066:
19006:
18875:
18857:
18834:
18804:
17809:
17795:
17777:
17763:
17680:
17669:
17656:
17616:
17543:
17531:
17517:
17493:
17420:
17379:
17357:
17267:
17229:
17207:
17098:
17064:
17022:
15929:{\displaystyle =i\delta ^{3}(x-y)}
15871:
14433:
14404:
14264:
14246:
14220:
14071:
14067:
13655:
13377:
13374:
13371:
13368:
13365:
13362:
13359:
13356:
13353:
13337:is the Euclidean action, given by
13321:
13318:
13315:
13312:
13309:
13306:
13303:
13300:
13297:
13259:
13182:
13179:
13176:
13173:
13170:
13167:
13164:
13161:
13158:
12845:
12783:
12584:
12130:
11704:
11543:
11535:
11319:
10341:
10333:
10309:
10301:
10130:
10122:
10104:
10096:
9643:
9607:
9599:
9089:
9084:
7836:
7127:
6894:
6640:
6515:This means that the propagator is
6355:
5967:
5526:
5459:
5431:
5427:
5377:
5169:
4974:
4277:
4067:
4052:
3883:
3553:
3448:
2705:
2697:
2681:
2673:
2583:
2347:Path integral in quantum mechanics
2026:
1462:
1454:
1286:
1278:
413:Sum-over-histories (path integral)
99:
73:
29:Part of a series of articles about
14:
26392:
25174:
24999:Path Integrals on Group Manifolds
24679:Quantum Theory for Mathematicians
24605:. American Institute of Physics.
23981:
23931:
23725:Path integrals in polymer science
23631:
23186:
23145:
23123:can be translated back as either
20973:Then, from the properties of the
19767:because of the assumption of the
19394:), and if we assume that after a
14599:
14452:
14237:
14177:can be extracted by studying it.
13942:
13848:{\displaystyle \mathbf {x} (0)=x}
13221:
13057:
13007:
12869:
12819:
12678:
12467:
12378:
12296:
12180:
11998:
11925:
8895:
8588:
8255:
8124:
7876:
7684:
7634:
7449:
7422:
7312:
7217:the propagator can be written as
7042:
6869:
6699:
6618:
3625:{\displaystyle \mathbf {x} (0)=x}
3155:
2726:
1500:superposition of different states
50:
25979:
25978:
25197:Feynman's Infinite Quantum Paths
24198:. The Nucleus. pp. 189–209.
23796:"Dynamic Vortex Mott Transition"
21405:If we expand this equation as a
19498:, it goes to zero faster than a
18303:of a free Schrödinger particle.
16590:Taking the Fourier transform in
16181:with an arbitrary initial state
14442:
14410:
14332:
14312:
14273:
14252:
14229:
13996:
13962:
13914:
13826:
13804:
13693:
13661:
13622:
13585:
13557:
13482:
13441:
13398:
13387:
13265:
13240:
13203:
13192:
13123:
11375:and the equations of motion for
3809:
3753:
3733:
3700:
3689:
3652:is a normalization factor. Here
3603:
3581:
3559:
3514:
3485:
3474:
3454:
3425:
3373:
2331:(corresponding to the classical
2293:assigns to all these amplitudes
2107:, we shall then in general have
16:Formulation of quantum mechanics
26299:Feynman Prize in Nanotechnology
26137:The Feynman Lectures on Physics
26050:Wheeler–Feynman absorber theory
25001:. Singapore: World Scientific.
24724:. Singapore: World Scientific.
24263:Journal of Mathematical Physics
24183:
24168:
24157:
24146:
24120:
24079:
24074:Feynman, Hibbs & Styer 2010
24055:
24044:
24032:
24020:
24009:
23997:
23975:
23964:
23751:
23738:
23715:Wheeler–Feynman absorber theory
23544:causal dynamical triangulations
23513:Einstein–Podolsky–Rosen paradox
21416:
18520:. The terms can be recombined:
14256:
13745:gives a rigorous foundation to
13703:{\displaystyle f(\mathbf {x} )}
10561:Canonical commutation relations
10206:which, integrating by parts in
9280:
8795:
2228:from the following postulates:
2222:canonical commutation relations
2162:has for its classical analogue
1063:Wheeler–Feynman absorber theory
25928:Relativistic quantum mechanics
25158:. Cambridge University Press.
24959:Foundations of Physics Letters
24893:. Cambridge University Press.
24870:. Cambridge University Press.
23920:
23909:
23898:
23870:
23843:
23787:
23775:
23486:
23480:
23468:
23453:
23441:
23432:
23399:
23387:
23358:
23346:
23294:
23282:
23253:
23241:
23110:
23104:
23092:
23077:
23065:
23056:
22986:
22980:
22943:
22940:
22934:
22928:
22922:
22916:
22859:
22853:
22803:
22797:
22760:
22757:
22751:
22745:
22739:
22733:
22724:
22718:
22681:
22675:
22649:
22643:
22606:
22603:
22597:
22591:
22585:
22579:
22570:
22564:
22527:
22521:
22518:
22512:
22472:
22466:
22426:
22420:
22417:
22411:
22396:
22390:
22387:
22381:
22360:
22354:
22351:
22345:
22307:
22301:
22298:
22292:
22289:
22283:
22212:
22206:
22200:
22194:
22181:
22165:
22150:
22144:
22128:
22122:
22085:
22079:
22053:
22047:
22044:
22038:
22008:
22005:
21999:
21993:
21987:
21975:
21970:
21964:
21953:
21950:
21944:
21938:
21935:
21929:
21900:
21894:
21770:
21764:
21723:
21717:
21629:
21623:
21620:
21614:
21526:
21520:
21494:
21491:
21485:
21475:
21358:
21253:
21241:
21189:
21183:
21177:
21171:
21162:
21156:
21116:
21110:
21046:
21040:
21031:
21025:
21019:
21013:
20954:
20948:
20939:
20926:
20857:
20844:
20775:
20765:
20759:
20753:
20679:
20666:
20609:
20596:
20539:
20533:
20377:
20371:
20308:
20302:
20279:
20273:
20152:
20146:
20127:
20121:
20048:
20035:
20026:
20013:
20000:
19994:
19974:
19968:
19962:
19949:
19937:
19924:
19846:
19840:
19801:
19795:
19606:
19600:
19584:
19578:
19543:
19537:
19476:
19470:
19428:
19422:
19330:
19324:
19233:
19227:
19219:
19213:
19195:
19189:
19128:
19122:
19114:
19108:
19090:
19084:
19024:
19012:
18886:
18880:
18845:
18839:
18821:
18815:
18539:
18533:
18456:
18386:
18352:
18346:
18258:
18243:
18224:
18212:
18206:
18104:
18043:
18009:
18003:
17929:
17892:
17886:
17747:
17741:
17716:can be done for each value of
17645:
17632:
17600:
17588:
17535:
17527:
17521:
17501:
17477:
17465:
17409:
17396:
17361:
17341:
17271:
17263:
17249:
17243:
17211:
17191:
17068:
17060:
17046:
17040:
17026:
17006:
16926:
16887:
16875:
16767:
16735:
16723:
16627:
16615:
16553:
16540:
16521:
16503:
16467:
16458:
16360:
16354:
16340:
16337:
16331:
16325:
16304:
16298:
16277:
16259:
16253:
16247:
16221:
16215:
16145:
16139:
16117:
16111:
16097:
16091:
16047:
16029:
15923:
15911:
15892:
15889:
15883:
15864:
15858:
15852:
14607:
14559:
14446:
14438:
14351:
14348:
14342:
14322:
14316:
14308:
14277:
14269:
14233:
14225:
14161:
14131:
14119:
14110:
14095:
14083:
14039:
14027:
14000:
13992:
13975:
13972:
13966:
13958:
13927:
13924:
13918:
13910:
13887:
13875:
13836:
13830:
13697:
13689:
13634:
13614:
13589:
13581:
13567:
13561:
13495:
13492:
13486:
13478:
13462:
13457:
13451:
13433:
13408:
13383:
13253:
13250:
13244:
13236:
13213:
13188:
13133:
13127:
13101:
13089:
13046:
12996:
12969:, the time-evolution operator
12883:
12877:
12856:
12850:
12833:
12827:
12806:
12800:
12794:
12788:
12734:
12731:
12725:
12707:
12701:
12695:
12665:
12662:
12649:
12643:
12630:
12627:
12614:
12608:
12595:
12589:
12564:
12558:
12537:
12531:
12503:
12497:
12484:
12478:
12451:
12445:
12436:
12421:
12395:
12389:
12362:
12356:
12347:
12332:
12313:
12307:
12279:
12236:
12233:
12227:
12209:
12203:
12197:
12167:
12164:
12153:
12147:
12141:
12135:
12110:
12104:
12083:
12077:
12032:
12026:
12015:
12009:
11982:
11976:
11967:
11959:
11942:
11936:
11908:
11828:
11822:
11696:
11690:
11676:
11670:
11607:
11595:
11426:
11405:
11349:
11339:
11333:
11327:
11260:
11254:
11245:
11233:
11214:
11208:
11146:
11140:
11131:
11119:
11100:
11094:
11088:
11082:
11046:
11040:
11031:
11019:
11010:
10998:
10963:
10957:
10930:
10924:
10915:
10903:
10894:
10888:
10853:
10847:
10804:
10798:
10789:
10777:
10731:
10725:
10716:
10704:
10537:Stationary-phase approximation
10480:
10474:
10364:
10358:
10256:
10250:
10239:
10233:
10074:
10068:
10057:
10051:
9998:
9956:
9934:
9928:
9917:
9911:
9798:
9777:
9755:
9749:
9738:
9732:
9664:
9658:
9537:
9524:
9509:
9503:
9483:
9471:
9459:
9441:
9352:
9346:
9287:
9281:
9270:
9264:
9240:
9234:
9152:
9140:
9126:
9120:
9109:
9097:
9070:
9052:
8993:
8987:
8647:
8641:
8563:
8511:
8099:
8047:
7937:
7923:
7910:
7897:
7746:
7710:
7599:
7547:
7487:
7461:
7391:
7365:
7351:
7325:
7282:
7230:
7071:
7045:
6948:
6922:
6824:
6798:
6748:
6738:
6584:
6532:
6491:
6465:
6415:
6405:
6273:
6247:
6207:
6181:
6169:
6133:
5883:
5857:
5843:
5824:
5793:
5767:
5753:
5734:
5706:
5700:
5350:
5344:
5330:
5324:
5313:
5307:
5281:
5263:
5257:
5251:
5232:
5226:
5157:
5145:
5080:
5067:
5051:
5033:
4965:
4953:
4896:
4878:
4812:
4799:
4782:
4764:
4693:
4681:
4675:
4605:
4599:
4587:
4533:
4526:
4514:
4501:
4489:
4483:
4381:
4363:
4247:
4241:
4232:
4220:
4162:
4156:
4142:
4136:
4122:
4104:
3984:
3978:
3964:
3958:
3944:
3926:
3772:
3769:
3763:
3743:
3737:
3729:
3710:
3685:
3613:
3607:
3527:
3524:
3518:
3510:
3495:
3470:
3435:
3429:
3403:
3391:
3205:
3113:
3107:
3091:
3073:
3041:
3020:
2794:
2788:
2779:
2773:
1862:
1850:
1798:
1786:
1687:
1681:
1622:
1610:
1591:
1585:
1576:
1564:
1227:
1221:
1212:
1200:
1187:
1181:
983:of a fluctuating field near a
563:Relativistic quantum mechanics
95:
88:
69:
1:
26148:The Character of Physical Law
25906:Quantum statistical mechanics
25683:Quantum differential calculus
25605:Delayed-choice quantum eraser
25388:Symmetry in quantum mechanics
25181:Path integral on Scholarpedia
24628:. New York: Springer-Verlag.
24469:(2005) . Brown, L. M. (ed.).
23905:Chaichian & Demichev 2001
23794:Vinokur, V. M. (2015-02-27).
23768:
15994:
14869:Spontaneous symmetry breaking
14829:Symmetry in quantum mechanics
14722:statistical mechanics model.
13072:. (This change is known as a
5016:. This is closely related to
3307:with the integration measure
2236:for an event is given by the
1640:{\displaystyle e^{i{\big }}.}
985:second-order phase transition
603:Quantum statistical mechanics
24935:. New York: Academic Press.
24438:10.1016/0370-2693(79)90280-6
24251:10.1016/0003-4916(83)90202-6
23850:Wood, Charlie (2023-02-06).
23333:can be translated to either
16486:For a free particle of mass
16195:constructs the final state:
13811:{\displaystyle \mathbf {x} }
11887:Matrix elements of the kind
9582:As mentioned, the spread in
9022:Duru–Kleinert transformation
4747:The Fourier transform gives
3588:{\displaystyle \mathbf {x} }
2905:considered (see below), and
1698:{\displaystyle e^{-ipq(t)},}
7:
25708:Quantum stochastic calculus
25698:Quantum measurement problem
25620:Mach–Zehnder interferometer
25136:. Oxford University Press.
24603:Memories Of Richard Feynman
24175:Caldeira & Leggett 1983
23886:writings.stephenwolfram.com
23687:
23663:and pre-exponential factor
20448:complications here!), then
18706:. Indeed, it is sometimes
16602:produces another Gaussian:
13769:indicates that the measure
11391:Defining the time order to
1902:basis from an intermediate
903:stationary action principle
573:Quantum information science
10:
26397:
26095:Quantum cellular automaton
25086:Cambridge University Press
24086:Zinn-Justin, Jean (2009).
23731:
23654:with the effective action
23008:
21436:
21423:local quantum field theory
20444:analogue, because we have
20440:, unlike its Wick-rotated
19286:
15957:careful limiting procedure
14864:Explicit symmetry breaking
14170:{\displaystyle {\hat {H}}}
14045:{\displaystyle \psi (x,t)}
9031:
5503:
5500:Simple harmonic oscillator
2355:
2088:, which remains finite as
1342:{\displaystyle {\dot {q}}}
18:
26267:
26238:
26118:
26075:Path integral formulation
26032:
25974:
25936:
25888:
25768:Quantum complexity theory
25746:Quantum cellular automata
25721:
25653:
25587:
25500:
25464:
25451:Path integral formulation
25418:
25283:
25240:
24695:10.1007/978-1-4614-7116-5
24553:. New York: McGraw-Hill.
24538:10.1103/RevModPhys.20.367
24518:Reviews of Modern Physics
24113:10.4249/scholarpedia.8674
21433:Ward–Takahashi identities
19388:translationally invariant
19369:Schwinger–Dyson equations
19363:(the left-hand side is a
19283:Schwinger–Dyson equations
15020:Bargmann–Wigner equations
11650:Measure-theoretic factors
11636:operator ordering problem
2191:becomes extremely small.
1506:, or different values of
1502:with different values of
895:path integral formulation
26100:Rogers Commission Report
26060:Hellmann–Feynman theorem
25835:Quantum machine learning
25815:Quantum key distribution
25805:Quantum image processing
25795:Quantum error correction
25645:Wheeler's delayed choice
25193:via animation on YouTube
24885:Ryder, Lewis H. (1985).
24784:. New York: Birkhäuser.
24004:Duru & Kleinert 1979
21550:only depends locally on
19778:of the source fields is
19769:translational invariance
19299:Euler–Lagrange equations
19289:Schwinger–Dyson equation
18754:vacuum expectation value
18695:mathematically precise.
17702:Schwinger representation
15981:constrained quantization
14542:statistical field theory
13789:{\displaystyle \mu _{x}}
13734:{\displaystyle \mu _{x}}
12906:Euclidean path integrals
11630:Particle in curved space
9028:The Schrödinger equation
8666:One may finally expand
3672:is the action, given by
2216:Feynman's interpretation
1146:Euler–Lagrange equations
1085:Quantum action principle
1049:in quantum mechanics by
1010:, quantum field theory,
981:statistical field theory
608:Quantum machine learning
361:Wheeler's delayed-choice
26326:The Challenger Disaster
26070:Feynman parametrization
25751:Quantum finite automata
24909:Schulman, L S. (1981).
24677:Hall, Brian C. (2013).
24509:Feynman, R. P. (1948).
24153:Sinha & Sorkin 1991
24127:Pössel, Markus (2006).
22828:Now for the case where
21439:Ward–Takahashi identity
19759:) is an element of the
18943:conditional probability
18935:also satisfy condition
18720:Analytically continuing
16821:-space representation.
15045:Electroweak interaction
15040:Quantum electrodynamics
15015:Wheeler–DeWitt equation
14902:Background field method
11575:{\displaystyle \left=1}
10547:Euler–Lagrange equation
9317:Trotter product formula
7010:for some normalization
2648:This process is called
2530:can be divided up into
2364:Trotter product formula
2352:Time-slicing derivation
1142:Legendre transformation
1111:superposition principle
1073:) as a starting point.
936:between very different
318:Leggett–Garg inequality
26371:Differential equations
26065:Feynman slash notation
25855:Quantum neural network
24866:Rivers, R. J. (1987).
24838:Mazzucchi, S. (2009).
24062:Glimm & Jaffe 1981
23645:
23565:, the tunneling rate (
23493:
23409:
23368:
23327:
23304:
23263:
23222:
23199:
23158:
23117:
22999:
22877:
22816:
22490:
22320:
22220:
22093:
22016:
21907:
21842:
21739:
21642:
21533:
21396:
21285:
21205:
21070:
20964:
20689:
20415:
20216:
20069:
19881:
19723:
19629:
19485:
19484:{\displaystyle e^{-H}}
19440:
19392:nonlinear sigma models
19354:
19257:
18901:
18593:
18502:
18290:
18141:
17964:
17848:
17691:
17554:
17435:
17302:
17149:
16968:
16808:
16675:
16581:
16477:
16403:is only a function of
16379:
16164:
15965:operator product rules
15930:
15050:Quantum chromodynamics
14927:Effective field theory
14701:
14627:
14459:
14418:
14365:
14171:
14138:
14046:
14007:
13849:
13812:
13796:is supported on paths
13790:
13763:
13735:
13704:
13670:
13514:
13331:
13274:
13066:
13016:
12963:
12937:
12895:
12743:
12245:
12053:
11850:
11719:
11620:
11576:
11491:
11366:
11287:
11186:
11059:
10943:
10821:
10744:
10645:
10513:
10404:
10197:
10015:
9815:
9680:
9573:
9400:
9294:
9003:
8911:
8835:
8657:
8472:
7960:
7840:
7501:
7208:
7131:
7088:
7001:
6898:
6644:
6506:
6359:
6297:
5896:
5613:
5490:
5392:
5193:
5108:
4998:
4916:
4841:
4738:
4650:
4557:
4436:
4292:
4082:
3897:
3806:
3779:
3666:
3646:
3626:
3589:
3567:
3535:
3278:
3189:
3120:
2855:
2709:
2685:
2639:
2521:
2199:
1973:
1874:
1810:
1719:, and so it takes the
1699:
1641:
1478:
1417:
1343:
1311:
1250:
1128:. The Lagrangian is a
1081:
1059:John Archibald Wheeler
1024:quantum chromodynamics
950:Feynman path integrals
930:canonical quantization
109:
26356:Statistical mechanics
26203:The Meaning of It All
26090:One-electron universe
26055:Bethe–Feynman formula
25880:Quantum teleportation
25408:Wave–particle duality
25047:10.1073/pnas.14.2.178
24458:. MIT OpenCourseWare.
23646:
23515:without resorting to
23494:
23410:
23369:
23328:
23305:
23264:
23223:
23200:
23159:
23118:
23030:Ordering prescription
23024:renormalization group
23000:
22878:
22817:
22491:
22321:
22221:
22094:
22017:
21908:
21843:
21740:
21643:
21534:
21397:
21286:
21206:
21071:
20965:
20690:
20442:statistical mechanics
20416:
20217:
20070:
19882:
19773:generating functional
19724:
19630:
19486:
19441:
19365:functional derivative
19355:
19266:where the functional
19258:
18902:
18704:statistical mechanics
18632:Functionals of fields
18594:
18503:
18291:
18142:
17965:
17849:
17692:
17555:
17436:
17322:central limit theorem
17303:
17150:
16969:
16809:
16676:
16582:
16478:
16380:
16165:
15931:
15005:Klein–Gordon equation
14947:LSZ reduction formula
14702:
14640:is evolved from time
14628:
14470:statistical mechanics
14460:
14419:
14366:
14172:
14139:
14047:
14008:
13850:
13813:
13791:
13764:
13736:
13705:
13671:
13515:
13332:
13275:
13067:
13017:
12964:
12938:
12896:
12744:
12246:
12054:
11851:
11720:
11621:
11577:
11492:
11367:
11288:
11187:
11060:
10944:
10822:
10745:
10646:
10514:
10405:
10198:
10016:
9816:
9681:
9574:
9401:
9295:
9004:
8912:
8836:
8658:
8498:, thereby obtaining
8473:
7969:Using the identities
7961:
7820:
7502:
7209:
7111:
7089:
7002:
6878:
6624:
6507:
6339:
6298:
5897:
5622:Write its trajectory
5614:
5491:
5393:
5194:
5109:
4999:
4917:
4857:central limit theorem
4842:
4739:
4651:
4558:
4437:
4293:
4083:
3898:
3800:
3780:
3667:
3647:
3627:
3590:
3568:
3536:
3279:
3163:
3121:
2975:in the initial state
2856:
2689:
2665:
2640:
2522:
2301:, or argument of the
2245:probability amplitude
2033:
1974:
1885:change basis back to
1875:
1811:
1700:
1642:
1510:, and the quantities
1479:
1418:
1344:
1312:
1251:
1079:
1037:, who introduced the
997:analytic continuation
901:that generalizes the
303:Elitzur–Vaidman
293:Davisson–Germer
110:
26376:Mathematical physics
26366:Quantum field theory
26105:Feynman checkerboard
26085:Sticky bead argument
25911:Quantum field theory
25840:Quantum metamaterial
25785:Quantum cryptography
25515:Consistent histories
24889:Quantum Field Theory
24842:. World Scientific.
24475:. World Scientific.
23700:Feynman checkerboard
23576:
23419:
23378:
23337:
23314:
23273:
23232:
23209:
23168:
23127:
23043:
22893:
22842:
22506:
22336:
22329:Then, we would have
22263:
22109:
22032:
21923:
21865:
21858:is a local integral
21755:
21671:
21594:
21564:or something. Here,
21469:
21347:
21323:is diffeomorphic to
21310:is a functional and
21221:
21086:
20983:
20975:functional integrals
20713:
20527:
20232:
20085:
19897:
19789:
19653:
19521:
19506:for large values of
19456:
19404:
19308:
19003:
18781:
18758:polynomially bounded
18736:quantum field theory
18693:functional integrals
18669:such given function
18527:
18340:
18187:
17984:
17880:
17735:
17582:
17459:
17335:
17185:
17000:
16862:
16710:
16609:
16497:
16426:
16415:, the integral is a
16202:
16023:
15849:
15088:Theory of everything
14942:Lattice field theory
14912:Correlation function
14734:Quantum field theory
14726:Quantum field theory
14658:
14555:
14468:The connection with
14428:
14399:
14196:
14152:
14059:
14021:
13869:
13822:
13800:
13773:
13753:
13718:
13683:
13538:
13344:
13288:
13083:
13026:
12973:
12947:
12927:
12760:
12262:
12066:
11891:
11738:
11661:
11640:nonholonomic mapping
11592:
11518:
11402:
11395:the operator order:
11314:
11202:
11079:
10954:
10844:
10771:
10672:
10587:
10431:
10217:
10035:
9895:
9716:
9593:
9435:
9326:
9046:
8961:
8851:
8696:
8505:
8041:
7537:
7224:
7108:
7017:
6522:
6317:
5912:
5687:
5521:
5419:
5213:
5124:
5027:
4932:
4869:
4758:
4666:
4577:
4474:
4357:
4098:
3920:
3843:
3679:
3656:
3636:
3599:
3577:
3548:
3385:
3136:
3014:
2662:
2574:
2567:, of fixed duration
2408:
2401:, the time sequence
2333:correlation function
2329:transition amplitude
2318:Schrödinger equation
1947:
1833:
1766:
1661:
1533:
1433:
1371:
1324:
1266:
1166:
1020:lattice gauge theory
989:Schrödinger equation
977:quantum field theory
897:is a description in
568:Quantum field theory
480:Consistent histories
117:Schrödinger equation
44:
26351:Concepts in physics
26045:Feynman–Kac formula
25896:Quantum fluctuation
25865:Quantum programming
25825:Quantum logic gates
25810:Quantum information
25790:Quantum electronics
25265:Classical mechanics
25038:1928PNAS...14..178V
24971:1991FoPhL...4..303S
24687:2013qtm..book.....H
24530:1948RvMP...20..367F
24481:2005ftna.book.....B
24430:1979PhLB...84..185D
24348:1972CMaPh..28...47D
24285:1995JMP....36.2237C
24243:1983AnPhy.149..374C
24104:2009SchpJ...4.8674Z
23720:Feynman–Kac formula
22248:insisting upon the
21661:for more details.)
20910:
20828:
20702:is a functional of
20659:
20589:
20495:is a functional of
18686:field configuration
17918:
17767:
17620:
17497:
17281:
17159:time. The function
17103:
17078:
16173:This is called the
16127:
15067:Incomplete theories
14385:and ending at time
14304:
13860:Feynman–Kac formula
12962:{\displaystyle -it}
12581:
12127:
11506:stochastic calculus
11500:This is called the
9694:Equations of motion
9191:
9162:
9093:
8303:
8285:
7745:
7727:
6719:
6386:
6168:
6150:
6010:
5964:
5360:
5012:the time increment
4172:
4023:
3994:
3334:, e.g. its center,
2989:in the final state
2932:functional integral
2759:
2196:Dirac (1933), p. 69
2158:s. ... We see that
1734:to change basis to
1107:de Broglie relation
924:, because manifest
922:theoretical physics
911:functional integral
907:classical mechanics
356:Stern–Gerlach
153:Classical mechanics
26280:Cargo cult science
25949:in popular culture
25731:Quantum algorithms
25579:Von Neumann–Wigner
25559:Objective collapse
25270:Old quantum theory
24979:10.1007/BF00665892
24356:10.1007/BF02099371
24332:DeWitt-Morette, C.
23991:221A Lecture Notes
23674:dissipative system
23641:
23559:Quantum tunnelling
23489:
23405:
23364:
23326:{\displaystyle qp}
23323:
23300:
23259:
23221:{\displaystyle qp}
23218:
23195:
23154:
23113:
23036:operator formalism
22995:
22873:
22812:
22486:
22316:
22216:
22089:
22012:
21903:
21838:
21735:
21638:
21587:Let's also assume
21542:for some function
21529:
21443:Now how about the
21392:
21329:. However, if the
21281:
21201:
21066:
20960:
20889:
20807:
20685:
20638:
20568:
20520:. For example, if
20411:
20212:
20065:
19877:
19719:
19625:
19512:integrate by parts
19481:
19449:which now becomes
19436:
19376:functional measure
19350:
19253:
19159:
19054:
18897:
18730:Expectation values
18712:partition function
18700:partition function
18654:, where the field
18589:
18498:
18286:
18137:
17960:
17904:
17844:
17753:
17687:
17606:
17550:
17483:
17431:
17298:
17235:
17145:
17087:
17032:
16964:
16804:
16671:
16577:
16473:
16375:
16160:
16083:
15977:Grassmann variable
15926:
14952:Partition function
14879:Topological charge
14799:General relativity
14794:Special relativity
14716:diffusion equation
14697:
14623:
14491:canonical ensemble
14487:partition function
14455:
14414:
14361:
14283:
14167:
14134:
14042:
14003:
13845:
13808:
13786:
13759:
13731:
13700:
13666:
13510:
13327:
13270:
13062:
13012:
12959:
12933:
12891:
12739:
12523:
12241:
12069:
12049:
11846:
11715:
11619:{\displaystyle =i}
11616:
11572:
11487:
11362:
11283:
11182:
11055:
10939:
10817:
10740:
10641:
10509:
10400:
10193:
10011:
9811:
9676:
9640:
9569:
9396:
9290:
9171:
9112:
9076:
8999:
8907:
8888:
8831:
8653:
8468:
8289:
8271:
8034:, this amounts to
7956:
7954:
7731:
7713:
7699:
7497:
7204:
7084:
6997:
6995:
6705:
6502:
6372:
6370:
6293:
6291:
6154:
6136:
6063:
6026:
5982:
5936:
5892:
5663:and the action as
5609:
5581:
5544:
5486:
5388:
5316:
5189:
5104:
4994:
4912:
4837:
4734:
4659:and the result is
4646:
4553:
4432:
4288:
4207:
4182:
4128:
4078:
4009:
3950:
3893:
3807:
3775:
3662:
3642:
3622:
3585:
3563:
3531:
3274:
3116:
2851:
2731:
2635:
2517:
1969:
1870:
1806:
1695:
1637:
1474:
1413:
1339:
1307:
1246:
1122:special relativity
1082:
993:diffusion equation
926:Lorentz covariance
544:Von Neumann–Wigner
524:Objective-collapse
323:Mach–Zehnder
313:Leggett inequality
308:Franck–Hertz
158:Old quantum theory
105:
26361:Quantum mechanics
26338:
26337:
26330:
26309:
26255:Charles Hirshberg
26110:Feynman sprinkler
25992:
25991:
25966:Quantum mysticism
25944:Schrödinger's cat
25875:Quantum simulator
25845:Quantum metrology
25773:Quantum computing
25736:Quantum amplifier
25713:Quantum spacetime
25678:Quantum cosmology
25668:Quantum chemistry
25383:Scattering theory
25331:Zero-point energy
25326:Degenerate levels
25234:Quantum mechanics
25143:978-0-19-856674-8
25120:978-0-691-14034-6
25095:978-0-521-55001-7
25008:978-981-02-3355-6
24942:978-0-8218-6941-3
24920:978-0-486-44528-1
24900:978-0-521-33859-2
24877:978-0-521-25979-8
24849:978-981-283-690-8
24815:978-981-238-107-1
24791:978-0-8176-4790-2
24769:978-0-19-851572-2
24750:978-981-283-726-4
24731:978-981-02-0656-7
24704:978-1-4614-7115-8
24654:978-3-540-57135-3
24635:978-0-387-90562-4
24599:Gell-Mann, Murray
24590:978-0-486-47722-0
24567:
24560:978-0-07-020650-2
24505:
24498:978-981-256-366-8
24462:
24380:Dirac, Paul A. M.
24323:978-0-7503-0801-4
24231:Annals of Physics
23678:Langevin equation
23634:
23563:WKB approximation
23554:Quantum tunneling
23483:
23471:
23456:
23444:
23430:
23402:
23390:
23361:
23349:
23297:
23285:
23256:
23244:
23193:
23152:
23107:
23095:
23080:
23068:
23054:
22970:
22787:
22708:
22633:
22554:
22252:), but just that
22185:
21658:anomaly (physics)
21576:as well, such as
21448:Noether's theorem
21427:causally complete
21361:
21146:
21120:
21023:
20943:
20912:
20861:
20830:
20743:
20661:
20591:
20486:Fourier transform
20406:
19966:
19618:
19555:
19342:
19248:
19186:
19144:
19081:
19033:
18892:
18584:
18493:
18481:
18459:
18423:
18411:
18389:
18281:
18278:
18261:
18203:
18132:
18129:
18107:
18071:
18068:
18046:
17975:partial fractions
17955:
17932:
17839:
17660:
17424:
17125:
17116:
16959:
16947:
16929:
16872:
16799:
16787:
16770:
16720:
16665:
16571:
15839:
15838:
14932:Expectation value
14907:BRST quantization
14854:Poincaré symmetry
14809:Yang–Mills theory
14789:Quantum mechanics
14712:Erwin Schrödinger
14602:
14339:
14260:
14240:
14164:
14113:
14078:
13762:{\displaystyle x}
13629:
13608:
13549:
13448:
13430:
13405:
13210:
13115:
13049:
12999:
12936:{\displaystyle t}
12889:
12872:
12822:
12722:
12681:
12481:
12470:
12448:
12392:
12381:
12359:
12310:
12299:
12224:
12183:
12012:
12001:
11979:
11939:
11928:
11807:
11788:
11763:
11731:For instance, if
11559:
11555:
11476:
11453:
11423:
11383:corresponding to
11281:
11180:
11167:
11053:
10990:
10937:
10880:
10815:
10738:
10693:
10622:
10472:
10348:
10325:
10321:
10296:
10157:
10146:
10142:
10111:
9995:
9980:
9795:
9639:
9614:
9555:
9389:
9377:
9219:
9207:
9014:İsmail Hakkı Duru
8997:
8927:Coulomb potential
8921:Coulomb potential
8903:
8887:
8799:
8780:
8722:
8632:
8606:
8592:
8452:
8371:
8259:
8222:
8168:
8142:
8128:
7948:
7879:
7802:
7698:
7687:
7664:
7650:
7492:
7491:
7425:
7396:
7395:
7315:
7308:
7199:
7165:
7080:
7076:
7075:
6989:
6960:
6872:
6865:
6783:
6728:
6703:
6660:
6659:
6621:
6614:
6450:
6395:
6369:
6333:
6285:
6277:
6118:
6062:
6041:
6025:
5926:
5887:
5797:
5697:
5580:
5559:
5543:
5471:
5438:
5181:
5140:
5098:
4986:
4948:
4830:
4727:
4678:
4639:
4590:
4517:
4486:
4343:of this Gaussian
4254:
4206:
4173:
4048:
4036:
3879:
3867:
3798:
3760:
3707:
3665:{\displaystyle S}
3645:{\displaystyle Z}
3492:
3417:
3256:
3208:
3158:
3086:
3067:
3038:
3003:is the classical
2930:, this becomes a
2729:
2630:
2537:smaller segments
2210:stationary points
2045:is a function of
1721:Fourier transform
1469:
1445:
1398:
1336:
1302:
1298:
1152:has an extremum.
915:quantum amplitude
899:quantum mechanics
891:
890:
598:Scattering theory
578:Quantum computing
351:Schrödinger's cat
283:Bell's inequality
91:
66:
35:Quantum mechanics
26388:
26328:
26320:
26307:
26260:
26251:
26231:
26220:
26209:
26198:
26187:
26176:
26165:
26154:
26143:
26132:
26019:
26012:
26005:
25996:
25995:
25982:
25981:
25693:Quantum geometry
25688:Quantum dynamics
25545:Superdeterminism
25441:Matrix mechanics
25296:Bra–ket notation
25227:
25220:
25213:
25204:
25203:
25169:
25147:
25124:
25112:
25098:
25083:
25069:
25059:
25049:
25018:Van Vleck, J. H.
25012:
24990:
24956:
24946:
24924:
24904:
24892:
24881:
24862:
24853:
24834:
24832:
24830:quant-ph/0004090
24819:
24795:
24778:Klauder, John R.
24773:
24754:
24735:
24716:
24673:
24671:
24658:
24639:
24627:
24616:
24594:
24571:Feynman, R. P.;
24565:
24564:
24552:
24541:
24515:
24503:
24502:
24460:
24459:
24447:
24445:
24444:
24415:
24398:
24388:
24375:
24327:
24304:
24293:10.1063/1.531039
24278:
24276:funct-an/9602005
24269:(5): 2137–2340.
24254:
24218:
24199:
24177:
24172:
24166:
24161:
24155:
24150:
24144:
24143:
24141:
24140:
24124:
24118:
24117:
24115:
24083:
24077:
24076:, pp. 29–31
24071:
24065:
24059:
24053:
24048:
24042:
24036:
24030:
24024:
24018:
24013:
24007:
24001:
23995:
23994:
23988:
23979:
23973:
23968:
23962:
23956:
23950:
23949:
23947:
23935:
23929:
23924:
23918:
23913:
23907:
23902:
23896:
23895:
23893:
23892:
23878:Wolfram, Stephen
23874:
23868:
23867:
23865:
23864:
23847:
23841:
23840:
23838:
23826:
23817:
23816:
23814:
23813:
23807:
23801:. Archived from
23800:
23791:
23785:
23779:
23762:
23755:
23749:
23742:
23705:Berezin integral
23671:
23662:
23650:
23648:
23647:
23642:
23640:
23636:
23635:
23630:
23629:
23628:
23612:
23596:
23595:
23594:
23568:
23498:
23496:
23495:
23490:
23485:
23484:
23476:
23473:
23472:
23464:
23458:
23457:
23449:
23446:
23445:
23437:
23431:
23423:
23414:
23412:
23411:
23406:
23404:
23403:
23395:
23392:
23391:
23383:
23373:
23371:
23370:
23365:
23363:
23362:
23354:
23351:
23350:
23342:
23332:
23330:
23329:
23324:
23309:
23307:
23306:
23301:
23299:
23298:
23290:
23287:
23286:
23278:
23268:
23266:
23265:
23260:
23258:
23257:
23249:
23246:
23245:
23237:
23227:
23225:
23224:
23219:
23204:
23202:
23201:
23196:
23194:
23189:
23181:
23163:
23161:
23160:
23155:
23153:
23148:
23140:
23122:
23120:
23119:
23114:
23109:
23108:
23100:
23097:
23096:
23088:
23082:
23081:
23073:
23070:
23069:
23061:
23055:
23047:
23004:
23002:
23001:
22996:
22976:
22972:
22971:
22969:
22958:
22908:
22907:
22882:
22880:
22879:
22874:
22866:
22862:
22834:
22821:
22819:
22818:
22813:
22793:
22789:
22788:
22786:
22775:
22714:
22710:
22709:
22707:
22696:
22674:
22673:
22664:
22663:
22639:
22635:
22634:
22632:
22621:
22560:
22556:
22555:
22553:
22542:
22495:
22493:
22492:
22487:
22479:
22475:
22465:
22464:
22455:
22454:
22325:
22323:
22322:
22317:
22275:
22274:
22255:
22243:
22232:
22225:
22223:
22222:
22217:
22193:
22192:
22186:
22184:
22177:
22176:
22157:
22143:
22142:
22121:
22120:
22098:
22096:
22095:
22090:
22078:
22077:
22068:
22067:
22021:
22019:
22018:
22013:
21974:
21973:
21912:
21910:
21909:
21904:
21886:
21885:
21857:
21847:
21845:
21844:
21839:
21831:
21827:
21826:
21825:
21812:
21811:
21802:
21801:
21744:
21742:
21741:
21736:
21716:
21712:
21711:
21710:
21683:
21682:
21654:
21647:
21645:
21644:
21639:
21606:
21605:
21567:
21563:
21553:
21549:
21545:
21538:
21536:
21535:
21530:
21519:
21518:
21509:
21508:
21484:
21483:
21401:
21399:
21398:
21393:
21373:
21372:
21363:
21362:
21357:
21352:
21339:
21328:
21318:
21314:
21309:
21305:
21301:
21290:
21288:
21287:
21282:
21271:
21270:
21240:
21239:
21231:
21230:
21210:
21208:
21207:
21202:
21152:
21148:
21147:
21145:
21134:
21121:
21119:
21102:
21101:
21100:
21090:
21075:
21073:
21072:
21067:
21059:
21058:
21053:
21049:
21024:
21022:
21005:
21004:
21003:
20993:
20969:
20967:
20966:
20961:
20944:
20942:
20938:
20937:
20915:
20913:
20911:
20909:
20908:
20907:
20897:
20884:
20883:
20882:
20881:
20867:
20862:
20860:
20856:
20855:
20833:
20831:
20829:
20827:
20826:
20825:
20815:
20802:
20801:
20800:
20799:
20785:
20783:
20782:
20749:
20745:
20744:
20742:
20731:
20705:
20701:
20694:
20692:
20691:
20686:
20678:
20677:
20662:
20660:
20658:
20657:
20656:
20646:
20633:
20632:
20631:
20630:
20616:
20608:
20607:
20592:
20590:
20588:
20587:
20586:
20576:
20563:
20562:
20561:
20560:
20546:
20519:
20515:
20511:
20505:
20498:
20494:
20483:
20475:
20474:
20435:
20428:
20420:
20418:
20417:
20412:
20407:
20405:
20404:
20403:
20402:
20398:
20370:
20369:
20347:
20346:
20336:
20335:
20334:
20333:
20329:
20301:
20300:
20266:
20265:
20255:
20250:
20249:
20221:
20219:
20218:
20213:
20208:
20207:
20202:
20198:
20197:
20196:
20195:
20194:
20177:
20176:
20175:
20174:
20142:
20141:
20120:
20119:
20118:
20117:
20105:
20104:
20074:
20072:
20071:
20066:
20061:
20060:
20055:
20051:
20047:
20046:
20025:
20024:
19989:
19988:
19967:
19965:
19961:
19960:
19936:
19935:
19916:
19912:
19911:
19901:
19886:
19884:
19883:
19878:
19873:
19872:
19871:
19867:
19839:
19838:
19816:
19815:
19777:
19765:affine structure
19754:
19747:
19740:
19728:
19726:
19725:
19720:
19715:
19711:
19710:
19709:
19701:
19700:
19676:
19672:
19671:
19646:this looks like
19641:
19634:
19632:
19631:
19626:
19624:
19620:
19619:
19617:
19609:
19599:
19598:
19588:
19560:
19556:
19554:
19546:
19529:
19509:
19497:
19490:
19488:
19487:
19482:
19480:
19479:
19445:
19443:
19442:
19437:
19432:
19431:
19421:
19420:
19386:turns out to be
19385:
19381:
19359:
19357:
19356:
19351:
19343:
19341:
19333:
19323:
19322:
19312:
19274:
19262:
19260:
19259:
19254:
19249:
19247:
19246:
19245:
19240:
19236:
19223:
19222:
19212:
19211:
19188:
19187:
19184:
19175:
19174:
19158:
19142:
19141:
19140:
19135:
19131:
19118:
19117:
19107:
19106:
19083:
19082:
19079:
19070:
19069:
19053:
19031:
18995:
18994:
18992:
18991:
18984:
18981:
18956:
18938:
18934:
18925:As a probability
18919:
18915:
18906:
18904:
18903:
18898:
18893:
18891:
18890:
18889:
18879:
18878:
18861:
18860:
18850:
18849:
18848:
18838:
18837:
18808:
18807:
18797:
18773:
18772:
18763:
18748:
18742:is given by the
18717:
18679:
18664:
18653:
18615:
18605:
18598:
18596:
18595:
18590:
18585:
18583:
18573:
18572:
18560:
18559:
18546:
18519:
18507:
18505:
18504:
18499:
18494:
18492:
18482:
18480:
18479:
18467:
18466:
18461:
18460:
18452:
18448:
18443:
18442:
18429:
18424:
18422:
18412:
18410:
18409:
18397:
18396:
18391:
18390:
18382:
18378:
18373:
18372:
18359:
18332:
18323:
18316:
18301:Green's function
18295:
18293:
18292:
18287:
18282:
18280:
18279:
18277:
18269:
18268:
18263:
18262:
18254:
18250:
18236:
18235:
18219:
18205:
18204:
18201:
18179:
18166:
18162:
18146:
18144:
18143:
18138:
18133:
18131:
18130:
18128:
18127:
18115:
18114:
18109:
18108:
18100:
18096:
18091:
18090:
18077:
18072:
18070:
18069:
18067:
18066:
18054:
18053:
18048:
18047:
18039:
18035:
18030:
18029:
18016:
17999:
17998:
17969:
17967:
17966:
17961:
17956:
17954:
17953:
17952:
17940:
17939:
17934:
17933:
17925:
17917:
17912:
17899:
17872:
17865:
17853:
17851:
17850:
17845:
17840:
17838:
17831:
17830:
17817:
17812:
17803:
17802:
17798:
17790:
17789:
17780:
17766:
17761:
17727:
17723:
17719:
17715:
17696:
17694:
17693:
17688:
17683:
17674:
17673:
17672:
17661:
17659:
17654:
17653:
17652:
17630:
17619:
17614:
17574:
17569:
17559:
17557:
17556:
17551:
17546:
17534:
17520:
17496:
17491:
17451:
17447:
17440:
17438:
17437:
17432:
17427:
17426:
17425:
17423:
17418:
17417:
17416:
17394:
17384:
17383:
17382:
17360:
17327:
17319:
17314:
17307:
17305:
17304:
17299:
17294:
17293:
17280:
17270:
17258:
17234:
17233:
17232:
17210:
17177:
17154:
17152:
17151:
17146:
17141:
17140:
17126:
17124:
17123:
17118:
17117:
17109:
17105:
17102:
17101:
17095:
17077:
17067:
17055:
17025:
16988:
16985:, the parameter
16984:
16980:
16973:
16971:
16970:
16965:
16960:
16958:
16948:
16946:
16938:
16937:
16936:
16931:
16930:
16922:
16914:
16902:
16894:
16874:
16873:
16870:
16854:
16847:
16839:
16835:
16831:
16827:
16820:
16813:
16811:
16810:
16805:
16800:
16798:
16788:
16786:
16778:
16777:
16772:
16771:
16763:
16759:
16750:
16742:
16722:
16721:
16718:
16702:
16687:
16680:
16678:
16677:
16672:
16667:
16666:
16664:
16656:
16655:
16654:
16638:
16601:
16586:
16584:
16583:
16578:
16573:
16572:
16570:
16562:
16561:
16560:
16532:
16489:
16482:
16480:
16479:
16474:
16451:
16450:
16438:
16437:
16414:
16402:
16384:
16382:
16381:
16376:
16364:
16363:
16324:
16323:
16314:
16313:
16246:
16245:
16236:
16235:
16214:
16213:
16194:
16180:
16169:
16167:
16166:
16161:
16149:
16148:
16126:
16106:
16015:
16011:
16007:
15946:
15942:
15935:
15933:
15932:
15927:
15910:
15909:
15879:
15878:
15831:
15824:
15817:
14922:Effective action
14849:Lorentz symmetry
14774:Electromagnetism
14744:
14730:
14729:
14706:
14704:
14703:
14698:
14696:
14692:
14691:
14650:
14646:
14639:
14636:where the state
14632:
14630:
14629:
14624:
14610:
14605:
14604:
14603:
14598:
14587:
14562:
14539:
14538:
14536:
14535:
14530:
14527:
14511:
14509:
14508:
14503:
14500:
14484:
14464:
14462:
14461:
14456:
14445:
14437:
14436:
14423:
14421:
14420:
14415:
14413:
14408:
14407:
14394:
14384:
14370:
14368:
14367:
14362:
14341:
14340:
14335:
14330:
14315:
14303:
14302:
14301:
14291:
14276:
14268:
14267:
14261:
14258:
14255:
14250:
14249:
14242:
14241:
14236:
14232:
14224:
14223:
14213:
14176:
14174:
14173:
14168:
14166:
14165:
14157:
14143:
14141:
14140:
14135:
14115:
14114:
14106:
14079:
14077:
14066:
14051:
14049:
14048:
14043:
14012:
14010:
14009:
14004:
13999:
13991:
13990:
13965:
13957:
13956:
13946:
13945:
13941:
13917:
13854:
13852:
13851:
13846:
13829:
13817:
13815:
13814:
13809:
13807:
13795:
13793:
13792:
13787:
13785:
13784:
13768:
13766:
13765:
13760:
13749:. The subscript
13740:
13738:
13737:
13732:
13730:
13729:
13709:
13707:
13706:
13701:
13696:
13675:
13673:
13672:
13667:
13664:
13659:
13658:
13651:
13650:
13643:
13642:
13637:
13631:
13630:
13625:
13620:
13617:
13609:
13601:
13588:
13577:
13576:
13560:
13550:
13542:
13519:
13517:
13516:
13511:
13502:
13498:
13485:
13471:
13470:
13465:
13450:
13449:
13444:
13439:
13436:
13431:
13423:
13407:
13406:
13401:
13396:
13390:
13382:
13381:
13380:
13336:
13334:
13333:
13328:
13326:
13325:
13324:
13279:
13277:
13276:
13271:
13268:
13263:
13262:
13243:
13235:
13234:
13225:
13224:
13220:
13212:
13211:
13206:
13201:
13195:
13187:
13186:
13185:
13143:
13142:
13126:
13116:
13108:
13071:
13069:
13068:
13063:
13061:
13060:
13056:
13051:
13050:
13042:
13021:
13019:
13018:
13013:
13011:
13010:
13006:
13001:
13000:
12992:
12968:
12966:
12965:
12960:
12942:
12940:
12939:
12934:
12900:
12898:
12897:
12892:
12890:
12888:
12887:
12886:
12873:
12865:
12849:
12848:
12838:
12837:
12836:
12823:
12815:
12787:
12786:
12776:
12748:
12746:
12745:
12740:
12738:
12737:
12724:
12723:
12715:
12682:
12674:
12661:
12660:
12642:
12641:
12626:
12625:
12607:
12606:
12588:
12587:
12580:
12579:
12578:
12553:
12552:
12551:
12516:
12515:
12506:
12501:
12500:
12496:
12495:
12483:
12482:
12474:
12471:
12463:
12450:
12449:
12441:
12435:
12434:
12425:
12424:
12420:
12419:
12407:
12406:
12394:
12393:
12385:
12382:
12374:
12361:
12360:
12352:
12346:
12345:
12336:
12335:
12331:
12330:
12312:
12311:
12303:
12300:
12292:
12282:
12277:
12276:
12250:
12248:
12247:
12242:
12240:
12239:
12226:
12225:
12217:
12184:
12176:
12163:
12134:
12133:
12126:
12125:
12124:
12099:
12098:
12097:
12058:
12056:
12055:
12050:
12045:
12044:
12035:
12030:
12029:
12025:
12014:
12013:
12005:
12002:
11994:
11981:
11980:
11972:
11963:
11962:
11958:
11941:
11940:
11932:
11929:
11921:
11911:
11906:
11905:
11878:
11874:
11869:
11868:
11867:
11855:
11853:
11852:
11847:
11835:
11831:
11815:
11814:
11809:
11808:
11800:
11796:
11795:
11790:
11789:
11781:
11777:
11776:
11764:
11756:
11724:
11722:
11721:
11716:
11708:
11707:
11700:
11699:
11625:
11623:
11622:
11617:
11581:
11579:
11578:
11573:
11565:
11561:
11560:
11558:
11557:
11556:
11548:
11541:
11533:
11496:
11494:
11493:
11488:
11477:
11475:
11467:
11459:
11454:
11452:
11444:
11436:
11425:
11424:
11416:
11386:
11382:
11378:
11371:
11369:
11368:
11363:
11357:
11356:
11323:
11322:
11306:
11292:
11290:
11289:
11284:
11282:
11277:
11276:
11275:
11270:
11269:
11229:
11228:
11221:
11191:
11189:
11188:
11183:
11181:
11173:
11168:
11163:
11162:
11161:
11156:
11155:
11115:
11114:
11107:
11071:
11064:
11062:
11061:
11056:
11054:
11049:
11014:
10991:
10989:
10981:
10973:
10948:
10946:
10945:
10940:
10938:
10933:
10898:
10881:
10879:
10871:
10863:
10836:
10826:
10824:
10823:
10818:
10816:
10811:
10763:
10762:
10761:
10749:
10747:
10746:
10741:
10739:
10734:
10699:
10694:
10692:
10684:
10676:
10664:
10650:
10648:
10647:
10642:
10633:
10632:
10627:
10623:
10621:
10613:
10605:
10579:
10572:
10568:
10556:
10552:
10544:
10532:
10528:
10518:
10516:
10515:
10510:
10502:
10498:
10497:
10496:
10487:
10483:
10473:
10471:
10463:
10455:
10448:
10447:
10423:
10409:
10407:
10406:
10401:
10391:
10390:
10378:
10374:
10354:
10350:
10349:
10347:
10339:
10331:
10326:
10324:
10323:
10322:
10314:
10307:
10299:
10297:
10295:
10284:
10266:
10265:
10232:
10231:
10209:
10202:
10200:
10199:
10194:
10184:
10183:
10171:
10167:
10159:
10158:
10150:
10147:
10145:
10144:
10143:
10135:
10128:
10120:
10112:
10110:
10102:
10094:
10084:
10083:
10050:
10049:
10027:
10020:
10018:
10017:
10012:
10002:
10001:
9997:
9996:
9988:
9982:
9981:
9973:
9944:
9943:
9910:
9909:
9887:
9872:
9861:
9834:
9820:
9818:
9817:
9812:
9802:
9801:
9797:
9796:
9788:
9765:
9764:
9731:
9730:
9705:
9701:
9685:
9683:
9682:
9677:
9671:
9667:
9651:
9650:
9641:
9632:
9615:
9613:
9605:
9597:
9585:
9578:
9576:
9575:
9570:
9557:
9556:
9554:
9546:
9545:
9544:
9519:
9513:
9512:
9427:
9423:
9419:
9405:
9403:
9402:
9397:
9395:
9394:
9390:
9385:
9384:
9379:
9378:
9370:
9366:
9356:
9355:
9314:
9310:
9306:
9299:
9297:
9296:
9291:
9257:
9256:
9249:
9248:
9227:
9226:
9221:
9220:
9212:
9208:
9200:
9198:
9197:
9190:
9179:
9161:
9135:
9092:
9087:
9008:
9006:
9005:
9000:
8998:
8996:
8982:
8974:
8953:
8949:
8948:
8946:
8945:
8940:
8937:
8916:
8914:
8913:
8908:
8901:
8894:
8890:
8889:
8880:
8863:
8862:
8840:
8838:
8837:
8832:
8800:
8797:
8794:
8793:
8792:
8788:
8781:
8773:
8746:
8745:
8724:
8723:
8718:
8704:
8688:
8682:
8676:
8662:
8660:
8659:
8654:
8634:
8633:
8628:
8614:
8608:
8607:
8599:
8597:
8593:
8591:
8583:
8575:
8562:
8561:
8549:
8548:
8536:
8535:
8523:
8522:
8497:
8486:
8477:
8475:
8474:
8469:
8464:
8463:
8459:
8458:
8454:
8453:
8451:
8450:
8449:
8421:
8420:
8419:
8401:
8400:
8391:
8390:
8377:
8372:
8370:
8369:
8368:
8340:
8339:
8338:
8310:
8308:
8304:
8302:
8297:
8284:
8279:
8260:
8258:
8250:
8242:
8225:
8224:
8223:
8215:
8209:
8205:
8204:
8203:
8170:
8169:
8164:
8150:
8144:
8143:
8135:
8133:
8129:
8127:
8119:
8111:
8098:
8097:
8085:
8084:
8072:
8071:
8059:
8058:
8033:
8024:
8022:
8021:
8018:
8015:
8002:
7993:
7991:
7990:
7987:
7984:
7965:
7963:
7962:
7957:
7955:
7946:
7945:
7944:
7935:
7934:
7922:
7921:
7909:
7908:
7896:
7895:
7886:
7885:
7881:
7880:
7875:
7871:
7870:
7857:
7839:
7834:
7813:
7809:
7808:
7804:
7803:
7801:
7787:
7786:
7785:
7776:
7775:
7744:
7739:
7726:
7721:
7708:
7700:
7691:
7688:
7680:
7666:
7665:
7657:
7655:
7651:
7649:
7623:
7615:
7598:
7597:
7585:
7584:
7572:
7571:
7559:
7558:
7529:
7506:
7504:
7503:
7498:
7493:
7490:
7486:
7485:
7473:
7472:
7438:
7430:
7429:
7427:
7426:
7421:
7420:
7419:
7406:
7397:
7394:
7390:
7389:
7377:
7376:
7354:
7350:
7349:
7337:
7336:
7320:
7319:
7317:
7316:
7311:
7310:
7309:
7306:
7296:
7281:
7280:
7268:
7267:
7255:
7254:
7242:
7241:
7213:
7211:
7210:
7205:
7200:
7198:
7190:
7176:
7171:
7167:
7166:
7164:
7163:
7154:
7153:
7144:
7130:
7125:
7093:
7091:
7090:
7085:
7078:
7077:
7074:
7070:
7069:
7057:
7056:
7028:
7027:
7006:
7004:
7003:
6998:
6996:
6992:
6991:
6990:
6982:
6976:
6972:
6971:
6970:
6965:
6961:
6959:
6951:
6947:
6946:
6934:
6933:
6917:
6897:
6892:
6874:
6873:
6868:
6867:
6866:
6863:
6853:
6841:
6837:
6836:
6832:
6831:
6827:
6823:
6822:
6810:
6809:
6797:
6796:
6784:
6782:
6781:
6780:
6768:
6767:
6757:
6756:
6755:
6736:
6729:
6721:
6718:
6713:
6704:
6702:
6691:
6677:
6676:
6661:
6655:
6654:
6646:
6643:
6638:
6623:
6622:
6617:
6616:
6615:
6612:
6602:
6583:
6582:
6570:
6569:
6557:
6556:
6544:
6543:
6511:
6509:
6508:
6503:
6498:
6494:
6490:
6489:
6477:
6476:
6464:
6463:
6451:
6449:
6448:
6447:
6435:
6434:
6424:
6423:
6422:
6403:
6396:
6388:
6385:
6380:
6371:
6362:
6358:
6353:
6335:
6334:
6331:
6309:
6302:
6300:
6299:
6294:
6292:
6283:
6282:
6278:
6276:
6272:
6271:
6259:
6258:
6236:
6235:
6234:
6225:
6224:
6206:
6205:
6193:
6192:
6167:
6162:
6149:
6144:
6131:
6119:
6111:
6103:
6092:
6088:
6087:
6086:
6077:
6076:
6064:
6055:
6049:
6048:
6043:
6042:
6034:
6027:
6018:
6009:
6008:
6007:
5997:
5996:
5995:
5971:
5970:
5963:
5962:
5961:
5951:
5950:
5949:
5928:
5927:
5924:
5901:
5899:
5898:
5893:
5888:
5886:
5882:
5881:
5869:
5868:
5846:
5842:
5841:
5813:
5811:
5810:
5798:
5796:
5792:
5791:
5779:
5778:
5756:
5746:
5745:
5723:
5721:
5720:
5699:
5698:
5695:
5679:
5662:
5632:
5618:
5616:
5615:
5610:
5605:
5604:
5595:
5594:
5582:
5573:
5567:
5566:
5561:
5560:
5552:
5545:
5536:
5530:
5529:
5495:
5493:
5492:
5487:
5482:
5481:
5472:
5467:
5466:
5457:
5449:
5448:
5439:
5437:
5426:
5411:
5407:
5397:
5395:
5394:
5389:
5381:
5380:
5373:
5372:
5359:
5339:
5306:
5305:
5250:
5249:
5225:
5224:
5205:
5198:
5196:
5195:
5190:
5182:
5177:
5176:
5167:
5141:
5139:
5128:
5113:
5111:
5110:
5105:
5100:
5099:
5097:
5089:
5088:
5087:
5062:
5015:
5010:
5003:
5001:
5000:
4995:
4987:
4982:
4981:
4972:
4949:
4947:
4936:
4921:
4919:
4918:
4913:
4846:
4844:
4843:
4838:
4833:
4832:
4831:
4829:
4821:
4820:
4819:
4797:
4750:
4743:
4741:
4740:
4735:
4730:
4729:
4728:
4723:
4722:
4721:
4708:
4680:
4679:
4671:
4655:
4653:
4652:
4647:
4642:
4641:
4640:
4635:
4634:
4633:
4620:
4598:
4597:
4592:
4591:
4583:
4569:
4562:
4560:
4559:
4554:
4549:
4548:
4544:
4525:
4524:
4519:
4518:
4510:
4488:
4487:
4479:
4466:
4465:
4463:
4462:
4457:
4454:
4441:
4439:
4438:
4433:
4428:
4427:
4409:
4408:
4396:
4395:
4349:
4338:
4334:
4323:
4308:
4304:
4297:
4295:
4294:
4289:
4281:
4280:
4273:
4269:
4265:
4264:
4259:
4255:
4250:
4215:
4208:
4199:
4181:
4171:
4151:
4087:
4085:
4084:
4079:
4071:
4070:
4063:
4059:
4055:
4049:
4044:
4043:
4038:
4037:
4029:
4025:
4022:
4017:
3993:
3973:
3912:
3902:
3900:
3899:
3894:
3886:
3880:
3875:
3874:
3869:
3868:
3860:
3856:
3835:
3828:
3821:
3817:
3799:
3784:
3782:
3781:
3776:
3762:
3761:
3756:
3751:
3736:
3709:
3708:
3703:
3698:
3692:
3671:
3669:
3668:
3663:
3651:
3649:
3648:
3643:
3631:
3629:
3628:
3623:
3606:
3594:
3592:
3591:
3586:
3584:
3572:
3570:
3569:
3564:
3562:
3557:
3556:
3540:
3538:
3537:
3532:
3517:
3509:
3508:
3499:
3498:
3494:
3493:
3488:
3483:
3477:
3457:
3452:
3451:
3445:
3444:
3428:
3418:
3410:
3366:
3365:
3363:
3362:
3359:
3356:
3333:
3329:
3322:
3306:
3299:
3283:
3281:
3280:
3275:
3273:
3269:
3268:
3264:
3257:
3252:
3251:
3250:
3232:
3231:
3221:
3216:
3215:
3210:
3209:
3201:
3188:
3177:
3159:
3151:
3125:
3123:
3122:
3117:
3100:
3099:
3094:
3088:
3087:
3079:
3076:
3068:
3060:
3040:
3039:
3031:
3002:
2995:
2988:
2981:
2974:
2967:
2966:
2929:
2919:
2915:
2911:
2904:
2889:
2878:
2860:
2858:
2857:
2852:
2847:
2846:
2829:
2828:
2815:
2811:
2803:
2802:
2769:
2768:
2758:
2757:
2756:
2746:
2745:
2744:
2730:
2722:
2708:
2700:
2684:
2676:
2644:
2642:
2641:
2636:
2631:
2629:
2618:
2617:
2616:
2604:
2603:
2593:
2566:
2555:
2536:
2526:
2524:
2523:
2518:
2516:
2515:
2503:
2502:
2484:
2483:
2471:
2470:
2446:
2445:
2433:
2432:
2420:
2419:
2400:
2393:
2386:
2379:
2259:
2255:
2207:
2197:
2190:
2186:
2182:
2178:
2177:
2176:
2167:
2161:
2157:
2153:
2149:
2142:
2138:
2134:
2127:
2120:
2106:
2102:
2095:
2091:
2087:
2044:
2040:
2022:
2011:
2003:
1999:
1995:
1985:
1978:
1976:
1975:
1970:
1965:
1964:
1940:representation:
1939:
1928:
1924:
1920:
1916:
1912:
1905:
1901:
1897:
1888:
1879:
1877:
1876:
1871:
1866:
1865:
1815:
1813:
1812:
1807:
1802:
1801:
1754:
1750:
1747:change basis to
1744:
1733:
1718:
1704:
1702:
1701:
1696:
1691:
1690:
1646:
1644:
1643:
1638:
1633:
1632:
1631:
1630:
1600:
1599:
1560:
1559:
1550:
1549:
1525:
1521:
1517:
1513:
1509:
1505:
1494:
1490:
1483:
1481:
1480:
1475:
1470:
1468:
1460:
1452:
1447:
1446:
1438:
1422:
1420:
1419:
1414:
1400:
1399:
1391:
1363:
1348:
1346:
1345:
1340:
1338:
1337:
1329:
1316:
1314:
1313:
1308:
1303:
1301:
1300:
1299:
1291:
1284:
1276:
1255:
1253:
1252:
1247:
1236:
1235:
1196:
1195:
1126:reference frames
1104:
946:coordinate space
883:
876:
869:
510:Superdeterminism
163:Bra–ket notation
114:
112:
111:
106:
98:
93:
92:
84:
72:
67:
65:
54:
26:
25:
26396:
26395:
26391:
26390:
26389:
26387:
26386:
26385:
26341:
26340:
26339:
26334:
26318:
26263:
26258:
26249:
26234:
26229:
26218:
26207:
26196:
26185:
26174:
26163:
26152:
26141:
26130:
26114:
26040:Feynman diagram
26028:
26026:Richard Feynman
26023:
25993:
25988:
25970:
25956:Wigner's friend
25932:
25923:Quantum gravity
25884:
25870:Quantum sensing
25850:Quantum network
25830:Quantum machine
25800:Quantum imaging
25763:Quantum circuit
25758:Quantum channel
25717:
25663:Quantum biology
25649:
25625:Elitzur–Vaidman
25600:Davisson–Germer
25583:
25535:Hidden-variable
25525:de Broglie–Bohm
25502:Interpretations
25496:
25460:
25414:
25301:Complementarity
25279:
25236:
25231:
25177:
25172:
25166:
25152:Deshmukh, P. C.
25144:
25130:Zinn Justin, J.
25121:
25096:
25009:
24954:
24943:
24921:
24901:
24878:
24850:
24816:
24800:Kleinert, Hagen
24792:
24770:
24751:
24732:
24705:
24655:
24636:
24613:
24591:
24561:
24513:
24499:
24442:
24440:
24418:Physics Letters
24413:
24407:Kleinert, Hagen
24386:
24324:
24223:Caldeira, A. O.
24215:
24186:
24181:
24180:
24173:
24169:
24162:
24158:
24151:
24147:
24138:
24136:
24133:Einstein Online
24125:
24121:
24088:"Path integral"
24084:
24080:
24072:
24068:
24060:
24056:
24049:
24045:
24041:, Theorem 20.3.
24037:
24033:
24029:, Section 20.3.
24025:
24021:
24014:
24010:
24002:
23998:
23986:
23984:"Path Integral"
23980:
23976:
23969:
23965:
23957:
23953:
23936:
23932:
23925:
23921:
23914:
23910:
23903:
23899:
23890:
23888:
23875:
23871:
23862:
23860:
23857:Quanta Magazine
23848:
23844:
23827:
23820:
23811:
23809:
23805:
23798:
23792:
23788:
23780:
23776:
23771:
23766:
23765:
23761:, Section 20.6.
23756:
23752:
23743:
23739:
23734:
23729:
23690:
23682:Brownian motion
23670:
23664:
23661:
23655:
23618:
23617:
23613:
23611:
23607:
23603:
23590:
23589:
23585:
23577:
23574:
23573:
23566:
23556:
23532:
23530:Quantum gravity
23505:
23475:
23474:
23463:
23462:
23448:
23447:
23436:
23435:
23422:
23420:
23417:
23416:
23394:
23393:
23382:
23381:
23379:
23376:
23375:
23353:
23352:
23341:
23340:
23338:
23335:
23334:
23315:
23312:
23311:
23289:
23288:
23277:
23276:
23274:
23271:
23270:
23248:
23247:
23236:
23235:
23233:
23230:
23229:
23210:
23207:
23206:
23182:
23180:
23169:
23166:
23165:
23141:
23139:
23128:
23125:
23124:
23099:
23098:
23087:
23086:
23072:
23071:
23060:
23059:
23046:
23044:
23041:
23040:
23032:
23016:
23011:
22962:
22957:
22950:
22946:
22903:
22899:
22894:
22891:
22890:
22886:Alternatively,
22849:
22845:
22843:
22840:
22839:
22829:
22779:
22774:
22767:
22763:
22700:
22695:
22688:
22684:
22669:
22665:
22659:
22655:
22625:
22620:
22613:
22609:
22546:
22541:
22534:
22530:
22507:
22504:
22503:
22499:Alternatively,
22460:
22456:
22450:
22446:
22442:
22438:
22337:
22334:
22333:
22270:
22269:
22264:
22261:
22260:
22253:
22250:gauge principle
22234:
22230:
22188:
22187:
22172:
22168:
22161:
22156:
22138:
22134:
22116:
22112:
22110:
22107:
22106:
22073:
22069:
22063:
22059:
22033:
22030:
22029:
21963:
21959:
21924:
21921:
21920:
21881:
21877:
21866:
21863:
21862:
21855:
21821:
21817:
21807:
21803:
21794:
21790:
21786:
21782:
21756:
21753:
21752:
21703:
21699:
21695:
21691:
21678:
21677:
21672:
21669:
21668:
21652:
21601:
21600:
21595:
21592:
21591:
21574:antiderivations
21565:
21558:
21551:
21547:
21543:
21514:
21510:
21504:
21500:
21479:
21478:
21470:
21467:
21466:
21441:
21435:
21419:
21368:
21367:
21353:
21351:
21350:
21348:
21345:
21344:
21337:
21331:target manifold
21324:
21312:
21311:
21307:
21299:
21295:
21266:
21262:
21232:
21226:
21225:
21224:
21222:
21219:
21218:
21138:
21133:
21126:
21122:
21103:
21096:
21095:
21091:
21089:
21087:
21084:
21083:
21054:
21006:
20999:
20998:
20994:
20992:
20991:
20987:
20986:
20984:
20981:
20980:
20933:
20929:
20919:
20914:
20903:
20899:
20898:
20893:
20885:
20877:
20873:
20872:
20868:
20866:
20851:
20847:
20837:
20832:
20821:
20817:
20816:
20811:
20803:
20795:
20791:
20790:
20786:
20784:
20778:
20774:
20735:
20730:
20723:
20719:
20714:
20711:
20710:
20703:
20699:
20673:
20669:
20652:
20648:
20647:
20642:
20634:
20626:
20622:
20621:
20617:
20615:
20603:
20599:
20582:
20578:
20577:
20572:
20564:
20556:
20552:
20551:
20547:
20545:
20528:
20525:
20524:
20517:
20513:
20507:
20503:
20496:
20492:
20481:
20471:
20461:
20450:
20449:
20426:
20425:
20365:
20364:
20363:
20359:
20355:
20351:
20342:
20341:
20337:
20296:
20295:
20294:
20290:
20286:
20282:
20261:
20260:
20256:
20254:
20245:
20241:
20233:
20230:
20229:
20203:
20190:
20186:
20185:
20181:
20170:
20166:
20165:
20161:
20160:
20156:
20155:
20137:
20133:
20113:
20109:
20100:
20096:
20092:
20088:
20086:
20083:
20082:
20056:
20042:
20038:
20020:
20016:
20009:
20005:
20004:
19984:
19980:
19956:
19952:
19931:
19927:
19917:
19907:
19903:
19902:
19900:
19898:
19895:
19894:
19834:
19833:
19832:
19828:
19824:
19820:
19811:
19810:
19790:
19787:
19786:
19775:
19752:
19746:
19738:
19737:
19702:
19696:
19695:
19694:
19690:
19686:
19664:
19660:
19656:
19654:
19651:
19650:
19644:deWitt notation
19639:
19610:
19594:
19593:
19589:
19587:
19574:
19570:
19547:
19530:
19528:
19524:
19522:
19519:
19518:
19507:
19495:
19463:
19459:
19457:
19454:
19453:
19416:
19415:
19411:
19407:
19405:
19402:
19401:
19379:
19378:
19334:
19318:
19317:
19313:
19311:
19309:
19306:
19305:
19291:
19285:
19273:
19267:
19241:
19207:
19206:
19202:
19198:
19183:
19179:
19170:
19169:
19165:
19161:
19160:
19148:
19143:
19136:
19102:
19101:
19097:
19093:
19078:
19074:
19065:
19064:
19060:
19056:
19055:
19037:
19032:
19030:
19004:
19001:
19000:
18985:
18982:
18971:
18970:
18968:
18958:
18946:
18936:
18932:
18927:
18913:
18911:
18874:
18873:
18869:
18865:
18856:
18855:
18851:
18833:
18832:
18828:
18824:
18803:
18802:
18798:
18796:
18782:
18779:
18778:
18766:
18765:
18761:
18746:
18732:
18715:
18670:
18655:
18649:
18634:
18607:
18603:
18568:
18564:
18555:
18551:
18550:
18545:
18528:
18525:
18524:
18518:
18512:
18475:
18471:
18462:
18451:
18450:
18449:
18447:
18438:
18434:
18433:
18428:
18405:
18401:
18392:
18381:
18380:
18379:
18377:
18368:
18364:
18363:
18358:
18341:
18338:
18337:
18331:
18325:
18318:
18311:
18270:
18264:
18253:
18252:
18251:
18249:
18231:
18227:
18223:
18218:
18200:
18196:
18188:
18185:
18184:
18174:
18168:
18164:
18157:
18151:
18123:
18119:
18110:
18099:
18098:
18097:
18095:
18086:
18082:
18081:
18076:
18062:
18058:
18049:
18038:
18037:
18036:
18034:
18025:
18021:
18020:
18015:
17994:
17990:
17985:
17982:
17981:
17948:
17944:
17935:
17924:
17923:
17922:
17913:
17908:
17903:
17898:
17881:
17878:
17877:
17867:
17864:
17858:
17826:
17822:
17821:
17816:
17808:
17794:
17785:
17781:
17776:
17772:
17768:
17762:
17757:
17736:
17733:
17732:
17725:
17721:
17717:
17705:
17679:
17668:
17655:
17648:
17644:
17631:
17629:
17625:
17621:
17615:
17610:
17583:
17580:
17579:
17572:
17564:
17542:
17530:
17516:
17492:
17487:
17460:
17457:
17456:
17449:
17445:
17419:
17412:
17408:
17395:
17393:
17389:
17385:
17378:
17371:
17367:
17356:
17336:
17333:
17332:
17325:
17317:
17312:
17286:
17282:
17266:
17259:
17239:
17228:
17221:
17217:
17206:
17186:
17183:
17182:
17160:
17119:
17108:
17107:
17106:
17104:
17097:
17096:
17091:
17083:
17079:
17063:
17056:
17036:
17021:
17001:
16998:
16997:
16986:
16982:
16978:
16939:
16932:
16921:
16920:
16919:
16915:
16913:
16903:
16895:
16893:
16869:
16865:
16863:
16860:
16859:
16849:
16848:is replaced by
16845:
16837:
16833:
16829:
16825:
16818:
16779:
16773:
16762:
16761:
16760:
16758:
16751:
16743:
16741:
16717:
16713:
16711:
16708:
16707:
16689:
16685:
16657:
16650:
16646:
16639:
16637:
16633:
16610:
16607:
16606:
16591:
16563:
16556:
16552:
16533:
16531:
16527:
16498:
16495:
16494:
16487:
16446:
16442:
16433:
16429:
16427:
16424:
16423:
16404:
16389:
16347:
16343:
16319:
16315:
16294:
16290:
16241:
16237:
16231:
16227:
16209:
16205:
16203:
16200:
16199:
16188:
16182:
16178:
16132:
16128:
16107:
16087:
16024:
16021:
16020:
16013:
16009:
16005:
15997:
15953:continuum limit
15944:
15940:
15905:
15901:
15874:
15870:
15850:
15847:
15846:
15835:
15806:
15805:
15804:
15802:
15106:
15098:
15097:
15093:Quantum gravity
15068:
15060:
15059:
15055:Higgs mechanism
15035:
15025:
15024:
15010:Proca equations
14995:
14987:
14986:
14972:Renormalization
14937:Feynman diagram
14892:
14884:
14883:
14824:
14814:
14813:
14764:
14749:
14747:Feynman diagram
14728:
14681:
14677:
14673:
14659:
14656:
14655:
14648:
14641:
14637:
14606:
14588:
14586:
14582:
14578:
14558:
14556:
14553:
14552:
14531:
14528:
14523:
14516:
14515:
14513:
14504:
14501:
14498:
14497:
14495:
14494:
14476:
14441:
14432:
14431:
14429:
14426:
14425:
14409:
14403:
14402:
14400:
14397:
14396:
14393:
14386:
14379:
14331:
14329:
14328:
14311:
14297:
14293:
14292:
14287:
14272:
14263:
14262:
14257:
14251:
14245:
14244:
14228:
14219:
14218:
14214:
14212:
14208:
14197:
14194:
14193:
14187:
14156:
14155:
14153:
14150:
14149:
14105:
14104:
14070:
14065:
14060:
14057:
14056:
14022:
14019:
14018:
13995:
13986:
13982:
13961:
13952:
13948:
13937:
13913:
13900:
13896:
13870:
13867:
13866:
13825:
13823:
13820:
13819:
13803:
13801:
13798:
13797:
13780:
13776:
13774:
13771:
13770:
13754:
13751:
13750:
13725:
13721:
13719:
13716:
13715:
13692:
13684:
13681:
13680:
13660:
13654:
13653:
13638:
13633:
13632:
13621:
13619:
13618:
13613:
13600:
13596:
13592:
13584:
13556:
13555:
13551:
13541:
13539:
13536:
13535:
13481:
13466:
13461:
13460:
13440:
13438:
13437:
13432:
13422:
13421:
13417:
13397:
13395:
13394:
13386:
13352:
13351:
13347:
13345:
13342:
13341:
13296:
13295:
13291:
13289:
13286:
13285:
13264:
13258:
13257:
13239:
13230:
13226:
13216:
13202:
13200:
13199:
13191:
13157:
13156:
13152:
13148:
13144:
13122:
13121:
13117:
13107:
13084:
13081:
13080:
13052:
13041:
13040:
13033:
13029:
13027:
13024:
13023:
13022:is replaced by
13002:
12991:
12990:
12980:
12976:
12974:
12971:
12970:
12948:
12945:
12944:
12928:
12925:
12924:
12921:
12908:
12864:
12863:
12859:
12844:
12843:
12839:
12814:
12813:
12809:
12782:
12781:
12777:
12775:
12761:
12758:
12757:
12714:
12713:
12673:
12672:
12668:
12656:
12652:
12637:
12633:
12621:
12617:
12602:
12598:
12583:
12582:
12574:
12570:
12554:
12547:
12543:
12527:
12511:
12507:
12502:
12491:
12487:
12473:
12472:
12462:
12458:
12454:
12440:
12439:
12430:
12426:
12415:
12411:
12402:
12398:
12384:
12383:
12373:
12369:
12365:
12351:
12350:
12341:
12337:
12326:
12322:
12302:
12301:
12291:
12287:
12283:
12278:
12272:
12268:
12263:
12260:
12259:
12216:
12215:
12175:
12174:
12170:
12156:
12129:
12128:
12120:
12116:
12100:
12093:
12089:
12073:
12067:
12064:
12063:
12040:
12036:
12031:
12018:
12004:
12003:
11993:
11989:
11985:
11971:
11970:
11951:
11931:
11930:
11920:
11916:
11912:
11907:
11901:
11897:
11892:
11889:
11888:
11885:
11872:
11871:
11863:
11861:
11860:
11810:
11799:
11798:
11797:
11791:
11780:
11779:
11778:
11769:
11765:
11755:
11754:
11750:
11739:
11736:
11735:
11703:
11702:
11683:
11679:
11662:
11659:
11658:
11652:
11632:
11593:
11590:
11589:
11547:
11546:
11542:
11534:
11532:
11525:
11521:
11519:
11516:
11515:
11468:
11460:
11458:
11445:
11437:
11435:
11415:
11414:
11403:
11400:
11399:
11384:
11380:
11376:
11352:
11348:
11318:
11317:
11315:
11312:
11311:
11297:
11271:
11265:
11264:
11263:
11224:
11223:
11222:
11220:
11203:
11200:
11199:
11172:
11157:
11151:
11150:
11149:
11110:
11109:
11108:
11106:
11080:
11077:
11076:
11069:
11015:
11013:
10982:
10974:
10972:
10955:
10952:
10951:
10899:
10897:
10872:
10864:
10862:
10845:
10842:
10841:
10834:
10810:
10772:
10769:
10768:
10757:
10755:
10754:
10700:
10698:
10685:
10677:
10675:
10673:
10670:
10669:
10655:
10628:
10614:
10606:
10604:
10600:
10599:
10588:
10585:
10584:
10577:
10570:
10566:
10563:
10554:
10550:
10542:
10539:
10530:
10526:
10492:
10488:
10464:
10456:
10454:
10453:
10449:
10443:
10439:
10438:
10434:
10432:
10429:
10428:
10414:
10383:
10379:
10340:
10332:
10330:
10313:
10312:
10308:
10300:
10298:
10288:
10283:
10282:
10278:
10274:
10270:
10246:
10242:
10227:
10223:
10218:
10215:
10214:
10207:
10176:
10172:
10149:
10148:
10134:
10133:
10129:
10121:
10119:
10103:
10095:
10093:
10089:
10085:
10064:
10060:
10045:
10041:
10036:
10033:
10032:
10025:
9987:
9986:
9972:
9971:
9949:
9945:
9924:
9920:
9905:
9901:
9896:
9893:
9892:
9874:
9863:
9836:
9825:
9787:
9786:
9770:
9766:
9745:
9741:
9726:
9722:
9717:
9714:
9713:
9703:
9699:
9696:
9646:
9642:
9630:
9629:
9625:
9606:
9598:
9596:
9594:
9591:
9590:
9583:
9547:
9540:
9536:
9520:
9518:
9514:
9490:
9486:
9436:
9433:
9432:
9425:
9421:
9410:
9380:
9369:
9368:
9367:
9365:
9361:
9357:
9333:
9329:
9327:
9324:
9323:
9312:
9308:
9304:
9244:
9243:
9222:
9211:
9210:
9209:
9199:
9193:
9192:
9180:
9175:
9167:
9163:
9136:
9116:
9088:
9080:
9047:
9044:
9043:
9036:
9030:
8983:
8975:
8973:
8962:
8959:
8958:
8951:
8941:
8938:
8933:
8932:
8930:
8929:
8923:
8878:
8871:
8867:
8858:
8854:
8852:
8849:
8848:
8796:
8772:
8771:
8767:
8754:
8750:
8729:
8725:
8705:
8703:
8699:
8697:
8694:
8693:
8684:
8678:
8667:
8615:
8613:
8609:
8598:
8584:
8576:
8574:
8570:
8569:
8557:
8553:
8544:
8540:
8531:
8527:
8518:
8514:
8506:
8503:
8502:
8488:
8482:
8433:
8429:
8422:
8406:
8402:
8396:
8392:
8386:
8382:
8378:
8376:
8352:
8348:
8341:
8322:
8318:
8311:
8309:
8298:
8293:
8280:
8275:
8270:
8266:
8265:
8261:
8251:
8243:
8241:
8237:
8233:
8232:
8214:
8210:
8187:
8183:
8176:
8172:
8171:
8151:
8149:
8145:
8134:
8120:
8112:
8110:
8106:
8105:
8093:
8089:
8080:
8076:
8067:
8063:
8054:
8050:
8042:
8039:
8038:
8019:
8016:
8013:
8012:
8010:
8004:
7988:
7985:
7982:
7981:
7979:
7970:
7953:
7952:
7940:
7936:
7930:
7926:
7917:
7913:
7904:
7900:
7891:
7887:
7866:
7862:
7858:
7856:
7852:
7848:
7847:
7835:
7824:
7811:
7810:
7788:
7781:
7777:
7771:
7767:
7740:
7735:
7722:
7717:
7709:
7707:
7689:
7679:
7678:
7674:
7673:
7656:
7624:
7616:
7614:
7610:
7609:
7602:
7593:
7589:
7580:
7576:
7567:
7563:
7554:
7550:
7540:
7538:
7535:
7534:
7527:
7520:
7511:
7481:
7477:
7468:
7464:
7439:
7431:
7428:
7415:
7411:
7407:
7405:
7401:
7385:
7381:
7372:
7368:
7355:
7345:
7341:
7332:
7328:
7321:
7318:
7305:
7301:
7297:
7295:
7291:
7276:
7272:
7263:
7259:
7250:
7246:
7237:
7233:
7225:
7222:
7221:
7191:
7177:
7175:
7159:
7155:
7149:
7145:
7143:
7136:
7132:
7126:
7115:
7109:
7106:
7105:
7065:
7061:
7052:
7048:
7032:
7026:
7018:
7015:
7014:
6994:
6993:
6981:
6977:
6966:
6952:
6942:
6938:
6929:
6925:
6918:
6916:
6912:
6911:
6904:
6900:
6899:
6893:
6882:
6862:
6858:
6854:
6852:
6848:
6839:
6838:
6818:
6814:
6805:
6801:
6792:
6788:
6776:
6772:
6763:
6759:
6758:
6751:
6747:
6737:
6735:
6734:
6730:
6720:
6714:
6709:
6695:
6690:
6689:
6685:
6684:
6672:
6668:
6647:
6645:
6639:
6628:
6611:
6607:
6603:
6601:
6597:
6587:
6578:
6574:
6565:
6561:
6552:
6548:
6539:
6535:
6525:
6523:
6520:
6519:
6485:
6481:
6472:
6468:
6459:
6455:
6443:
6439:
6430:
6426:
6425:
6418:
6414:
6404:
6402:
6401:
6397:
6387:
6381:
6376:
6360:
6354:
6343:
6330:
6326:
6318:
6315:
6314:
6307:
6290:
6289:
6267:
6263:
6254:
6250:
6237:
6230:
6226:
6220:
6216:
6201:
6197:
6188:
6184:
6163:
6158:
6145:
6140:
6132:
6130:
6126:
6110:
6101:
6100:
6082:
6078:
6072:
6068:
6053:
6044:
6033:
6032:
6031:
6016:
6015:
6011:
6003:
5999:
5998:
5991:
5987:
5986:
5966:
5965:
5957:
5953:
5952:
5945:
5941:
5940:
5929:
5923:
5919:
5915:
5913:
5910:
5909:
5877:
5873:
5864:
5860:
5847:
5837:
5833:
5814:
5812:
5806:
5802:
5787:
5783:
5774:
5770:
5757:
5741:
5737:
5724:
5722:
5716:
5712:
5694:
5690:
5688:
5685:
5684:
5674:
5664:
5648:
5634:
5623:
5600:
5596:
5590:
5586:
5571:
5562:
5551:
5550:
5549:
5534:
5525:
5524:
5522:
5519:
5518:
5512:
5502:
5477:
5473:
5462:
5458:
5456:
5444:
5440:
5430:
5425:
5420:
5417:
5416:
5409:
5406:
5402:
5376:
5375:
5365:
5361:
5340:
5320:
5301:
5297:
5245:
5241:
5220:
5216:
5214:
5211:
5210:
5203:
5172:
5168:
5166:
5132:
5127:
5125:
5122:
5121:
5090:
5083:
5079:
5063:
5061:
5057:
5028:
5025:
5024:
5013:
5008:
4977:
4973:
4971:
4940:
4935:
4933:
4930:
4929:
4870:
4867:
4866:
4822:
4815:
4811:
4798:
4796:
4792:
4788:
4759:
4756:
4755:
4748:
4717:
4713:
4709:
4707:
4703:
4699:
4670:
4669:
4667:
4664:
4663:
4629:
4625:
4621:
4619:
4615:
4611:
4593:
4582:
4581:
4580:
4578:
4575:
4574:
4567:
4540:
4536:
4532:
4520:
4509:
4508:
4507:
4478:
4477:
4475:
4472:
4471:
4458:
4455:
4450:
4449:
4447:
4446:
4423:
4419:
4404:
4400:
4391:
4387:
4358:
4355:
4354:
4348:
4344:
4336:
4325:
4310:
4306:
4302:
4276:
4275:
4260:
4216:
4214:
4210:
4209:
4197:
4193:
4189:
4177:
4152:
4132:
4099:
4096:
4095:
4066:
4065:
4051:
4039:
4028:
4027:
4026:
4024:
4018:
4013:
4005:
4001:
3974:
3954:
3921:
3918:
3917:
3910:
3882:
3870:
3859:
3858:
3857:
3855:
3844:
3841:
3840:
3830:
3823:
3819:
3815:
3812:
3788:
3752:
3750:
3749:
3732:
3699:
3697:
3696:
3688:
3680:
3677:
3676:
3657:
3654:
3653:
3637:
3634:
3633:
3602:
3600:
3597:
3596:
3580:
3578:
3575:
3574:
3558:
3552:
3551:
3549:
3546:
3545:
3513:
3504:
3500:
3484:
3482:
3481:
3473:
3463:
3459:
3453:
3447:
3446:
3424:
3423:
3419:
3409:
3386:
3383:
3382:
3376:
3360:
3357:
3355:
3344:
3339:
3338:
3336:
3335:
3331:
3328:
3324:
3320:
3314:
3308:
3305:
3301:
3298:
3292:
3240:
3236:
3227:
3223:
3222:
3220:
3211:
3200:
3199:
3198:
3197:
3193:
3178:
3167:
3150:
3149:
3145:
3137:
3134:
3133:
3095:
3090:
3089:
3078:
3077:
3072:
3059:
3030:
3029:
3015:
3012:
3011:
3000:
2994:
2990:
2987:
2983:
2980:
2976:
2973:
2969:
2963:
2956:
2949:
2942:
2936:
2935:
2924:
2917:
2913:
2910:
2906:
2891:
2880:
2865:
2842:
2838:
2824:
2820:
2798:
2797:
2764:
2763:
2752:
2748:
2747:
2740:
2736:
2735:
2721:
2720:
2716:
2701:
2693:
2677:
2669:
2663:
2660:
2659:
2619:
2612:
2608:
2599:
2595:
2594:
2592:
2575:
2572:
2571:
2557:
2554:
2543:
2538:
2531:
2511:
2507:
2492:
2488:
2479:
2475:
2460:
2456:
2441:
2437:
2428:
2424:
2415:
2411:
2409:
2406:
2405:
2399:
2395:
2392:
2388:
2385:
2381:
2378:
2374:
2360:
2354:
2349:
2337:Feynman diagram
2272:along the path.
2257:
2251:
2238:squared modulus
2218:
2205:
2203:Planck constant
2198:
2195:
2188:
2184:
2175:
2170:
2169:
2168:
2165:
2164:
2163:
2159:
2155:
2151:
2148:
2144:
2140:
2136:
2133:
2129:
2126:
2122:
2116:
2104:
2101:
2097:
2093:
2089:
2086:
2077:
2068:
2061:
2054:
2046:
2042:
2036:
2029:
2027:Classical limit
2013:
2006:
2001:
1997:
1996:. While in the
1987:
1983:
1954:
1950:
1948:
1945:
1944:
1937:
1926:
1922:
1918:
1914:
1910:
1903:
1899:
1895:
1889:at a later time
1886:
1840:
1836:
1834:
1831:
1830:
1773:
1769:
1767:
1764:
1763:
1752:
1748:
1735:
1724:
1709:
1668:
1664:
1662:
1659:
1658:
1626:
1625:
1595:
1594:
1555:
1554:
1545:
1544:
1540:
1536:
1534:
1531:
1530:
1523:
1519:
1515:
1511:
1507:
1503:
1492:
1488:
1461:
1453:
1451:
1437:
1436:
1434:
1431:
1430:
1390:
1389:
1372:
1369:
1368:
1350:
1328:
1327:
1325:
1322:
1321:
1290:
1289:
1285:
1277:
1275:
1267:
1264:
1263:
1231:
1230:
1191:
1190:
1167:
1164:
1163:
1099:
1087:
1069:(rather than a
1055:Richard Feynman
1043:Brownian motion
1039:Wiener integral
1028:Stephen Wolfram
1008:polymer physics
887:
858:
857:
856:
621:
613:
612:
558:
557:Advanced topics
550:
549:
548:
500:Hidden-variable
490:de Broglie–Bohm
469:
467:Interpretations
459:
458:
457:
427:
419:
418:
417:
375:
367:
366:
365:
332:
288:CHSH inequality
277:
269:
268:
267:
196:Complementarity
190:
182:
181:
180:
148:
119:
94:
83:
82:
68:
58:
53:
45:
42:
41:
24:
17:
12:
11:
5:
26394:
26384:
26383:
26378:
26373:
26368:
26363:
26358:
26353:
26336:
26335:
26333:
26332:
26322:
26311:
26301:
26296:
26289:
26282:
26277:
26271:
26269:
26265:
26264:
26262:
26261:
26252:
26242:
26240:
26236:
26235:
26233:
26232:
26221:
26210:
26199:
26188:
26177:
26166:
26155:
26144:
26133:
26122:
26120:
26116:
26115:
26113:
26112:
26107:
26102:
26097:
26092:
26087:
26082:
26077:
26072:
26067:
26062:
26057:
26052:
26047:
26042:
26036:
26034:
26030:
26029:
26022:
26021:
26014:
26007:
25999:
25990:
25989:
25987:
25986:
25975:
25972:
25971:
25969:
25968:
25963:
25958:
25953:
25952:
25951:
25940:
25938:
25934:
25933:
25931:
25930:
25925:
25920:
25919:
25918:
25908:
25903:
25901:Casimir effect
25898:
25892:
25890:
25886:
25885:
25883:
25882:
25877:
25872:
25867:
25862:
25860:Quantum optics
25857:
25852:
25847:
25842:
25837:
25832:
25827:
25822:
25817:
25812:
25807:
25802:
25797:
25792:
25787:
25782:
25781:
25780:
25770:
25765:
25760:
25755:
25754:
25753:
25743:
25738:
25733:
25727:
25725:
25719:
25718:
25716:
25715:
25710:
25705:
25700:
25695:
25690:
25685:
25680:
25675:
25670:
25665:
25659:
25657:
25651:
25650:
25648:
25647:
25642:
25637:
25635:Quantum eraser
25632:
25627:
25622:
25617:
25612:
25607:
25602:
25597:
25591:
25589:
25585:
25584:
25582:
25581:
25576:
25571:
25566:
25561:
25556:
25551:
25550:
25549:
25548:
25547:
25532:
25527:
25522:
25517:
25512:
25506:
25504:
25498:
25497:
25495:
25494:
25489:
25484:
25479:
25474:
25468:
25466:
25462:
25461:
25459:
25458:
25453:
25448:
25443:
25438:
25433:
25428:
25422:
25420:
25416:
25415:
25413:
25412:
25411:
25410:
25405:
25395:
25390:
25385:
25380:
25375:
25370:
25365:
25360:
25355:
25350:
25345:
25340:
25335:
25334:
25333:
25328:
25323:
25318:
25308:
25306:Density matrix
25303:
25298:
25293:
25287:
25285:
25281:
25280:
25278:
25277:
25272:
25267:
25262:
25261:
25260:
25250:
25244:
25242:
25238:
25237:
25230:
25229:
25222:
25215:
25207:
25201:
25200:
25194:
25188:
25183:
25176:
25175:External links
25173:
25171:
25170:
25165:978-1316512258
25164:
25148:
25142:
25126:
25119:
25105:(2010-02-21).
25099:
25094:
25070:
25032:(2): 178–188.
25014:
25007:
24991:
24965:(4): 303–335.
24947:
24941:
24925:
24919:
24906:
24899:
24882:
24876:
24863:
24854:
24848:
24835:
24820:
24814:
24796:
24790:
24774:
24768:
24755:
24749:
24736:
24730:
24717:
24703:
24674:
24669:hep-th/9302097
24659:
24653:
24640:
24634:
24617:
24612:978-0883188705
24611:
24595:
24589:
24568:
24559:
24542:
24524:(2): 367–387.
24506:
24497:
24467:Feynman, R. P.
24463:
24448:
24424:(2): 185–188.
24399:
24376:
24328:
24322:
24310:"Introduction"
24305:
24259:Cartier, P. C.
24255:
24237:(2): 374–456.
24227:Leggett, A. J.
24219:
24213:
24200:
24187:
24185:
24182:
24179:
24178:
24167:
24164:Gell-Mann 1993
24156:
24145:
24119:
24078:
24066:
24054:
24043:
24031:
24019:
24008:
23996:
23974:
23963:
23951:
23930:
23927:Van Vleck 1928
23919:
23908:
23897:
23880:(2020-04-14).
23869:
23842:
23818:
23786:
23773:
23772:
23770:
23767:
23764:
23763:
23750:
23736:
23735:
23733:
23730:
23728:
23727:
23722:
23717:
23712:
23707:
23702:
23697:
23691:
23689:
23686:
23668:
23659:
23652:
23651:
23639:
23633:
23627:
23624:
23621:
23616:
23610:
23606:
23602:
23599:
23593:
23588:
23584:
23581:
23555:
23552:
23531:
23528:
23504:
23501:
23488:
23482:
23479:
23470:
23467:
23461:
23455:
23452:
23443:
23440:
23434:
23429:
23426:
23401:
23398:
23389:
23386:
23360:
23357:
23348:
23345:
23322:
23319:
23296:
23293:
23284:
23281:
23255:
23252:
23243:
23240:
23217:
23214:
23192:
23188:
23185:
23179:
23176:
23173:
23151:
23147:
23144:
23138:
23135:
23132:
23112:
23106:
23103:
23094:
23091:
23085:
23079:
23076:
23067:
23064:
23058:
23053:
23050:
23031:
23028:
23015:
23012:
23010:
23007:
23006:
23005:
22994:
22991:
22988:
22985:
22982:
22979:
22975:
22968:
22965:
22961:
22956:
22953:
22949:
22945:
22942:
22939:
22936:
22933:
22930:
22927:
22924:
22921:
22918:
22915:
22911:
22906:
22902:
22898:
22884:
22883:
22872:
22869:
22865:
22861:
22858:
22855:
22852:
22848:
22823:
22822:
22811:
22808:
22805:
22802:
22799:
22796:
22792:
22785:
22782:
22778:
22773:
22770:
22766:
22762:
22759:
22756:
22753:
22750:
22747:
22744:
22741:
22738:
22735:
22732:
22729:
22726:
22723:
22720:
22717:
22713:
22706:
22703:
22699:
22694:
22691:
22687:
22683:
22680:
22677:
22672:
22668:
22662:
22658:
22654:
22651:
22648:
22645:
22642:
22638:
22631:
22628:
22624:
22619:
22616:
22612:
22608:
22605:
22602:
22599:
22596:
22593:
22590:
22587:
22584:
22581:
22578:
22575:
22572:
22569:
22566:
22563:
22559:
22552:
22549:
22545:
22540:
22537:
22533:
22529:
22526:
22523:
22520:
22517:
22514:
22511:
22497:
22496:
22485:
22482:
22478:
22474:
22471:
22468:
22463:
22459:
22453:
22449:
22445:
22441:
22437:
22434:
22431:
22428:
22425:
22422:
22419:
22416:
22413:
22410:
22407:
22404:
22401:
22398:
22395:
22392:
22389:
22386:
22383:
22380:
22377:
22374:
22371:
22368:
22365:
22362:
22359:
22356:
22353:
22350:
22347:
22344:
22341:
22327:
22326:
22315:
22312:
22309:
22306:
22303:
22300:
22297:
22294:
22291:
22288:
22285:
22282:
22278:
22273:
22268:
22227:
22226:
22214:
22211:
22208:
22205:
22202:
22199:
22196:
22191:
22183:
22180:
22175:
22171:
22167:
22164:
22160:
22155:
22152:
22149:
22146:
22141:
22137:
22133:
22130:
22127:
22124:
22119:
22115:
22100:
22099:
22087:
22084:
22081:
22076:
22072:
22066:
22062:
22058:
22055:
22052:
22049:
22046:
22043:
22040:
22037:
22023:
22022:
22010:
22007:
22004:
22001:
21998:
21995:
21992:
21989:
21986:
21983:
21980:
21977:
21972:
21969:
21966:
21962:
21958:
21955:
21952:
21949:
21946:
21943:
21940:
21937:
21934:
21931:
21928:
21914:
21913:
21902:
21899:
21896:
21893:
21889:
21884:
21880:
21876:
21873:
21870:
21849:
21848:
21837:
21834:
21830:
21824:
21820:
21816:
21810:
21806:
21800:
21797:
21793:
21789:
21785:
21781:
21778:
21775:
21772:
21769:
21766:
21763:
21760:
21748:which implies
21746:
21745:
21734:
21731:
21728:
21725:
21722:
21719:
21715:
21709:
21706:
21702:
21698:
21694:
21690:
21686:
21681:
21676:
21649:
21648:
21637:
21634:
21631:
21628:
21625:
21622:
21619:
21616:
21613:
21609:
21604:
21599:
21540:
21539:
21528:
21525:
21522:
21517:
21513:
21507:
21503:
21499:
21496:
21493:
21490:
21487:
21482:
21477:
21474:
21456:gauge symmetry
21437:Main article:
21434:
21431:
21418:
21415:
21403:
21402:
21391:
21388:
21385:
21382:
21379:
21376:
21371:
21366:
21360:
21356:
21292:
21291:
21280:
21277:
21274:
21269:
21265:
21261:
21258:
21255:
21252:
21249:
21246:
21243:
21238:
21235:
21229:
21212:
21211:
21200:
21197:
21194:
21191:
21188:
21185:
21182:
21179:
21176:
21173:
21170:
21167:
21164:
21161:
21158:
21155:
21151:
21144:
21141:
21137:
21132:
21129:
21125:
21118:
21115:
21112:
21109:
21106:
21099:
21094:
21077:
21076:
21065:
21062:
21057:
21052:
21048:
21045:
21042:
21039:
21036:
21033:
21030:
21027:
21021:
21018:
21015:
21012:
21009:
21002:
20997:
20990:
20971:
20970:
20959:
20956:
20953:
20950:
20947:
20941:
20936:
20932:
20928:
20925:
20922:
20918:
20906:
20902:
20896:
20892:
20888:
20880:
20876:
20871:
20865:
20859:
20854:
20850:
20846:
20843:
20840:
20836:
20824:
20820:
20814:
20810:
20806:
20798:
20794:
20789:
20781:
20777:
20773:
20770:
20767:
20764:
20761:
20758:
20755:
20752:
20748:
20741:
20738:
20734:
20729:
20726:
20722:
20718:
20696:
20695:
20684:
20681:
20676:
20672:
20668:
20665:
20655:
20651:
20645:
20641:
20637:
20629:
20625:
20620:
20614:
20611:
20606:
20602:
20598:
20595:
20585:
20581:
20575:
20571:
20567:
20559:
20555:
20550:
20544:
20541:
20538:
20535:
20532:
20499:, then for an
20469:
20459:
20424:Basically, if
20422:
20421:
20410:
20401:
20397:
20394:
20391:
20388:
20385:
20382:
20379:
20376:
20373:
20368:
20362:
20358:
20354:
20350:
20345:
20340:
20332:
20328:
20325:
20322:
20319:
20316:
20313:
20310:
20307:
20304:
20299:
20293:
20289:
20285:
20281:
20278:
20275:
20272:
20269:
20264:
20259:
20253:
20248:
20244:
20240:
20237:
20223:
20222:
20211:
20206:
20201:
20193:
20189:
20184:
20180:
20173:
20169:
20164:
20159:
20154:
20151:
20148:
20145:
20140:
20136:
20132:
20129:
20126:
20123:
20116:
20112:
20108:
20103:
20099:
20095:
20091:
20076:
20075:
20064:
20059:
20054:
20050:
20045:
20041:
20037:
20034:
20031:
20028:
20023:
20019:
20015:
20012:
20008:
20002:
19999:
19996:
19993:
19987:
19983:
19979:
19976:
19973:
19970:
19964:
19959:
19955:
19951:
19948:
19945:
19942:
19939:
19934:
19930:
19926:
19923:
19920:
19915:
19910:
19906:
19888:
19887:
19876:
19870:
19866:
19863:
19860:
19857:
19854:
19851:
19848:
19845:
19842:
19837:
19831:
19827:
19823:
19819:
19814:
19809:
19806:
19803:
19800:
19797:
19794:
19741:
19730:
19729:
19718:
19714:
19708:
19705:
19699:
19693:
19689:
19685:
19682:
19679:
19675:
19670:
19667:
19663:
19659:
19636:
19635:
19623:
19616:
19613:
19608:
19605:
19602:
19597:
19592:
19586:
19583:
19580:
19577:
19573:
19569:
19566:
19563:
19559:
19553:
19550:
19545:
19542:
19539:
19536:
19533:
19527:
19510:, then we can
19492:
19491:
19478:
19475:
19472:
19469:
19466:
19462:
19447:
19446:
19435:
19430:
19427:
19424:
19419:
19414:
19410:
19361:
19360:
19349:
19346:
19340:
19337:
19332:
19329:
19326:
19321:
19316:
19287:Main article:
19284:
19281:
19271:
19264:
19263:
19252:
19244:
19239:
19235:
19232:
19229:
19226:
19221:
19218:
19215:
19210:
19205:
19201:
19197:
19194:
19191:
19182:
19178:
19173:
19168:
19164:
19157:
19154:
19151:
19147:
19139:
19134:
19130:
19127:
19124:
19121:
19116:
19113:
19110:
19105:
19100:
19096:
19092:
19089:
19086:
19077:
19073:
19068:
19063:
19059:
19052:
19049:
19046:
19043:
19040:
19036:
19029:
19026:
19023:
19020:
19017:
19014:
19011:
19008:
18926:
18923:
18908:
18907:
18896:
18888:
18885:
18882:
18877:
18872:
18868:
18864:
18859:
18854:
18847:
18844:
18841:
18836:
18831:
18827:
18823:
18820:
18817:
18814:
18811:
18806:
18801:
18795:
18792:
18789:
18786:
18774:, is given by
18731:
18728:
18648:of the field:
18633:
18630:
18622:interpretation
18600:
18599:
18588:
18582:
18579:
18576:
18571:
18567:
18563:
18558:
18554:
18549:
18544:
18541:
18538:
18535:
18532:
18516:
18509:
18508:
18497:
18491:
18488:
18485:
18478:
18474:
18470:
18465:
18458:
18455:
18446:
18441:
18437:
18432:
18427:
18421:
18418:
18415:
18408:
18404:
18400:
18395:
18388:
18385:
18376:
18371:
18367:
18362:
18357:
18354:
18351:
18348:
18345:
18329:
18297:
18296:
18285:
18276:
18273:
18267:
18260:
18257:
18248:
18245:
18242:
18239:
18234:
18230:
18226:
18222:
18217:
18214:
18211:
18208:
18199:
18195:
18192:
18172:
18155:
18148:
18147:
18136:
18126:
18122:
18118:
18113:
18106:
18103:
18094:
18089:
18085:
18080:
18075:
18065:
18061:
18057:
18052:
18045:
18042:
18033:
18028:
18024:
18019:
18014:
18011:
18008:
18005:
18002:
17997:
17993:
17989:
17971:
17970:
17959:
17951:
17947:
17943:
17938:
17931:
17928:
17921:
17916:
17911:
17907:
17902:
17897:
17894:
17891:
17888:
17885:
17862:
17855:
17854:
17843:
17837:
17834:
17829:
17825:
17820:
17815:
17811:
17807:
17801:
17797:
17793:
17788:
17784:
17779:
17775:
17771:
17765:
17760:
17756:
17752:
17749:
17746:
17743:
17740:
17698:
17697:
17686:
17682:
17678:
17671:
17667:
17664:
17658:
17651:
17647:
17643:
17640:
17637:
17634:
17628:
17624:
17618:
17613:
17609:
17605:
17602:
17599:
17596:
17593:
17590:
17587:
17561:
17560:
17549:
17545:
17541:
17537:
17533:
17529:
17526:
17523:
17519:
17515:
17512:
17509:
17506:
17503:
17500:
17495:
17490:
17486:
17482:
17479:
17476:
17473:
17470:
17467:
17464:
17442:
17441:
17430:
17422:
17415:
17411:
17407:
17404:
17401:
17398:
17392:
17388:
17381:
17377:
17374:
17370:
17366:
17363:
17359:
17355:
17352:
17349:
17346:
17343:
17340:
17309:
17308:
17297:
17292:
17289:
17285:
17279:
17276:
17273:
17269:
17265:
17262:
17257:
17254:
17251:
17248:
17245:
17242:
17238:
17231:
17227:
17224:
17220:
17216:
17213:
17209:
17205:
17202:
17199:
17196:
17193:
17190:
17156:
17155:
17144:
17139:
17136:
17132:
17129:
17122:
17115:
17112:
17100:
17094:
17090:
17086:
17082:
17076:
17073:
17070:
17066:
17062:
17059:
17054:
17051:
17048:
17045:
17042:
17039:
17035:
17031:
17028:
17024:
17020:
17017:
17014:
17011:
17008:
17005:
16975:
16974:
16963:
16957:
16954:
16951:
16945:
16942:
16935:
16928:
16925:
16918:
16912:
16909:
16906:
16901:
16898:
16892:
16889:
16886:
16883:
16880:
16877:
16868:
16815:
16814:
16803:
16797:
16794:
16791:
16785:
16782:
16776:
16769:
16766:
16757:
16754:
16749:
16746:
16740:
16737:
16734:
16731:
16728:
16725:
16716:
16682:
16681:
16670:
16663:
16660:
16653:
16649:
16645:
16642:
16636:
16632:
16629:
16626:
16623:
16620:
16617:
16614:
16588:
16587:
16576:
16569:
16566:
16559:
16555:
16551:
16548:
16545:
16542:
16539:
16536:
16530:
16526:
16523:
16520:
16517:
16514:
16511:
16508:
16505:
16502:
16484:
16483:
16472:
16469:
16466:
16463:
16460:
16457:
16454:
16449:
16445:
16441:
16436:
16432:
16386:
16385:
16374:
16371:
16368:
16362:
16359:
16356:
16353:
16350:
16346:
16342:
16339:
16336:
16333:
16330:
16327:
16322:
16318:
16312:
16309:
16306:
16303:
16300:
16297:
16293:
16289:
16286:
16283:
16279:
16276:
16273:
16270:
16267:
16264:
16261:
16258:
16255:
16252:
16249:
16244:
16240:
16234:
16230:
16226:
16223:
16220:
16217:
16212:
16208:
16186:
16171:
16170:
16159:
16156:
16153:
16147:
16144:
16141:
16138:
16135:
16131:
16125:
16122:
16119:
16116:
16113:
16110:
16105:
16102:
16099:
16096:
16093:
16090:
16086:
16082:
16079:
16076:
16073:
16070:
16067:
16064:
16061:
16058:
16055:
16052:
16049:
16046:
16043:
16040:
16037:
16034:
16031:
16028:
15996:
15993:
15937:
15936:
15925:
15922:
15919:
15916:
15913:
15908:
15904:
15900:
15897:
15894:
15891:
15888:
15885:
15882:
15877:
15873:
15869:
15866:
15863:
15860:
15857:
15854:
15837:
15836:
15834:
15833:
15826:
15819:
15811:
15808:
15807:
15800:
15799:
15794:
15789:
15784:
15779:
15774:
15769:
15764:
15759:
15754:
15749:
15744:
15739:
15734:
15729:
15724:
15719:
15714:
15709:
15704:
15699:
15694:
15689:
15684:
15679:
15674:
15669:
15664:
15659:
15654:
15649:
15644:
15639:
15634:
15629:
15624:
15619:
15614:
15609:
15604:
15599:
15594:
15589:
15584:
15579:
15574:
15569:
15564:
15559:
15554:
15549:
15544:
15539:
15534:
15529:
15524:
15519:
15514:
15509:
15504:
15499:
15494:
15489:
15484:
15479:
15474:
15469:
15464:
15459:
15454:
15449:
15444:
15439:
15434:
15429:
15424:
15419:
15414:
15409:
15404:
15399:
15394:
15389:
15384:
15379:
15374:
15369:
15364:
15359:
15354:
15349:
15344:
15339:
15334:
15329:
15324:
15319:
15314:
15309:
15304:
15299:
15294:
15289:
15284:
15279:
15274:
15269:
15264:
15259:
15254:
15249:
15244:
15239:
15234:
15229:
15224:
15219:
15214:
15209:
15204:
15199:
15194:
15189:
15184:
15179:
15174:
15169:
15164:
15159:
15154:
15149:
15144:
15139:
15134:
15129:
15124:
15119:
15114:
15108:
15107:
15104:
15103:
15100:
15099:
15096:
15095:
15090:
15085:
15080:
15075:
15069:
15066:
15065:
15062:
15061:
15058:
15057:
15052:
15047:
15042:
15036:
15033:Standard Model
15031:
15030:
15027:
15026:
15023:
15022:
15017:
15012:
15007:
15002:
15000:Dirac equation
14996:
14993:
14992:
14989:
14988:
14985:
14984:
14982:Wick's theorem
14979:
14974:
14969:
14967:Regularization
14964:
14959:
14954:
14949:
14944:
14939:
14934:
14929:
14924:
14919:
14914:
14909:
14904:
14899:
14893:
14890:
14889:
14886:
14885:
14882:
14881:
14876:
14874:Noether charge
14871:
14866:
14861:
14859:Gauge symmetry
14856:
14851:
14846:
14841:
14836:
14831:
14825:
14820:
14819:
14816:
14815:
14812:
14811:
14806:
14801:
14796:
14791:
14786:
14781:
14776:
14771:
14765:
14762:
14761:
14758:
14757:
14751:
14750:
14745:
14737:
14736:
14727:
14724:
14708:
14707:
14695:
14690:
14687:
14684:
14680:
14676:
14672:
14669:
14666:
14663:
14634:
14633:
14622:
14619:
14616:
14613:
14609:
14601:
14597:
14594:
14591:
14585:
14581:
14577:
14574:
14571:
14568:
14565:
14561:
14521:
14454:
14451:
14448:
14444:
14440:
14435:
14412:
14406:
14391:
14372:
14371:
14360:
14357:
14353:
14350:
14347:
14344:
14338:
14334:
14327:
14324:
14321:
14318:
14314:
14310:
14307:
14300:
14296:
14290:
14286:
14282:
14279:
14275:
14271:
14266:
14254:
14248:
14239:
14235:
14231:
14227:
14222:
14217:
14211:
14207:
14204:
14201:
14186:
14183:
14163:
14160:
14146:
14145:
14133:
14130:
14127:
14124:
14121:
14118:
14112:
14109:
14103:
14100:
14097:
14094:
14091:
14088:
14085:
14082:
14076:
14073:
14069:
14064:
14041:
14038:
14035:
14032:
14029:
14026:
14015:
14014:
14002:
13998:
13994:
13989:
13985:
13981:
13977:
13974:
13971:
13968:
13964:
13960:
13955:
13951:
13944:
13940:
13936:
13933:
13929:
13926:
13923:
13920:
13916:
13912:
13909:
13906:
13903:
13899:
13895:
13892:
13889:
13886:
13883:
13880:
13877:
13874:
13844:
13841:
13838:
13835:
13832:
13828:
13806:
13783:
13779:
13758:
13743:Norbert Wiener
13728:
13724:
13712:Wiener measure
13699:
13695:
13691:
13688:
13677:
13676:
13663:
13657:
13649:
13646:
13641:
13636:
13628:
13624:
13616:
13612:
13607:
13604:
13599:
13595:
13591:
13587:
13583:
13580:
13575:
13572:
13569:
13566:
13563:
13559:
13554:
13548:
13545:
13522:
13521:
13509:
13506:
13501:
13497:
13494:
13491:
13488:
13484:
13480:
13477:
13474:
13469:
13464:
13459:
13456:
13453:
13447:
13443:
13435:
13429:
13426:
13420:
13416:
13413:
13410:
13404:
13400:
13393:
13389:
13385:
13379:
13376:
13373:
13370:
13367:
13364:
13361:
13358:
13355:
13350:
13323:
13320:
13317:
13314:
13311:
13308:
13305:
13302:
13299:
13294:
13282:
13281:
13267:
13261:
13255:
13252:
13249:
13246:
13242:
13238:
13233:
13229:
13223:
13219:
13215:
13209:
13205:
13198:
13194:
13190:
13184:
13181:
13178:
13175:
13172:
13169:
13166:
13163:
13160:
13155:
13151:
13147:
13141:
13138:
13135:
13132:
13129:
13125:
13120:
13114:
13111:
13106:
13103:
13100:
13097:
13094:
13091:
13088:
13059:
13055:
13048:
13045:
13039:
13036:
13032:
13009:
13005:
12998:
12995:
12989:
12986:
12983:
12979:
12958:
12955:
12952:
12932:
12923:If we replace
12920:
12917:
12907:
12904:
12903:
12902:
12885:
12882:
12879:
12876:
12871:
12868:
12862:
12858:
12855:
12852:
12847:
12842:
12835:
12832:
12829:
12826:
12821:
12818:
12812:
12808:
12805:
12802:
12799:
12796:
12793:
12790:
12785:
12780:
12774:
12771:
12768:
12765:
12751:
12750:
12736:
12733:
12730:
12727:
12721:
12718:
12712:
12709:
12706:
12703:
12700:
12697:
12694:
12691:
12688:
12685:
12680:
12677:
12671:
12667:
12664:
12659:
12655:
12651:
12648:
12645:
12640:
12636:
12632:
12629:
12624:
12620:
12616:
12613:
12610:
12605:
12601:
12597:
12594:
12591:
12586:
12577:
12573:
12569:
12566:
12563:
12560:
12557:
12550:
12546:
12542:
12539:
12536:
12533:
12530:
12526:
12522:
12519:
12514:
12510:
12505:
12499:
12494:
12490:
12486:
12480:
12477:
12469:
12466:
12461:
12457:
12453:
12447:
12444:
12438:
12433:
12429:
12423:
12418:
12414:
12410:
12405:
12401:
12397:
12391:
12388:
12380:
12377:
12372:
12368:
12364:
12358:
12355:
12349:
12344:
12340:
12334:
12329:
12325:
12321:
12318:
12315:
12309:
12306:
12298:
12295:
12290:
12286:
12281:
12275:
12271:
12267:
12253:
12252:
12238:
12235:
12232:
12229:
12223:
12220:
12214:
12211:
12208:
12205:
12202:
12199:
12196:
12193:
12190:
12187:
12182:
12179:
12173:
12169:
12166:
12162:
12159:
12155:
12152:
12149:
12146:
12143:
12140:
12137:
12132:
12123:
12119:
12115:
12112:
12109:
12106:
12103:
12096:
12092:
12088:
12085:
12082:
12079:
12076:
12072:
12059:take the form
12048:
12043:
12039:
12034:
12028:
12024:
12021:
12017:
12011:
12008:
12000:
11997:
11992:
11988:
11984:
11978:
11975:
11969:
11966:
11961:
11957:
11954:
11950:
11947:
11944:
11938:
11935:
11927:
11924:
11919:
11915:
11910:
11904:
11900:
11896:
11884:
11881:
11857:
11856:
11845:
11842:
11839:
11834:
11830:
11827:
11824:
11821:
11818:
11813:
11806:
11803:
11794:
11787:
11784:
11775:
11772:
11768:
11762:
11759:
11753:
11749:
11746:
11743:
11726:
11725:
11714:
11711:
11706:
11698:
11695:
11692:
11689:
11686:
11682:
11678:
11675:
11672:
11669:
11666:
11651:
11648:
11631:
11628:
11627:
11626:
11615:
11612:
11609:
11606:
11603:
11600:
11597:
11583:
11582:
11571:
11568:
11564:
11554:
11551:
11545:
11540:
11537:
11531:
11528:
11524:
11498:
11497:
11486:
11483:
11480:
11474:
11471:
11466:
11463:
11457:
11451:
11448:
11443:
11440:
11434:
11431:
11428:
11422:
11419:
11413:
11410:
11407:
11373:
11372:
11361:
11355:
11351:
11347:
11344:
11341:
11338:
11335:
11332:
11329:
11326:
11321:
11294:
11293:
11280:
11274:
11268:
11262:
11259:
11256:
11253:
11250:
11247:
11244:
11241:
11238:
11235:
11232:
11227:
11219:
11216:
11213:
11210:
11207:
11193:
11192:
11179:
11176:
11171:
11166:
11160:
11154:
11148:
11145:
11142:
11139:
11136:
11133:
11130:
11127:
11124:
11121:
11118:
11113:
11105:
11102:
11099:
11096:
11093:
11090:
11087:
11084:
11066:
11065:
11052:
11048:
11045:
11042:
11039:
11036:
11033:
11030:
11027:
11024:
11021:
11018:
11012:
11009:
11006:
11003:
11000:
10997:
10994:
10988:
10985:
10980:
10977:
10971:
10968:
10965:
10962:
10959:
10949:
10936:
10932:
10929:
10926:
10923:
10920:
10917:
10914:
10911:
10908:
10905:
10902:
10896:
10893:
10890:
10887:
10884:
10878:
10875:
10870:
10867:
10861:
10858:
10855:
10852:
10849:
10828:
10827:
10814:
10809:
10806:
10803:
10800:
10797:
10794:
10791:
10788:
10785:
10782:
10779:
10776:
10751:
10750:
10737:
10733:
10730:
10727:
10724:
10721:
10718:
10715:
10712:
10709:
10706:
10703:
10697:
10691:
10688:
10683:
10680:
10652:
10651:
10640:
10637:
10631:
10626:
10620:
10617:
10612:
10609:
10603:
10598:
10595:
10592:
10562:
10559:
10538:
10535:
10520:
10519:
10508:
10505:
10501:
10495:
10491:
10486:
10482:
10479:
10476:
10470:
10467:
10462:
10459:
10452:
10446:
10442:
10437:
10411:
10410:
10398:
10395:
10389:
10386:
10382:
10377:
10373:
10370:
10366:
10363:
10360:
10357:
10353:
10346:
10343:
10338:
10335:
10329:
10320:
10317:
10311:
10306:
10303:
10294:
10291:
10287:
10281:
10277:
10273:
10269:
10264:
10261:
10258:
10255:
10252:
10249:
10245:
10241:
10238:
10235:
10230:
10226:
10222:
10204:
10203:
10191:
10188:
10182:
10179:
10175:
10170:
10166:
10163:
10156:
10153:
10141:
10138:
10132:
10127:
10124:
10118:
10115:
10109:
10106:
10101:
10098:
10092:
10088:
10082:
10079:
10076:
10073:
10070:
10067:
10063:
10059:
10056:
10053:
10048:
10044:
10040:
10022:
10021:
10009:
10006:
10000:
9994:
9991:
9985:
9979:
9976:
9970:
9967:
9964:
9961:
9958:
9955:
9952:
9948:
9942:
9939:
9936:
9933:
9930:
9927:
9923:
9919:
9916:
9913:
9908:
9904:
9900:
9822:
9821:
9809:
9806:
9800:
9794:
9791:
9785:
9782:
9779:
9776:
9773:
9769:
9763:
9760:
9757:
9754:
9751:
9748:
9744:
9740:
9737:
9734:
9729:
9725:
9721:
9695:
9692:
9687:
9686:
9674:
9670:
9666:
9663:
9660:
9657:
9654:
9649:
9645:
9638:
9635:
9628:
9624:
9621:
9618:
9612:
9609:
9604:
9601:
9580:
9579:
9568:
9564:
9561:
9553:
9550:
9543:
9539:
9535:
9532:
9529:
9526:
9523:
9517:
9511:
9508:
9505:
9502:
9499:
9496:
9493:
9489:
9485:
9482:
9479:
9476:
9473:
9470:
9467:
9464:
9461:
9458:
9455:
9452:
9449:
9446:
9443:
9440:
9407:
9406:
9393:
9388:
9383:
9376:
9373:
9364:
9360:
9354:
9351:
9348:
9345:
9342:
9339:
9336:
9332:
9301:
9300:
9289:
9286:
9283:
9279:
9276:
9272:
9269:
9266:
9263:
9260:
9255:
9252:
9247:
9242:
9239:
9236:
9233:
9230:
9225:
9218:
9215:
9206:
9203:
9196:
9189:
9186:
9183:
9178:
9174:
9170:
9166:
9160:
9157:
9154:
9151:
9148:
9145:
9142:
9139:
9134:
9131:
9128:
9125:
9122:
9119:
9115:
9111:
9108:
9105:
9102:
9099:
9096:
9091:
9086:
9083:
9079:
9075:
9072:
9069:
9066:
9063:
9060:
9057:
9054:
9051:
9032:Main article:
9029:
9026:
9018:Hagen Kleinert
9010:
9009:
8995:
8992:
8989:
8986:
8981:
8978:
8972:
8969:
8966:
8922:
8919:
8918:
8917:
8906:
8900:
8897:
8893:
8886:
8883:
8877:
8874:
8870:
8866:
8861:
8857:
8842:
8841:
8830:
8827:
8824:
8821:
8818:
8815:
8812:
8809:
8806:
8803:
8791:
8787:
8784:
8779:
8776:
8770:
8766:
8763:
8760:
8757:
8753:
8749:
8744:
8741:
8738:
8735:
8732:
8728:
8721:
8717:
8714:
8711:
8708:
8702:
8677:in powers of
8664:
8663:
8652:
8649:
8646:
8643:
8640:
8637:
8631:
8627:
8624:
8621:
8618:
8612:
8605:
8602:
8596:
8590:
8587:
8582:
8579:
8573:
8568:
8565:
8560:
8556:
8552:
8547:
8543:
8539:
8534:
8530:
8526:
8521:
8517:
8513:
8510:
8479:
8478:
8467:
8462:
8457:
8448:
8445:
8442:
8439:
8436:
8432:
8428:
8425:
8418:
8415:
8412:
8409:
8405:
8399:
8395:
8389:
8385:
8381:
8375:
8367:
8364:
8361:
8358:
8355:
8351:
8347:
8344:
8337:
8334:
8331:
8328:
8325:
8321:
8317:
8314:
8307:
8301:
8296:
8292:
8288:
8283:
8278:
8274:
8269:
8264:
8257:
8254:
8249:
8246:
8240:
8236:
8231:
8228:
8221:
8218:
8213:
8208:
8202:
8199:
8196:
8193:
8190:
8186:
8182:
8179:
8175:
8167:
8163:
8160:
8157:
8154:
8148:
8141:
8138:
8132:
8126:
8123:
8118:
8115:
8109:
8104:
8101:
8096:
8092:
8088:
8083:
8079:
8075:
8070:
8066:
8062:
8057:
8053:
8049:
8046:
7967:
7966:
7951:
7943:
7939:
7933:
7929:
7925:
7920:
7916:
7912:
7907:
7903:
7899:
7894:
7890:
7884:
7878:
7874:
7869:
7865:
7861:
7855:
7851:
7846:
7843:
7838:
7833:
7830:
7827:
7823:
7819:
7816:
7814:
7812:
7807:
7800:
7797:
7794:
7791:
7784:
7780:
7774:
7770:
7766:
7763:
7760:
7757:
7754:
7751:
7748:
7743:
7738:
7734:
7730:
7725:
7720:
7716:
7712:
7706:
7703:
7697:
7694:
7686:
7683:
7677:
7672:
7669:
7663:
7660:
7654:
7648:
7645:
7642:
7639:
7636:
7633:
7630:
7627:
7622:
7619:
7613:
7608:
7605:
7603:
7601:
7596:
7592:
7588:
7583:
7579:
7575:
7570:
7566:
7562:
7557:
7553:
7549:
7546:
7543:
7542:
7525:
7518:
7508:
7507:
7496:
7489:
7484:
7480:
7476:
7471:
7467:
7463:
7460:
7457:
7454:
7451:
7448:
7445:
7442:
7437:
7434:
7424:
7418:
7414:
7410:
7404:
7400:
7393:
7388:
7384:
7380:
7375:
7371:
7367:
7364:
7361:
7358:
7353:
7348:
7344:
7340:
7335:
7331:
7327:
7324:
7314:
7304:
7300:
7294:
7290:
7287:
7284:
7279:
7275:
7271:
7266:
7262:
7258:
7253:
7249:
7245:
7240:
7236:
7232:
7229:
7215:
7214:
7203:
7197:
7194:
7189:
7186:
7183:
7180:
7174:
7170:
7162:
7158:
7152:
7148:
7142:
7139:
7135:
7129:
7124:
7121:
7118:
7114:
7095:
7094:
7083:
7073:
7068:
7064:
7060:
7055:
7051:
7047:
7044:
7041:
7038:
7035:
7031:
7025:
7022:
7008:
7007:
6988:
6985:
6980:
6975:
6969:
6964:
6958:
6955:
6950:
6945:
6941:
6937:
6932:
6928:
6924:
6921:
6915:
6910:
6907:
6903:
6896:
6891:
6888:
6885:
6881:
6877:
6871:
6861:
6857:
6851:
6847:
6844:
6842:
6840:
6835:
6830:
6826:
6821:
6817:
6813:
6808:
6804:
6800:
6795:
6791:
6787:
6779:
6775:
6771:
6766:
6762:
6754:
6750:
6746:
6743:
6740:
6733:
6727:
6724:
6717:
6712:
6708:
6701:
6698:
6694:
6688:
6683:
6680:
6675:
6671:
6667:
6664:
6658:
6653:
6650:
6642:
6637:
6634:
6631:
6627:
6620:
6610:
6606:
6600:
6596:
6593:
6590:
6588:
6586:
6581:
6577:
6573:
6568:
6564:
6560:
6555:
6551:
6547:
6542:
6538:
6534:
6531:
6528:
6527:
6513:
6512:
6501:
6497:
6493:
6488:
6484:
6480:
6475:
6471:
6467:
6462:
6458:
6454:
6446:
6442:
6438:
6433:
6429:
6421:
6417:
6413:
6410:
6407:
6400:
6394:
6391:
6384:
6379:
6375:
6368:
6365:
6357:
6352:
6349:
6346:
6342:
6338:
6329:
6325:
6322:
6310:, which gives
6304:
6303:
6288:
6281:
6275:
6270:
6266:
6262:
6257:
6253:
6249:
6246:
6243:
6240:
6233:
6229:
6223:
6219:
6215:
6212:
6209:
6204:
6200:
6196:
6191:
6187:
6183:
6180:
6177:
6174:
6171:
6166:
6161:
6157:
6153:
6148:
6143:
6139:
6135:
6129:
6125:
6122:
6117:
6114:
6109:
6106:
6104:
6102:
6099:
6096:
6091:
6085:
6081:
6075:
6071:
6067:
6061:
6058:
6052:
6047:
6040:
6037:
6030:
6024:
6021:
6014:
6006:
6002:
5994:
5990:
5985:
5981:
5978:
5975:
5969:
5960:
5956:
5948:
5944:
5939:
5935:
5932:
5930:
5922:
5918:
5917:
5903:
5902:
5891:
5885:
5880:
5876:
5872:
5867:
5863:
5859:
5856:
5853:
5850:
5845:
5840:
5836:
5832:
5829:
5826:
5823:
5820:
5817:
5809:
5805:
5801:
5795:
5790:
5786:
5782:
5777:
5773:
5769:
5766:
5763:
5760:
5755:
5752:
5749:
5744:
5740:
5736:
5733:
5730:
5727:
5719:
5715:
5711:
5708:
5705:
5702:
5693:
5672:
5646:
5620:
5619:
5608:
5603:
5599:
5593:
5589:
5585:
5579:
5576:
5570:
5565:
5558:
5555:
5548:
5542:
5539:
5533:
5528:
5501:
5498:
5497:
5496:
5485:
5480:
5476:
5470:
5465:
5461:
5455:
5452:
5447:
5443:
5436:
5433:
5429:
5424:
5404:
5399:
5398:
5387:
5384:
5379:
5371:
5368:
5364:
5358:
5355:
5352:
5349:
5346:
5343:
5338:
5335:
5332:
5329:
5326:
5323:
5319:
5315:
5312:
5309:
5304:
5300:
5296:
5293:
5290:
5287:
5283:
5280:
5277:
5274:
5271:
5268:
5265:
5262:
5259:
5256:
5253:
5248:
5244:
5240:
5237:
5234:
5231:
5228:
5223:
5219:
5200:
5199:
5188:
5185:
5180:
5175:
5171:
5165:
5162:
5159:
5156:
5153:
5150:
5147:
5144:
5138:
5135:
5131:
5115:
5114:
5103:
5096:
5093:
5086:
5082:
5078:
5075:
5072:
5069:
5066:
5060:
5056:
5053:
5050:
5047:
5044:
5041:
5038:
5035:
5032:
5005:
5004:
4993:
4990:
4985:
4980:
4976:
4970:
4967:
4964:
4961:
4958:
4955:
4952:
4946:
4943:
4939:
4923:
4922:
4911:
4908:
4905:
4902:
4898:
4895:
4892:
4889:
4886:
4883:
4880:
4877:
4874:
4848:
4847:
4836:
4828:
4825:
4818:
4814:
4810:
4807:
4804:
4801:
4795:
4791:
4787:
4784:
4781:
4778:
4775:
4772:
4769:
4766:
4763:
4745:
4744:
4733:
4726:
4720:
4716:
4712:
4706:
4702:
4698:
4695:
4692:
4689:
4686:
4683:
4677:
4674:
4657:
4656:
4645:
4638:
4632:
4628:
4624:
4618:
4614:
4610:
4607:
4604:
4601:
4596:
4589:
4586:
4564:
4563:
4552:
4547:
4543:
4539:
4535:
4531:
4528:
4523:
4516:
4513:
4506:
4503:
4500:
4497:
4494:
4491:
4485:
4482:
4443:
4442:
4431:
4426:
4422:
4418:
4415:
4412:
4407:
4403:
4399:
4394:
4390:
4386:
4383:
4380:
4377:
4374:
4371:
4368:
4365:
4362:
4346:
4335:with variance
4299:
4298:
4287:
4284:
4279:
4272:
4268:
4263:
4258:
4253:
4249:
4246:
4243:
4240:
4237:
4234:
4231:
4228:
4225:
4222:
4219:
4213:
4205:
4202:
4196:
4192:
4188:
4185:
4180:
4176:
4170:
4167:
4164:
4161:
4158:
4155:
4150:
4147:
4144:
4141:
4138:
4135:
4131:
4127:
4124:
4121:
4118:
4115:
4112:
4109:
4106:
4103:
4089:
4088:
4077:
4074:
4069:
4062:
4058:
4054:
4047:
4042:
4035:
4032:
4021:
4016:
4012:
4008:
4004:
4000:
3997:
3992:
3989:
3986:
3983:
3980:
3977:
3972:
3969:
3966:
3963:
3960:
3957:
3953:
3949:
3946:
3943:
3940:
3937:
3934:
3931:
3928:
3925:
3904:
3903:
3892:
3889:
3885:
3878:
3873:
3866:
3863:
3854:
3851:
3848:
3811:
3808:
3786:
3785:
3774:
3771:
3768:
3765:
3759:
3755:
3748:
3745:
3742:
3739:
3735:
3731:
3728:
3724:
3721:
3718:
3715:
3712:
3706:
3702:
3695:
3691:
3687:
3684:
3661:
3641:
3621:
3618:
3615:
3612:
3609:
3605:
3583:
3561:
3555:
3542:
3541:
3529:
3526:
3523:
3520:
3516:
3512:
3507:
3503:
3497:
3491:
3487:
3480:
3476:
3472:
3469:
3466:
3462:
3456:
3450:
3443:
3440:
3437:
3434:
3431:
3427:
3422:
3416:
3413:
3408:
3405:
3402:
3399:
3396:
3393:
3390:
3375:
3372:
3350:
3342:
3326:
3318:
3312:
3303:
3296:
3285:
3284:
3272:
3267:
3263:
3260:
3255:
3249:
3246:
3243:
3239:
3235:
3230:
3226:
3219:
3214:
3207:
3204:
3196:
3192:
3187:
3184:
3181:
3176:
3173:
3170:
3166:
3162:
3157:
3154:
3148:
3144:
3141:
3127:
3126:
3115:
3112:
3109:
3106:
3103:
3098:
3093:
3085:
3082:
3075:
3071:
3066:
3063:
3058:
3055:
3052:
3049:
3046:
3043:
3037:
3034:
3028:
3025:
3022:
3019:
2992:
2985:
2978:
2971:
2961:
2954:
2947:
2940:
2908:
2862:
2861:
2850:
2845:
2841:
2837:
2833:
2827:
2823:
2819:
2814:
2810:
2807:
2801:
2796:
2793:
2790:
2787:
2784:
2781:
2778:
2775:
2772:
2767:
2762:
2755:
2751:
2743:
2739:
2734:
2728:
2725:
2719:
2715:
2712:
2707:
2704:
2699:
2696:
2692:
2688:
2683:
2680:
2675:
2672:
2668:
2646:
2645:
2634:
2628:
2625:
2622:
2615:
2611:
2607:
2602:
2598:
2591:
2588:
2585:
2582:
2579:
2549:
2541:
2528:
2527:
2514:
2510:
2506:
2501:
2498:
2495:
2491:
2487:
2482:
2478:
2474:
2469:
2466:
2463:
2459:
2455:
2452:
2449:
2444:
2440:
2436:
2431:
2427:
2423:
2418:
2414:
2397:
2390:
2383:
2376:
2356:Main article:
2353:
2350:
2348:
2345:
2303:complex number
2274:
2273:
2248:
2241:
2217:
2214:
2193:
2171:
2146:
2131:
2124:
2099:
2082:
2073:
2066:
2059:
2050:
2028:
2025:
1980:
1979:
1968:
1963:
1960:
1957:
1953:
1890:
1881:
1880:
1869:
1864:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1839:
1822:
1817:
1816:
1805:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1776:
1772:
1755:
1706:
1705:
1694:
1689:
1686:
1683:
1680:
1677:
1674:
1671:
1667:
1648:
1647:
1636:
1629:
1624:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1598:
1593:
1590:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1558:
1553:
1548:
1543:
1539:
1485:
1484:
1473:
1467:
1464:
1459:
1456:
1450:
1444:
1441:
1424:
1423:
1412:
1409:
1406:
1403:
1397:
1394:
1388:
1385:
1382:
1379:
1376:
1335:
1332:
1318:
1317:
1306:
1297:
1294:
1288:
1283:
1280:
1274:
1271:
1257:
1256:
1245:
1242:
1239:
1234:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1194:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1157:discretization
1148:) is that the
1130:Lorentz scalar
1096:imaginary unit
1086:
1083:
1035:Norbert Wiener
889:
888:
886:
885:
878:
871:
863:
860:
859:
855:
854:
849:
844:
839:
834:
829:
824:
819:
814:
809:
804:
799:
794:
789:
784:
779:
774:
769:
764:
759:
754:
749:
744:
739:
734:
729:
724:
719:
714:
709:
704:
699:
694:
689:
684:
679:
674:
669:
664:
659:
654:
649:
644:
639:
634:
629:
623:
622:
619:
618:
615:
614:
611:
610:
605:
600:
595:
593:Density matrix
590:
585:
580:
575:
570:
565:
559:
556:
555:
552:
551:
547:
546:
541:
536:
531:
526:
521:
516:
515:
514:
513:
512:
497:
492:
487:
482:
477:
471:
470:
465:
464:
461:
460:
456:
455:
450:
445:
440:
435:
429:
428:
425:
424:
421:
420:
416:
415:
410:
405:
400:
395:
390:
384:
383:
382:
376:
373:
372:
369:
368:
364:
363:
358:
353:
347:
346:
345:
344:
343:
341:Delayed-choice
336:Quantum eraser
331:
330:
325:
320:
315:
310:
305:
300:
295:
290:
285:
279:
278:
275:
274:
271:
270:
266:
265:
264:
263:
253:
248:
243:
238:
233:
228:
226:Quantum number
223:
218:
213:
208:
203:
198:
192:
191:
188:
187:
184:
183:
179:
178:
173:
167:
166:
165:
160:
155:
149:
146:
145:
142:
141:
140:
139:
134:
129:
121:
120:
115:
104:
101:
97:
90:
87:
81:
78:
75:
71:
64:
61:
57:
52:
49:
38:
37:
31:
30:
15:
9:
6:
4:
3:
2:
26393:
26382:
26379:
26377:
26374:
26372:
26369:
26367:
26364:
26362:
26359:
26357:
26354:
26352:
26349:
26348:
26346:
26331:
26327:
26323:
26321:
26316:
26312:
26310:
26306:
26302:
26300:
26297:
26295:
26294:
26293:Tuva or Bust!
26290:
26288:
26287:
26283:
26281:
26278:
26276:
26273:
26272:
26270:
26266:
26256:
26253:
26247:
26244:
26243:
26241:
26237:
26227:
26226:
26222:
26216:
26215:
26211:
26205:
26204:
26200:
26194:
26193:
26189:
26183:
26182:
26178:
26172:
26171:
26167:
26161:
26160:
26156:
26150:
26149:
26145:
26139:
26138:
26134:
26128:
26124:
26123:
26121:
26117:
26111:
26108:
26106:
26103:
26101:
26098:
26096:
26093:
26091:
26088:
26086:
26083:
26081:
26078:
26076:
26073:
26071:
26068:
26066:
26063:
26061:
26058:
26056:
26053:
26051:
26048:
26046:
26043:
26041:
26038:
26037:
26035:
26031:
26027:
26020:
26015:
26013:
26008:
26006:
26001:
26000:
25997:
25985:
25977:
25976:
25973:
25967:
25964:
25962:
25959:
25957:
25954:
25950:
25947:
25946:
25945:
25942:
25941:
25939:
25935:
25929:
25926:
25924:
25921:
25917:
25914:
25913:
25912:
25909:
25907:
25904:
25902:
25899:
25897:
25894:
25893:
25891:
25887:
25881:
25878:
25876:
25873:
25871:
25868:
25866:
25863:
25861:
25858:
25856:
25853:
25851:
25848:
25846:
25843:
25841:
25838:
25836:
25833:
25831:
25828:
25826:
25823:
25821:
25820:Quantum logic
25818:
25816:
25813:
25811:
25808:
25806:
25803:
25801:
25798:
25796:
25793:
25791:
25788:
25786:
25783:
25779:
25776:
25775:
25774:
25771:
25769:
25766:
25764:
25761:
25759:
25756:
25752:
25749:
25748:
25747:
25744:
25742:
25739:
25737:
25734:
25732:
25729:
25728:
25726:
25724:
25720:
25714:
25711:
25709:
25706:
25704:
25701:
25699:
25696:
25694:
25691:
25689:
25686:
25684:
25681:
25679:
25676:
25674:
25673:Quantum chaos
25671:
25669:
25666:
25664:
25661:
25660:
25658:
25656:
25652:
25646:
25643:
25641:
25640:Stern–Gerlach
25638:
25636:
25633:
25631:
25628:
25626:
25623:
25621:
25618:
25616:
25613:
25611:
25608:
25606:
25603:
25601:
25598:
25596:
25593:
25592:
25590:
25586:
25580:
25577:
25575:
25574:Transactional
25572:
25570:
25567:
25565:
25564:Quantum logic
25562:
25560:
25557:
25555:
25552:
25546:
25543:
25542:
25541:
25538:
25537:
25536:
25533:
25531:
25528:
25526:
25523:
25521:
25518:
25516:
25513:
25511:
25508:
25507:
25505:
25503:
25499:
25493:
25490:
25488:
25485:
25483:
25480:
25478:
25475:
25473:
25470:
25469:
25467:
25463:
25457:
25454:
25452:
25449:
25447:
25444:
25442:
25439:
25437:
25434:
25432:
25429:
25427:
25424:
25423:
25421:
25417:
25409:
25406:
25404:
25401:
25400:
25399:
25398:Wave function
25396:
25394:
25391:
25389:
25386:
25384:
25381:
25379:
25376:
25374:
25373:Superposition
25371:
25369:
25368:Quantum state
25366:
25364:
25361:
25359:
25356:
25354:
25351:
25349:
25346:
25344:
25341:
25339:
25336:
25332:
25329:
25327:
25324:
25322:
25321:Excited state
25319:
25317:
25314:
25313:
25312:
25309:
25307:
25304:
25302:
25299:
25297:
25294:
25292:
25289:
25288:
25286:
25282:
25276:
25273:
25271:
25268:
25266:
25263:
25259:
25256:
25255:
25254:
25251:
25249:
25246:
25245:
25243:
25239:
25235:
25228:
25223:
25221:
25216:
25214:
25209:
25208:
25205:
25198:
25195:
25192:
25189:
25187:
25184:
25182:
25179:
25178:
25167:
25161:
25157:
25153:
25149:
25145:
25139:
25135:
25131:
25127:
25122:
25116:
25111:
25110:
25104:
25100:
25097:
25091:
25087:
25082:
25081:
25075:
25071:
25067:
25063:
25058:
25053:
25048:
25043:
25039:
25035:
25031:
25027:
25023:
25019:
25015:
25010:
25004:
25000:
24996:
24992:
24988:
24984:
24980:
24976:
24972:
24968:
24964:
24960:
24953:
24948:
24944:
24938:
24934:
24930:
24926:
24922:
24916:
24912:
24907:
24902:
24896:
24891:
24890:
24883:
24879:
24873:
24869:
24864:
24860:
24855:
24851:
24845:
24841:
24836:
24831:
24826:
24821:
24817:
24811:
24807:
24806:
24801:
24797:
24793:
24787:
24783:
24779:
24775:
24771:
24765:
24761:
24756:
24752:
24746:
24742:
24737:
24733:
24727:
24723:
24718:
24714:
24710:
24706:
24700:
24696:
24692:
24688:
24684:
24680:
24675:
24670:
24665:
24660:
24656:
24650:
24646:
24641:
24637:
24631:
24626:
24625:
24618:
24614:
24608:
24604:
24600:
24596:
24592:
24586:
24582:
24578:
24574:
24569:
24562:
24556:
24551:
24550:
24543:
24539:
24535:
24531:
24527:
24523:
24519:
24512:
24507:
24500:
24494:
24490:
24486:
24482:
24478:
24474:
24473:
24468:
24464:
24457:
24453:
24449:
24439:
24435:
24431:
24427:
24423:
24419:
24412:
24408:
24404:
24400:
24396:
24392:
24385:
24381:
24377:
24373:
24369:
24365:
24361:
24357:
24353:
24349:
24345:
24341:
24337:
24333:
24329:
24325:
24319:
24315:
24311:
24306:
24302:
24298:
24294:
24290:
24286:
24282:
24277:
24272:
24268:
24264:
24260:
24256:
24252:
24248:
24244:
24240:
24236:
24232:
24228:
24224:
24220:
24216:
24214:9783540769569
24210:
24206:
24201:
24197:
24193:
24192:Ahmad, Ishfaq
24189:
24188:
24176:
24171:
24165:
24160:
24154:
24149:
24134:
24130:
24123:
24114:
24109:
24105:
24101:
24097:
24093:
24089:
24082:
24075:
24070:
24064:, Chapter 19.
24063:
24058:
24052:
24047:
24040:
24035:
24028:
24023:
24017:
24012:
24006:, Chapter 13.
24005:
24000:
23992:
23985:
23978:
23972:
23967:
23960:
23955:
23946:
23941:
23934:
23928:
23923:
23917:
23912:
23906:
23901:
23887:
23883:
23879:
23873:
23859:
23858:
23853:
23846:
23837:
23832:
23825:
23823:
23808:on 2017-08-12
23804:
23797:
23790:
23783:
23782:Weinberg 2002
23778:
23774:
23760:
23754:
23747:
23741:
23737:
23726:
23723:
23721:
23718:
23716:
23713:
23711:
23708:
23706:
23703:
23701:
23698:
23696:
23693:
23692:
23685:
23683:
23679:
23675:
23667:
23658:
23637:
23614:
23608:
23604:
23600:
23597:
23586:
23582:
23572:
23571:
23570:
23564:
23560:
23551:
23549:
23545:
23541:
23537:
23536:Hilbert space
23527:
23525:
23520:
23518:
23514:
23510:
23500:
23477:
23465:
23459:
23450:
23438:
23427:
23424:
23396:
23384:
23355:
23343:
23320:
23317:
23291:
23279:
23250:
23238:
23215:
23212:
23190:
23183:
23177:
23174:
23171:
23149:
23142:
23136:
23133:
23130:
23101:
23089:
23083:
23074:
23062:
23051:
23048:
23037:
23027:
23025:
23021:
22992:
22989:
22983:
22977:
22973:
22966:
22963:
22959:
22954:
22951:
22947:
22937:
22931:
22925:
22919:
22913:
22909:
22904:
22900:
22896:
22889:
22888:
22887:
22870:
22867:
22863:
22856:
22850:
22846:
22838:
22837:
22836:
22832:
22826:
22809:
22806:
22800:
22794:
22790:
22783:
22780:
22776:
22771:
22768:
22764:
22754:
22748:
22742:
22736:
22730:
22727:
22721:
22715:
22711:
22704:
22701:
22697:
22692:
22689:
22685:
22678:
22670:
22666:
22660:
22652:
22646:
22640:
22636:
22629:
22626:
22622:
22617:
22614:
22610:
22600:
22594:
22588:
22582:
22576:
22573:
22567:
22561:
22557:
22550:
22547:
22543:
22538:
22535:
22531:
22524:
22515:
22509:
22502:
22501:
22500:
22483:
22480:
22476:
22469:
22461:
22457:
22451:
22443:
22439:
22435:
22432:
22423:
22414:
22408:
22402:
22393:
22384:
22378:
22375:
22369:
22366:
22357:
22348:
22342:
22332:
22331:
22330:
22313:
22310:
22304:
22295:
22286:
22280:
22276:
22266:
22259:
22258:
22257:
22251:
22247:
22241:
22237:
22209:
22203:
22197:
22178:
22173:
22153:
22147:
22139:
22135:
22131:
22125:
22117:
22113:
22105:
22104:
22103:
22082:
22074:
22070:
22064:
22056:
22050:
22041:
22035:
22028:
22027:
22026:
22002:
21996:
21990:
21984:
21981:
21978:
21967:
21960:
21956:
21947:
21941:
21932:
21926:
21919:
21918:
21917:
21897:
21891:
21887:
21882:
21878:
21874:
21871:
21868:
21861:
21860:
21859:
21852:
21835:
21832:
21828:
21822:
21818:
21814:
21808:
21804:
21798:
21791:
21787:
21783:
21779:
21776:
21767:
21761:
21751:
21750:
21749:
21732:
21729:
21726:
21720:
21713:
21707:
21704:
21700:
21696:
21692:
21688:
21684:
21674:
21667:
21666:
21665:
21662:
21660:
21659:
21635:
21632:
21626:
21617:
21611:
21607:
21597:
21590:
21589:
21588:
21585:
21583:
21582:supersymmetry
21579:
21575:
21571:
21561:
21555:
21523:
21515:
21511:
21505:
21497:
21488:
21472:
21465:
21464:
21463:
21461:
21457:
21452:
21449:
21446:
21440:
21430:
21428:
21424:
21414:
21412:
21408:
21407:Taylor series
21389:
21386:
21383:
21380:
21377:
21374:
21364:
21343:
21342:
21341:
21334:
21332:
21327:
21322:
21317:
21304:
21298:
21278:
21275:
21272:
21267:
21263:
21259:
21256:
21247:
21244:
21236:
21233:
21217:
21216:
21215:
21198:
21195:
21192:
21186:
21180:
21174:
21168:
21165:
21159:
21153:
21149:
21142:
21139:
21135:
21130:
21127:
21123:
21113:
21107:
21104:
21092:
21082:
21081:
21080:
21063:
21060:
21055:
21050:
21043:
21037:
21034:
21028:
21016:
21010:
21007:
20995:
20988:
20979:
20978:
20977:
20976:
20957:
20951:
20945:
20934:
20930:
20923:
20920:
20916:
20904:
20900:
20894:
20890:
20878:
20874:
20863:
20852:
20848:
20841:
20838:
20834:
20822:
20818:
20812:
20808:
20796:
20792:
20779:
20771:
20768:
20762:
20756:
20750:
20746:
20739:
20736:
20732:
20727:
20724:
20720:
20716:
20709:
20708:
20707:
20682:
20674:
20670:
20663:
20653:
20649:
20643:
20639:
20627:
20623:
20612:
20604:
20600:
20593:
20583:
20579:
20573:
20569:
20557:
20553:
20542:
20536:
20530:
20523:
20522:
20521:
20510:
20502:
20489:
20487:
20479:
20472:
20465:
20458:
20454:
20447:
20446:time ordering
20443:
20439:
20434:
20431:
20408:
20399:
20392:
20389:
20386:
20380:
20374:
20360:
20356:
20352:
20348:
20338:
20330:
20323:
20320:
20317:
20311:
20305:
20291:
20287:
20283:
20276:
20270:
20267:
20257:
20251:
20246:
20238:
20228:
20227:
20226:
20209:
20204:
20199:
20191:
20187:
20182:
20178:
20171:
20167:
20162:
20157:
20149:
20143:
20138:
20134:
20130:
20124:
20114:
20110:
20106:
20101:
20097:
20093:
20089:
20081:
20080:
20079:
20062:
20057:
20052:
20043:
20039:
20032:
20029:
20021:
20017:
20010:
20006:
19997:
19991:
19985:
19981:
19977:
19971:
19957:
19953:
19946:
19943:
19940:
19932:
19928:
19921:
19918:
19913:
19908:
19904:
19893:
19892:
19891:
19874:
19868:
19861:
19858:
19855:
19849:
19843:
19829:
19825:
19821:
19817:
19807:
19804:
19798:
19792:
19785:
19784:
19783:
19781:
19774:
19770:
19766:
19762:
19758:
19749:
19745:
19735:
19716:
19712:
19706:
19703:
19691:
19687:
19683:
19680:
19677:
19673:
19668:
19665:
19661:
19657:
19649:
19648:
19647:
19645:
19621:
19614:
19611:
19603:
19590:
19581:
19575:
19571:
19567:
19564:
19561:
19557:
19551:
19548:
19540:
19534:
19531:
19525:
19517:
19516:
19515:
19513:
19505:
19501:
19473:
19467:
19464:
19460:
19452:
19451:
19450:
19433:
19425:
19412:
19408:
19400:
19399:
19398:
19397:
19396:Wick rotation
19393:
19389:
19384:
19377:
19372:
19370:
19366:
19347:
19344:
19338:
19335:
19327:
19314:
19304:
19303:
19302:
19300:
19295:
19290:
19280:
19278:
19270:
19250:
19242:
19237:
19230:
19224:
19216:
19203:
19199:
19192:
19180:
19176:
19166:
19162:
19155:
19152:
19149:
19145:
19137:
19132:
19125:
19119:
19111:
19098:
19094:
19087:
19075:
19071:
19061:
19057:
19050:
19047:
19044:
19041:
19038:
19034:
19027:
19021:
19018:
19015:
19009:
18999:
18998:
18997:
18996:, this means
18989:
18979:
18975:
18966:
18962:
18954:
18950:
18945:, written as
18944:
18940:
18922:
18918:
18894:
18883:
18870:
18866:
18862:
18852:
18842:
18829:
18825:
18818:
18812:
18809:
18799:
18793:
18787:
18777:
18776:
18775:
18770:
18759:
18755:
18752:
18745:
18741:
18737:
18727:
18725:
18724:Wick rotation
18721:
18713:
18709:
18705:
18701:
18696:
18694:
18689:
18687:
18683:
18677:
18673:
18668:
18662:
18658:
18652:
18647:
18643:
18639:
18629:
18625:
18623:
18617:
18614:
18610:
18586:
18580:
18577:
18574:
18569:
18565:
18561:
18556:
18552:
18547:
18542:
18536:
18530:
18523:
18522:
18521:
18515:
18495:
18489:
18486:
18483:
18476:
18472:
18468:
18463:
18453:
18444:
18439:
18435:
18430:
18425:
18419:
18416:
18413:
18406:
18402:
18398:
18393:
18383:
18374:
18369:
18365:
18360:
18355:
18349:
18343:
18336:
18335:
18334:
18328:
18321:
18314:
18308:
18304:
18302:
18283:
18274:
18271:
18265:
18255:
18246:
18240:
18237:
18232:
18228:
18220:
18215:
18209:
18197:
18193:
18190:
18183:
18182:
18181:
18178:
18171:
18161:
18154:
18134:
18124:
18120:
18116:
18111:
18101:
18092:
18087:
18083:
18078:
18073:
18063:
18059:
18055:
18050:
18040:
18031:
18026:
18022:
18017:
18012:
18006:
18000:
17995:
17991:
17987:
17980:
17979:
17978:
17976:
17957:
17949:
17945:
17941:
17936:
17926:
17919:
17914:
17909:
17905:
17900:
17895:
17889:
17883:
17876:
17875:
17874:
17871:
17861:
17841:
17835:
17832:
17827:
17823:
17818:
17813:
17805:
17799:
17791:
17786:
17782:
17773:
17769:
17758:
17754:
17750:
17744:
17738:
17731:
17730:
17729:
17713:
17709:
17703:
17684:
17676:
17665:
17662:
17649:
17641:
17638:
17635:
17626:
17622:
17611:
17607:
17603:
17597:
17594:
17591:
17585:
17578:
17577:
17576:
17567:
17547:
17539:
17524:
17513:
17510:
17507:
17504:
17498:
17488:
17484:
17480:
17474:
17471:
17468:
17462:
17455:
17454:
17453:
17428:
17413:
17405:
17402:
17399:
17390:
17386:
17375:
17372:
17368:
17364:
17353:
17350:
17347:
17344:
17338:
17331:
17330:
17329:
17323:
17295:
17290:
17287:
17283:
17277:
17274:
17260:
17255:
17252:
17246:
17240:
17236:
17225:
17222:
17218:
17214:
17203:
17200:
17197:
17194:
17188:
17181:
17180:
17179:
17175:
17171:
17167:
17163:
17142:
17137:
17134:
17130:
17127:
17120:
17113:
17110:
17092:
17088:
17084:
17080:
17074:
17071:
17057:
17052:
17049:
17043:
17037:
17033:
17029:
17018:
17015:
17012:
17009:
17003:
16996:
16995:
16994:
16990:
16961:
16955:
16952:
16949:
16943:
16940:
16933:
16923:
16916:
16910:
16907:
16904:
16899:
16896:
16890:
16884:
16881:
16878:
16866:
16858:
16857:
16856:
16853:
16841:
16822:
16801:
16795:
16792:
16789:
16783:
16780:
16774:
16764:
16755:
16752:
16747:
16744:
16738:
16732:
16729:
16726:
16714:
16706:
16705:
16704:
16700:
16696:
16692:
16668:
16661:
16658:
16651:
16647:
16643:
16640:
16634:
16630:
16624:
16621:
16618:
16612:
16605:
16604:
16603:
16599:
16595:
16574:
16567:
16564:
16557:
16549:
16546:
16543:
16537:
16534:
16528:
16524:
16518:
16515:
16512:
16509:
16506:
16500:
16493:
16492:
16491:
16470:
16464:
16461:
16455:
16452:
16447:
16443:
16439:
16434:
16430:
16422:
16421:
16420:
16418:
16412:
16408:
16400:
16396:
16392:
16372:
16369:
16366:
16357:
16351:
16348:
16344:
16334:
16328:
16320:
16316:
16310:
16307:
16301:
16295:
16291:
16287:
16284:
16281:
16274:
16271:
16268:
16265:
16262:
16256:
16250:
16242:
16238:
16232:
16228:
16224:
16218:
16210:
16206:
16198:
16197:
16196:
16192:
16185:
16176:
16157:
16154:
16151:
16142:
16136:
16133:
16129:
16123:
16120:
16114:
16108:
16103:
16100:
16094:
16088:
16084:
16080:
16074:
16071:
16068:
16065:
16062:
16059:
16056:
16050:
16044:
16041:
16038:
16035:
16032:
16026:
16019:
16018:
16017:
16001:
15992:
15990:
15984:
15982:
15978:
15972:
15968:
15966:
15960:
15958:
15954:
15950:
15920:
15917:
15914:
15906:
15902:
15898:
15895:
15886:
15880:
15875:
15867:
15861:
15855:
15845:
15844:
15843:
15832:
15827:
15825:
15820:
15818:
15813:
15812:
15810:
15809:
15803:
15798:
15795:
15793:
15790:
15788:
15785:
15783:
15780:
15778:
15775:
15773:
15772:Zamolodchikov
15770:
15768:
15767:Zamolodchikov
15765:
15763:
15760:
15758:
15755:
15753:
15750:
15748:
15745:
15743:
15740:
15738:
15735:
15733:
15730:
15728:
15725:
15723:
15720:
15718:
15715:
15713:
15710:
15708:
15705:
15703:
15700:
15698:
15695:
15693:
15690:
15688:
15685:
15683:
15680:
15678:
15675:
15673:
15670:
15668:
15665:
15663:
15660:
15658:
15655:
15653:
15650:
15648:
15645:
15643:
15640:
15638:
15635:
15633:
15630:
15628:
15625:
15623:
15620:
15618:
15615:
15613:
15610:
15608:
15605:
15603:
15600:
15598:
15595:
15593:
15590:
15588:
15585:
15583:
15580:
15578:
15575:
15573:
15570:
15568:
15565:
15563:
15560:
15558:
15555:
15553:
15550:
15548:
15545:
15543:
15540:
15538:
15535:
15533:
15530:
15528:
15525:
15523:
15520:
15518:
15515:
15513:
15510:
15508:
15505:
15503:
15500:
15498:
15495:
15493:
15490:
15488:
15485:
15483:
15480:
15478:
15475:
15473:
15470:
15468:
15465:
15463:
15460:
15458:
15455:
15453:
15450:
15448:
15445:
15443:
15440:
15438:
15435:
15433:
15430:
15428:
15425:
15423:
15420:
15418:
15415:
15413:
15410:
15408:
15405:
15403:
15400:
15398:
15395:
15393:
15390:
15388:
15385:
15383:
15380:
15378:
15375:
15373:
15370:
15368:
15365:
15363:
15360:
15358:
15355:
15353:
15350:
15348:
15345:
15343:
15340:
15338:
15335:
15333:
15330:
15328:
15325:
15323:
15320:
15318:
15315:
15313:
15310:
15308:
15305:
15303:
15300:
15298:
15295:
15293:
15290:
15288:
15285:
15283:
15280:
15278:
15275:
15273:
15270:
15268:
15265:
15263:
15260:
15258:
15255:
15253:
15250:
15248:
15245:
15243:
15240:
15238:
15235:
15233:
15230:
15228:
15225:
15223:
15220:
15218:
15215:
15213:
15210:
15208:
15205:
15203:
15200:
15198:
15195:
15193:
15190:
15188:
15185:
15183:
15180:
15178:
15175:
15173:
15170:
15168:
15165:
15163:
15160:
15158:
15155:
15153:
15150:
15148:
15145:
15143:
15140:
15138:
15135:
15133:
15130:
15128:
15125:
15123:
15120:
15118:
15115:
15113:
15110:
15109:
15102:
15101:
15094:
15091:
15089:
15086:
15084:
15081:
15079:
15078:Supersymmetry
15076:
15074:
15073:String theory
15071:
15070:
15064:
15063:
15056:
15053:
15051:
15048:
15046:
15043:
15041:
15038:
15037:
15034:
15029:
15028:
15021:
15018:
15016:
15013:
15011:
15008:
15006:
15003:
15001:
14998:
14997:
14991:
14990:
14983:
14980:
14978:
14975:
14973:
14970:
14968:
14965:
14963:
14960:
14958:
14955:
14953:
14950:
14948:
14945:
14943:
14940:
14938:
14935:
14933:
14930:
14928:
14925:
14923:
14920:
14918:
14915:
14913:
14910:
14908:
14905:
14903:
14900:
14898:
14895:
14894:
14888:
14887:
14880:
14877:
14875:
14872:
14870:
14867:
14865:
14862:
14860:
14857:
14855:
14852:
14850:
14847:
14845:
14842:
14840:
14837:
14835:
14832:
14830:
14827:
14826:
14823:
14818:
14817:
14810:
14807:
14805:
14802:
14800:
14797:
14795:
14792:
14790:
14787:
14785:
14782:
14780:
14777:
14775:
14772:
14770:
14767:
14766:
14760:
14759:
14756:
14753:
14752:
14748:
14743:
14739:
14738:
14735:
14732:
14731:
14723:
14721:
14717:
14713:
14693:
14688:
14685:
14682:
14678:
14674:
14670:
14667:
14664:
14661:
14654:
14653:
14652:
14644:
14617:
14614:
14611:
14595:
14592:
14589:
14583:
14579:
14575:
14569:
14566:
14563:
14551:
14550:
14549:
14545:
14543:
14534:
14526:
14520:
14507:
14492:
14488:
14483:
14479:
14475:
14474:Wick rotation
14471:
14466:
14449:
14389:
14382:
14377:
14358:
14355:
14345:
14336:
14325:
14319:
14305:
14298:
14294:
14288:
14284:
14280:
14215:
14209:
14205:
14202:
14199:
14192:
14191:
14190:
14182:
14178:
14158:
14128:
14125:
14122:
14116:
14107:
14101:
14098:
14092:
14089:
14086:
14080:
14074:
14062:
14055:
14054:
14053:
14036:
14033:
14030:
14024:
13987:
13983:
13979:
13969:
13953:
13949:
13938:
13934:
13931:
13921:
13907:
13904:
13901:
13897:
13893:
13890:
13884:
13881:
13878:
13872:
13865:
13864:
13863:
13861:
13856:
13842:
13839:
13833:
13781:
13777:
13756:
13748:
13744:
13726:
13722:
13713:
13686:
13647:
13644:
13639:
13626:
13610:
13605:
13602:
13597:
13593:
13578:
13573:
13570:
13564:
13552:
13546:
13543:
13534:
13533:
13532:
13529:
13527:
13507:
13504:
13499:
13489:
13475:
13472:
13467:
13454:
13445:
13427:
13424:
13418:
13414:
13411:
13402:
13391:
13348:
13340:
13339:
13338:
13292:
13247:
13231:
13227:
13217:
13207:
13196:
13153:
13149:
13145:
13139:
13136:
13130:
13118:
13112:
13109:
13104:
13098:
13095:
13092:
13086:
13079:
13078:
13077:
13075:
13074:Wick rotation
13053:
13043:
13037:
13034:
13030:
13003:
12993:
12987:
12984:
12981:
12977:
12956:
12953:
12950:
12930:
12916:
12913:
12912:Wick rotation
12880:
12874:
12866:
12860:
12853:
12840:
12830:
12824:
12816:
12810:
12803:
12797:
12791:
12778:
12772:
12766:
12756:
12755:
12754:
12728:
12719:
12716:
12710:
12704:
12698:
12692:
12689:
12686:
12683:
12675:
12669:
12657:
12653:
12646:
12638:
12634:
12622:
12618:
12611:
12603:
12599:
12592:
12575:
12571:
12567:
12561:
12555:
12548:
12544:
12540:
12534:
12528:
12524:
12520:
12512:
12508:
12492:
12488:
12475:
12464:
12459:
12455:
12442:
12431:
12427:
12416:
12412:
12408:
12403:
12399:
12386:
12375:
12370:
12366:
12353:
12342:
12338:
12327:
12323:
12319:
12316:
12304:
12293:
12288:
12284:
12273:
12269:
12258:
12257:
12256:
12230:
12221:
12218:
12212:
12206:
12200:
12194:
12191:
12188:
12185:
12177:
12171:
12160:
12157:
12150:
12144:
12138:
12121:
12117:
12113:
12107:
12101:
12094:
12090:
12086:
12080:
12074:
12070:
12062:
12061:
12060:
12041:
12037:
12022:
12019:
12006:
11995:
11990:
11986:
11973:
11964:
11955:
11952:
11948:
11945:
11933:
11922:
11917:
11913:
11902:
11898:
11880:
11877:
11866:
11843:
11840:
11837:
11832:
11825:
11819:
11816:
11811:
11804:
11801:
11792:
11785:
11782:
11773:
11770:
11766:
11760:
11757:
11751:
11747:
11744:
11741:
11734:
11733:
11732:
11729:
11712:
11709:
11693:
11687:
11684:
11680:
11673:
11667:
11664:
11657:
11656:
11655:
11647:
11645:
11641:
11637:
11613:
11610:
11604:
11601:
11598:
11588:
11587:
11586:
11569:
11566:
11562:
11552:
11549:
11538:
11529:
11526:
11522:
11514:
11513:
11512:
11509:
11507:
11503:
11484:
11481:
11478:
11472:
11469:
11464:
11461:
11455:
11449:
11446:
11441:
11438:
11432:
11429:
11420:
11417:
11411:
11408:
11398:
11397:
11396:
11394:
11389:
11359:
11353:
11345:
11342:
11336:
11330:
11324:
11310:
11309:
11308:
11304:
11300:
11278:
11272:
11257:
11251:
11248:
11242:
11239:
11236:
11230:
11217:
11211:
11205:
11198:
11197:
11196:
11177:
11174:
11169:
11164:
11158:
11143:
11137:
11134:
11128:
11125:
11122:
11116:
11103:
11097:
11091:
11085:
11075:
11074:
11073:
11050:
11043:
11037:
11034:
11028:
11025:
11022:
11016:
11007:
11004:
11001:
10995:
10992:
10986:
10983:
10978:
10975:
10969:
10966:
10960:
10950:
10934:
10927:
10921:
10918:
10912:
10909:
10906:
10900:
10891:
10885:
10882:
10876:
10873:
10868:
10865:
10859:
10856:
10850:
10840:
10839:
10838:
10833:The quantity
10831:
10812:
10807:
10801:
10795:
10792:
10786:
10783:
10780:
10774:
10767:
10766:
10765:
10760:
10735:
10728:
10722:
10719:
10713:
10710:
10707:
10701:
10695:
10689:
10686:
10681:
10678:
10668:
10667:
10666:
10662:
10658:
10654:The quantity
10638:
10635:
10629:
10624:
10618:
10615:
10610:
10607:
10601:
10596:
10593:
10590:
10583:
10582:
10581:
10574:
10558:
10548:
10534:
10523:
10506:
10503:
10499:
10493:
10489:
10484:
10477:
10468:
10465:
10460:
10457:
10450:
10444:
10440:
10435:
10427:
10426:
10425:
10421:
10417:
10396:
10393:
10387:
10384:
10380:
10375:
10371:
10368:
10361:
10355:
10351:
10344:
10336:
10327:
10318:
10315:
10304:
10292:
10289:
10285:
10279:
10275:
10271:
10267:
10262:
10259:
10253:
10247:
10243:
10236:
10228:
10224:
10220:
10213:
10212:
10211:
10189:
10186:
10180:
10177:
10173:
10168:
10164:
10161:
10154:
10151:
10139:
10136:
10125:
10116:
10113:
10107:
10099:
10090:
10086:
10080:
10077:
10071:
10065:
10061:
10054:
10046:
10042:
10038:
10031:
10030:
10029:
10007:
10004:
9992:
9989:
9983:
9977:
9974:
9968:
9965:
9962:
9959:
9953:
9950:
9946:
9940:
9937:
9931:
9925:
9921:
9914:
9906:
9902:
9898:
9891:
9890:
9889:
9885:
9881:
9877:
9870:
9866:
9859:
9855:
9851:
9847:
9843:
9839:
9832:
9828:
9807:
9804:
9792:
9789:
9783:
9780:
9774:
9771:
9767:
9761:
9758:
9752:
9746:
9742:
9735:
9727:
9723:
9719:
9712:
9711:
9710:
9707:
9691:
9672:
9668:
9661:
9655:
9652:
9647:
9636:
9633:
9626:
9622:
9619:
9616:
9610:
9602:
9589:
9588:
9587:
9566:
9562:
9559:
9551:
9548:
9541:
9533:
9530:
9527:
9521:
9515:
9506:
9500:
9497:
9494:
9491:
9487:
9480:
9477:
9474:
9468:
9465:
9462:
9456:
9453:
9450:
9447:
9444:
9438:
9431:
9430:
9429:
9417:
9413:
9391:
9386:
9381:
9374:
9371:
9362:
9358:
9349:
9343:
9340:
9337:
9334:
9330:
9322:
9321:
9320:
9318:
9284:
9277:
9274:
9267:
9261:
9258:
9253:
9250:
9237:
9231:
9228:
9223:
9216:
9213:
9204:
9201:
9187:
9184:
9181:
9176:
9172:
9168:
9164:
9158:
9155:
9149:
9146:
9143:
9137:
9132:
9129:
9123:
9117:
9113:
9106:
9103:
9100:
9094:
9081:
9077:
9073:
9067:
9064:
9061:
9058:
9055:
9049:
9042:
9041:
9040:
9035:
9025:
9023:
9019:
9015:
8990:
8984:
8979:
8976:
8970:
8967:
8964:
8957:
8956:
8955:
8944:
8936:
8928:
8904:
8898:
8891:
8884:
8881:
8875:
8872:
8868:
8864:
8859:
8855:
8847:
8846:
8845:
8828:
8825:
8822:
8819:
8816:
8813:
8810:
8807:
8804:
8801:
8789:
8785:
8782:
8777:
8774:
8768:
8764:
8761:
8758:
8755:
8751:
8747:
8742:
8739:
8736:
8733:
8730:
8726:
8719:
8715:
8712:
8709:
8706:
8700:
8692:
8691:
8690:
8687:
8681:
8674:
8670:
8650:
8644:
8638:
8635:
8629:
8625:
8622:
8619:
8616:
8610:
8603:
8600:
8594:
8585:
8580:
8577:
8571:
8566:
8558:
8554:
8550:
8545:
8541:
8537:
8532:
8528:
8524:
8519:
8515:
8508:
8501:
8500:
8499:
8495:
8491:
8485:
8465:
8460:
8455:
8446:
8443:
8440:
8437:
8434:
8430:
8426:
8423:
8416:
8413:
8410:
8407:
8403:
8397:
8393:
8387:
8383:
8379:
8373:
8365:
8362:
8359:
8356:
8353:
8349:
8345:
8342:
8335:
8332:
8329:
8326:
8323:
8319:
8315:
8312:
8305:
8299:
8294:
8290:
8286:
8281:
8276:
8272:
8267:
8262:
8252:
8247:
8244:
8238:
8234:
8229:
8226:
8219:
8216:
8211:
8206:
8200:
8197:
8194:
8191:
8188:
8184:
8180:
8177:
8173:
8165:
8161:
8158:
8155:
8152:
8146:
8139:
8136:
8130:
8121:
8116:
8113:
8107:
8102:
8094:
8090:
8086:
8081:
8077:
8073:
8068:
8064:
8060:
8055:
8051:
8044:
8037:
8036:
8035:
8031:
8027:
8008:
8000:
7996:
7977:
7973:
7949:
7941:
7931:
7927:
7918:
7914:
7905:
7901:
7892:
7888:
7882:
7872:
7867:
7863:
7859:
7853:
7849:
7844:
7841:
7831:
7828:
7825:
7821:
7817:
7815:
7805:
7798:
7795:
7792:
7789:
7782:
7778:
7772:
7768:
7764:
7761:
7758:
7755:
7752:
7749:
7741:
7736:
7732:
7728:
7723:
7718:
7714:
7704:
7701:
7695:
7692:
7681:
7675:
7670:
7667:
7661:
7658:
7652:
7646:
7643:
7640:
7637:
7631:
7628:
7625:
7620:
7617:
7611:
7606:
7604:
7594:
7590:
7586:
7581:
7577:
7573:
7568:
7564:
7560:
7555:
7551:
7544:
7533:
7532:
7531:
7528:
7521:
7514:
7494:
7482:
7478:
7474:
7469:
7465:
7458:
7455:
7452:
7446:
7443:
7440:
7435:
7432:
7416:
7412:
7408:
7402:
7398:
7386:
7382:
7378:
7373:
7369:
7362:
7359:
7356:
7346:
7342:
7338:
7333:
7329:
7322:
7302:
7298:
7292:
7288:
7285:
7277:
7273:
7269:
7264:
7260:
7256:
7251:
7247:
7243:
7238:
7234:
7227:
7220:
7219:
7218:
7201:
7195:
7192:
7187:
7184:
7181:
7178:
7172:
7168:
7160:
7156:
7150:
7146:
7140:
7137:
7133:
7122:
7119:
7116:
7112:
7104:
7103:
7102:
7100:
7099:sinc function
7081:
7066:
7062:
7058:
7053:
7049:
7039:
7036:
7033:
7029:
7023:
7020:
7013:
7012:
7011:
6986:
6983:
6978:
6973:
6967:
6962:
6956:
6953:
6943:
6939:
6935:
6930:
6926:
6919:
6913:
6908:
6905:
6901:
6889:
6886:
6883:
6879:
6875:
6859:
6855:
6849:
6845:
6843:
6833:
6828:
6819:
6815:
6811:
6806:
6802:
6793:
6789:
6785:
6777:
6773:
6769:
6764:
6760:
6752:
6744:
6741:
6731:
6725:
6722:
6715:
6710:
6706:
6696:
6692:
6686:
6681:
6678:
6673:
6669:
6665:
6662:
6656:
6651:
6648:
6635:
6632:
6629:
6625:
6608:
6604:
6598:
6594:
6591:
6589:
6579:
6575:
6571:
6566:
6562:
6558:
6553:
6549:
6545:
6540:
6536:
6529:
6518:
6517:
6516:
6499:
6495:
6486:
6482:
6478:
6473:
6469:
6460:
6456:
6452:
6444:
6440:
6436:
6431:
6427:
6419:
6411:
6408:
6398:
6392:
6389:
6382:
6377:
6373:
6366:
6363:
6350:
6347:
6344:
6340:
6336:
6327:
6323:
6320:
6313:
6312:
6311:
6286:
6279:
6268:
6264:
6260:
6255:
6251:
6244:
6241:
6238:
6231:
6227:
6221:
6217:
6213:
6210:
6202:
6198:
6194:
6189:
6185:
6178:
6175:
6172:
6164:
6159:
6155:
6151:
6146:
6141:
6137:
6127:
6123:
6120:
6115:
6112:
6107:
6105:
6097:
6094:
6089:
6083:
6079:
6073:
6069:
6065:
6059:
6056:
6050:
6045:
6038:
6035:
6028:
6022:
6019:
6012:
6004:
6000:
5992:
5988:
5983:
5979:
5976:
5973:
5958:
5954:
5946:
5942:
5937:
5933:
5931:
5920:
5908:
5907:
5906:
5889:
5878:
5874:
5870:
5865:
5861:
5854:
5851:
5848:
5838:
5834:
5830:
5827:
5821:
5818:
5815:
5807:
5803:
5799:
5788:
5784:
5780:
5775:
5771:
5764:
5761:
5758:
5750:
5747:
5742:
5738:
5731:
5728:
5725:
5717:
5713:
5709:
5703:
5691:
5683:
5682:
5681:
5678:
5671:
5667:
5660:
5656:
5652:
5645:
5641:
5637:
5630:
5626:
5606:
5601:
5597:
5591:
5587:
5583:
5577:
5574:
5568:
5563:
5556:
5553:
5546:
5540:
5537:
5531:
5517:
5516:
5515:
5511:
5510:Mehler kernel
5507:
5483:
5478:
5474:
5468:
5463:
5453:
5450:
5445:
5441:
5434:
5422:
5415:
5414:
5413:
5385:
5382:
5369:
5366:
5362:
5356:
5353:
5347:
5341:
5336:
5333:
5327:
5321:
5317:
5310:
5302:
5298:
5294:
5291:
5288:
5285:
5278:
5275:
5272:
5269:
5266:
5260:
5254:
5246:
5242:
5238:
5235:
5229:
5221:
5217:
5209:
5208:
5207:
5186:
5183:
5178:
5173:
5163:
5160:
5154:
5151:
5148:
5142:
5136:
5133:
5129:
5120:
5119:
5118:
5101:
5094:
5091:
5084:
5076:
5073:
5070:
5064:
5058:
5054:
5048:
5045:
5042:
5039:
5036:
5030:
5023:
5022:
5021:
5019:
5018:Wick rotation
4991:
4988:
4983:
4978:
4968:
4962:
4959:
4956:
4950:
4944:
4941:
4937:
4928:
4927:
4926:
4909:
4906:
4903:
4900:
4893:
4890:
4887:
4884:
4881:
4875:
4872:
4865:
4864:
4863:
4860:
4858:
4852:
4834:
4826:
4823:
4816:
4808:
4805:
4802:
4793:
4789:
4785:
4779:
4776:
4773:
4770:
4767:
4761:
4754:
4753:
4752:
4731:
4724:
4718:
4714:
4710:
4704:
4700:
4696:
4690:
4687:
4684:
4672:
4662:
4661:
4660:
4643:
4636:
4630:
4626:
4622:
4616:
4612:
4608:
4602:
4594:
4584:
4573:
4572:
4571:
4550:
4545:
4541:
4537:
4529:
4521:
4511:
4504:
4498:
4495:
4492:
4480:
4470:
4469:
4468:
4461:
4453:
4429:
4424:
4420:
4416:
4413:
4410:
4405:
4401:
4397:
4392:
4388:
4384:
4378:
4375:
4372:
4369:
4366:
4360:
4353:
4352:
4351:
4342:
4332:
4328:
4321:
4317:
4313:
4285:
4282:
4270:
4266:
4261:
4256:
4251:
4244:
4238:
4235:
4229:
4226:
4223:
4217:
4211:
4203:
4200:
4194:
4190:
4186:
4183:
4178:
4174:
4168:
4165:
4159:
4153:
4148:
4145:
4139:
4133:
4129:
4125:
4119:
4116:
4113:
4110:
4107:
4101:
4094:
4093:
4092:
4075:
4072:
4060:
4056:
4045:
4040:
4033:
4030:
4019:
4014:
4010:
4006:
4002:
3998:
3995:
3990:
3987:
3981:
3975:
3970:
3967:
3961:
3955:
3951:
3947:
3941:
3938:
3935:
3932:
3929:
3923:
3916:
3915:
3914:
3907:
3890:
3887:
3876:
3871:
3864:
3861:
3852:
3849:
3846:
3839:
3838:
3837:
3833:
3826:
3810:Free particle
3804:
3766:
3757:
3746:
3740:
3726:
3722:
3719:
3716:
3713:
3704:
3693:
3682:
3675:
3674:
3673:
3659:
3639:
3619:
3616:
3610:
3521:
3505:
3501:
3489:
3478:
3467:
3464:
3460:
3441:
3438:
3432:
3420:
3414:
3411:
3406:
3400:
3397:
3394:
3388:
3381:
3380:
3379:
3374:Path integral
3371:
3368:
3353:
3349:
3345:
3321:
3311:
3295:
3290:
3270:
3265:
3261:
3258:
3253:
3247:
3244:
3241:
3237:
3233:
3228:
3224:
3217:
3212:
3202:
3194:
3190:
3185:
3182:
3179:
3174:
3171:
3168:
3164:
3160:
3152:
3146:
3142:
3139:
3132:
3131:
3130:
3110:
3104:
3101:
3096:
3083:
3080:
3069:
3064:
3061:
3056:
3053:
3050:
3047:
3044:
3035:
3032:
3026:
3023:
3017:
3010:
3009:
3008:
3006:
2997:
2964:
2957:
2950:
2943:
2933:
2927:
2923:In the limit
2921:
2902:
2898:
2894:
2890:and velocity
2887:
2883:
2876:
2872:
2868:
2848:
2843:
2839:
2835:
2831:
2825:
2821:
2817:
2812:
2808:
2805:
2791:
2785:
2782:
2776:
2770:
2760:
2753:
2749:
2741:
2737:
2732:
2723:
2717:
2713:
2710:
2702:
2694:
2690:
2686:
2678:
2670:
2666:
2658:
2657:
2656:
2653:
2651:
2632:
2626:
2623:
2620:
2613:
2609:
2605:
2600:
2596:
2589:
2586:
2580:
2577:
2570:
2569:
2568:
2564:
2560:
2552:
2548:
2544:
2534:
2512:
2508:
2504:
2499:
2496:
2493:
2489:
2485:
2480:
2476:
2472:
2467:
2464:
2461:
2457:
2453:
2450:
2447:
2442:
2438:
2434:
2429:
2425:
2421:
2416:
2412:
2404:
2403:
2402:
2372:
2367:
2365:
2359:
2344:
2342:
2338:
2334:
2330:
2325:
2323:
2319:
2315:
2310:
2309:(see below).
2308:
2304:
2300:
2296:
2292:
2291:path integral
2288:
2283:
2279:
2271:
2267:
2266:time integral
2264:given by the
2263:
2254:
2249:
2246:
2242:
2239:
2235:
2231:
2230:
2229:
2225:
2223:
2213:
2211:
2204:
2192:
2181:
2174:
2119:
2114:
2110:
2085:
2081:
2076:
2072:
2065:
2058:
2053:
2049:
2039:
2032:
2024:
2020:
2016:
2009:
1994:
1990:
1966:
1961:
1958:
1955:
1951:
1943:
1942:
1941:
1934:
1932:
1907:
1892:
1884:
1867:
1859:
1856:
1853:
1847:
1844:
1841:
1837:
1829:
1828:
1827:
1824:
1820:
1803:
1795:
1792:
1789:
1783:
1780:
1777:
1774:
1770:
1762:
1761:
1760:
1757:
1746:
1742:
1738:
1731:
1727:
1722:
1716:
1712:
1692:
1684:
1678:
1675:
1672:
1669:
1665:
1657:
1656:
1655:
1653:
1634:
1619:
1616:
1613:
1607:
1604:
1601:
1588:
1582:
1579:
1573:
1570:
1567:
1561:
1551:
1541:
1537:
1529:
1528:
1527:
1501:
1496:
1471:
1465:
1457:
1448:
1442:
1439:
1429:
1428:
1427:
1410:
1407:
1404:
1401:
1395:
1392:
1386:
1383:
1380:
1377:
1374:
1367:
1366:
1365:
1361:
1357:
1353:
1333:
1330:
1304:
1295:
1292:
1281:
1272:
1269:
1262:
1261:
1260:
1243:
1240:
1237:
1224:
1218:
1215:
1209:
1206:
1203:
1197:
1184:
1178:
1175:
1172:
1169:
1162:
1161:
1160:
1158:
1153:
1151:
1147:
1143:
1137:
1135:
1131:
1127:
1123:
1119:
1114:
1112:
1108:
1103:
1097:
1092:
1078:
1074:
1072:
1068:
1064:
1060:
1056:
1052:
1048:
1044:
1040:
1036:
1031:
1029:
1025:
1021:
1017:
1013:
1012:string theory
1009:
1004:
1002:
998:
994:
990:
986:
982:
978:
974:
969:
967:
963:
959:
955:
951:
947:
943:
939:
935:
931:
927:
923:
918:
916:
912:
908:
904:
900:
896:
884:
879:
877:
872:
870:
865:
864:
862:
861:
853:
850:
848:
845:
843:
840:
838:
835:
833:
830:
828:
825:
823:
820:
818:
815:
813:
810:
808:
805:
803:
800:
798:
795:
793:
790:
788:
785:
783:
780:
778:
775:
773:
770:
768:
765:
763:
760:
758:
755:
753:
750:
748:
745:
743:
740:
738:
735:
733:
730:
728:
725:
723:
720:
718:
715:
713:
710:
708:
705:
703:
700:
698:
695:
693:
690:
688:
685:
683:
680:
678:
675:
673:
670:
668:
665:
663:
660:
658:
655:
653:
650:
648:
645:
643:
640:
638:
635:
633:
630:
628:
625:
624:
617:
616:
609:
606:
604:
601:
599:
596:
594:
591:
589:
586:
584:
583:Quantum chaos
581:
579:
576:
574:
571:
569:
566:
564:
561:
560:
554:
553:
545:
542:
540:
539:Transactional
537:
535:
532:
530:
529:Quantum logic
527:
525:
522:
520:
517:
511:
508:
507:
506:
503:
502:
501:
498:
496:
493:
491:
488:
486:
483:
481:
478:
476:
473:
472:
468:
463:
462:
454:
451:
449:
446:
444:
441:
439:
436:
434:
431:
430:
423:
422:
414:
411:
409:
406:
404:
401:
399:
396:
394:
391:
389:
386:
385:
381:
378:
377:
371:
370:
362:
359:
357:
354:
352:
349:
348:
342:
339:
338:
337:
334:
333:
329:
326:
324:
321:
319:
316:
314:
311:
309:
306:
304:
301:
299:
296:
294:
291:
289:
286:
284:
281:
280:
273:
272:
262:
259:
258:
257:
256:Wave function
254:
252:
249:
247:
244:
242:
239:
237:
236:Superposition
234:
232:
229:
227:
224:
222:
219:
217:
214:
212:
209:
207:
204:
202:
199:
197:
194:
193:
186:
185:
177:
174:
172:
169:
168:
164:
161:
159:
156:
154:
151:
150:
144:
143:
138:
135:
133:
130:
128:
125:
124:
123:
122:
118:
85:
79:
62:
59:
55:
47:
40:
39:
36:
33:
32:
28:
27:
22:
21:line integral
26325:
26314:
26304:
26291:
26284:
26246:Joan Feynman
26223:
26212:
26201:
26190:
26179:
26168:
26157:
26146:
26135:
26080:Parton model
26074:
25703:Quantum mind
25615:Franck–Hertz
25477:Klein–Gordon
25450:
25426:Formulations
25419:Formulations
25348:Interference
25338:Entanglement
25316:Ground state
25311:Energy level
25284:Fundamentals
25248:Introduction
25155:
25133:
25108:
25079:
25074:Weinberg, S.
25029:
25025:
24998:
24962:
24958:
24932:
24910:
24888:
24867:
24858:
24839:
24804:
24781:
24759:
24740:
24721:
24678:
24644:
24623:
24602:
24580:
24577:Styer, D. F.
24573:Hibbs, A. R.
24548:
24521:
24517:
24489:10.1142/5852
24471:
24441:. Retrieved
24421:
24417:
24394:
24390:
24342:(1): 47–67.
24339:
24335:
24313:
24266:
24262:
24234:
24230:
24204:
24195:
24184:Bibliography
24170:
24159:
24148:
24137:. Retrieved
24132:
24122:
24095:
24092:Scholarpedia
24091:
24081:
24069:
24057:
24046:
24034:
24022:
24016:Feynman 1948
24011:
23999:
23990:
23977:
23966:
23959:Feynman 1948
23954:
23933:
23922:
23911:
23900:
23889:. Retrieved
23885:
23872:
23861:. Retrieved
23855:
23845:
23810:. Retrieved
23803:the original
23789:
23784:, Chapter 9.
23777:
23753:
23740:
23665:
23656:
23653:
23557:
23533:
23521:
23506:
23033:
23017:
22885:
22830:
22827:
22824:
22498:
22328:
22245:
22239:
22235:
22228:
22101:
22024:
21915:
21853:
21850:
21747:
21663:
21656:
21650:
21586:
21559:
21556:
21541:
21453:
21442:
21426:
21420:
21417:Localization
21410:
21404:
21335:
21325:
21321:target space
21315:
21302:
21296:
21293:
21213:
21078:
20972:
20697:
20508:
20490:
20467:
20463:
20456:
20452:
20432:
20429:
20423:
20224:
20077:
19889:
19779:
19757:source field
19755:(called the
19750:
19743:
19731:
19637:
19493:
19448:
19382:
19373:
19362:
19296:
19292:
19268:
19265:
18987:
18977:
18973:
18964:
18960:
18952:
18948:
18930:
18928:
18916:
18909:
18768:
18751:time-ordered
18733:
18707:
18697:
18690:
18685:
18684:is called a
18675:
18671:
18666:
18660:
18656:
18650:
18637:
18635:
18626:
18618:
18612:
18608:
18601:
18513:
18510:
18326:
18319:
18312:
18309:
18305:
18298:
18176:
18169:
18159:
18152:
18149:
17972:
17869:
17859:
17856:
17711:
17707:
17700:This is the
17699:
17565:
17562:
17443:
17310:
17173:
17169:
17165:
17161:
17157:
16991:
16976:
16851:
16842:
16823:
16816:
16698:
16694:
16690:
16683:
16597:
16593:
16589:
16485:
16410:
16406:
16398:
16394:
16390:
16387:
16190:
16183:
16172:
16002:
15998:
15985:
15973:
15969:
15961:
15948:
15938:
15840:
15801:
15647:Stueckelberg
15387:Jona-Lasinio
14977:Vacuum state
14962:Quantization
14804:Gauge theory
14784:Strong force
14769:Field theory
14719:
14709:
14651:is given by
14642:
14635:
14546:
14532:
14524:
14518:
14505:
14481:
14477:
14467:
14387:
14380:
14373:
14188:
14179:
14147:
14016:
13857:
13678:
13530:
13525:
13523:
13283:
12922:
12909:
12752:
12254:
11886:
11875:
11864:
11858:
11730:
11727:
11653:
11633:
11584:
11510:
11499:
11392:
11390:
11374:
11302:
11298:
11295:
11194:
11067:
10832:
10829:
10758:
10752:
10660:
10656:
10653:
10575:
10564:
10540:
10524:
10521:
10419:
10415:
10412:
10205:
10023:
9883:
9879:
9875:
9868:
9864:
9857:
9853:
9849:
9845:
9841:
9837:
9830:
9826:
9823:
9708:
9697:
9688:
9581:
9415:
9411:
9408:
9302:
9037:
9011:
8942:
8934:
8924:
8843:
8685:
8679:
8672:
8668:
8665:
8493:
8489:
8483:
8480:
8029:
8025:
8006:
7998:
7994:
7975:
7971:
7968:
7523:
7516:
7512:
7509:
7216:
7096:
7009:
6514:
6305:
5904:
5676:
5669:
5665:
5658:
5654:
5650:
5643:
5639:
5635:
5628:
5624:
5621:
5513:
5400:
5201:
5116:
5006:
4924:
4861:
4853:
4849:
4746:
4658:
4565:
4459:
4451:
4444:
4330:
4326:
4324:centered at
4319:
4315:
4311:
4300:
4090:
3908:
3905:
3831:
3824:
3813:
3803:Cornu Spiral
3543:
3377:
3369:
3351:
3347:
3340:
3316:
3309:
3293:
3286:
3128:
2998:
2959:
2952:
2945:
2938:
2925:
2922:
2900:
2896:
2892:
2885:
2881:
2874:
2870:
2866:
2863:
2654:
2650:time-slicing
2649:
2647:
2562:
2558:
2550:
2546:
2539:
2532:
2529:
2368:
2361:
2341:perturbative
2326:
2311:
2307:interference
2297:but varying
2295:equal weight
2294:
2290:
2281:
2275:
2252:
2226:
2219:
2200:
2179:
2172:
2117:
2112:
2108:
2083:
2079:
2074:
2070:
2063:
2056:
2051:
2047:
2037:
2034:
2030:
2018:
2014:
2007:
1992:
1988:
1981:
1935:
1908:
1893:
1883:which means
1882:
1825:
1818:
1758:
1740:
1736:
1729:
1725:
1714:
1710:
1707:
1651:
1649:
1497:
1486:
1425:
1359:
1355:
1351:
1319:
1258:
1154:
1138:
1115:
1101:
1088:
1032:
1005:
1001:random walks
970:
965:
952:), than the
949:
945:
919:
894:
892:
438:Klein–Gordon
412:
374:Formulations
211:Energy level
206:Entanglement
189:Fundamentals
176:Interference
127:Introduction
26329:(2013 film)
26319:(2001 play)
26308:(1996 film)
25961:EPR paradox
25741:Quantum bus
25610:Double-slit
25588:Experiments
25554:Many-worlds
25492:Schrödinger
25456:Phase space
25446:Schrödinger
25436:Interaction
25393:Uncertainty
25363:Nonlocality
25358:Measurement
25353:Decoherence
25343:Hamiltonian
25080:Foundations
24995:Tomé, W. A.
24452:Etingof, P.
24403:Duru, İ. H.
24098:(2). 8674.
23710:Propagators
23524:decoherence
23517:nonlocality
21462:, and that
18910:The symbol
18760:functional
16417:convolution
15787:Zinn-Justin
15637:Sommerfield
15562:Pomeranchuk
15532:Osterwalder
15527:Oppenheimer
15457:Łopuszański
15282:Fredenhagen
15083:Technicolor
14259:where
10764:, so that:
4341:convolution
3289:Riemann sum
2322:Hamiltonian
2234:probability
1759:Next comes
1134:four-vector
1091:Hamiltonian
1071:Hamiltonian
954:Hamiltonian
934:coordinates
827:von Neumann
812:Schrödinger
588:EPR paradox
519:Many-worlds
453:Schrödinger
408:Schrödinger
403:Phase-space
393:Interaction
298:Double-slit
276:Experiments
251:Uncertainty
221:Nonlocality
216:Measurement
201:Decoherence
171:Hamiltonian
26345:Categories
25889:Extensions
25723:Technology
25569:Relational
25520:Copenhagen
25431:Heisenberg
25378:Tunnelling
25241:Background
24443:2007-11-25
24139:2021-07-16
24051:Simon 1979
23982:Hilke, M.
23971:Dirac 1933
23916:Dirac 1933
23891:2024-06-21
23863:2024-06-21
23836:2003.12683
23812:2018-12-15
23769:References
23020:regulators
21570:derivation
21460:Lagrangian
19890:Note that
19761:dual space
19504:polynomial
19500:reciprocal
18744:functional
18646:functional
16175:propagator
15995:Propagator
15782:Zimmermann
15677:Vainshtein
15422:Kontsevich
15367:Iliopoulos
15342:Heisenberg
15167:Bogoliubov
15105:Scientists
14957:Propagator
14844:T-symmetry
14839:P-symmetry
14834:C-symmetry
14822:Symmetries
14779:Weak force
14763:Background
14017:where now
13714:, denoted
11642:explained
5504:See also:
4301:where the
3632:and where
3005:Lagrangian
2561:= 1, ...,
2285:elaborate
2278:integrates
2270:Lagrangian
1931:Paul Dirac
1118:Lagrangian
1067:Lagrangian
1051:Paul Dirac
1047:Lagrangian
973:stochastic
942:Lagrangian
822:Sommerfeld
737:Heisenberg
732:Gutzwiller
672:de Broglie
620:Scientists
534:Relational
485:Copenhagen
388:Heisenberg
246:Tunnelling
147:Background
26381:Integrals
26275:Namesakes
25595:Bell test
25465:Equations
25291:Born rule
25076:(2002) ,
24987:121370426
24929:Simon, B.
24713:117837329
24372:119669964
24301:119581543
24135:. 02-1020
24039:Hall 2013
24027:Hall 2013
23945:1004.3578
23759:Hall 2013
23680:to model
23632:ℏ
23609:−
23601:
23580:Γ
23481:^
23469:^
23454:^
23442:^
23400:^
23388:^
23359:^
23347:^
23295:^
23283:^
23254:^
23242:^
23187:ℏ
23146:ℏ
23137:−
23105:^
23093:^
23078:^
23066:^
22964:δ
22960:δ
22952:−
22932:φ
22897:∫
22781:δ
22777:δ
22769:−
22749:φ
22702:δ
22698:δ
22690:−
22671:μ
22661:μ
22657:∂
22627:δ
22623:δ
22615:−
22595:φ
22548:δ
22544:δ
22536:−
22462:μ
22452:μ
22448:∂
22430:⟩
22406:⟨
22400:⟩
22373:⟨
22364:⟩
22340:⟨
22305:φ
22277:φ
22267:∫
22210:φ
22179:φ
22174:μ
22170:∂
22163:∂
22159:∂
22154:−
22140:μ
22118:μ
22075:μ
22065:μ
22061:∂
22025:so that\
21997:φ
21982:−
21961:δ
21942:φ
21875:∫
21823:μ
21809:μ
21796:∂
21792:∫
21774:⟩
21759:⟨
21721:φ
21685:φ
21675:∫
21627:φ
21608:φ
21598:∫
21516:μ
21506:μ
21502:∂
21384:
21375:−
21359:^
21251:∂
21245:−
21140:δ
21136:δ
21128:−
21108:φ
21105:δ
21093:δ
21029:φ
21011:φ
21008:δ
20996:δ
20921:δ
20917:δ
20887:∂
20870:∂
20864:⋯
20839:δ
20835:δ
20805:∂
20788:∂
20769:−
20737:δ
20733:δ
20725:−
20664:φ
20636:∂
20619:∂
20613:⋯
20594:φ
20566:∂
20549:∂
20537:φ
20473:)⟩
20396:⟩
20393:φ
20384:⟨
20375:φ
20349:φ
20339:∫
20327:⟩
20324:φ
20315:⟨
20306:φ
20277:φ
20268:φ
20258:∫
20243:⟩
20236:⟨
20183:φ
20179:⋯
20163:φ
20107:⋯
20033:φ
20030:⋯
20011:φ
19944:δ
19941:⋯
19919:δ
19905:δ
19865:⟩
19862:φ
19853:⟨
19844:φ
19818:φ
19808:∫
19681:−
19642:. In the
19615:φ
19612:δ
19604:φ
19591:δ
19582:φ
19565:−
19552:φ
19549:δ
19541:φ
19532:δ
19494:for some
19474:φ
19465:−
19426:φ
19339:φ
19336:δ
19328:φ
19315:δ
19231:φ
19217:φ
19193:φ
19177:φ
19167:∫
19153:⊂
19146:∑
19126:φ
19112:φ
19088:φ
19072:φ
19062:∫
19048:∩
19042:⊂
19035:∑
19019:∣
19010:
18884:φ
18863:φ
18853:∫
18843:φ
18819:φ
18810:φ
18800:∫
18791:⟩
18785:⟨
18738:, if the
18682:spacetime
18581:ε
18562:−
18490:ε
18484:−
18457:→
18445:−
18420:ε
18387:→
18375:−
18259:→
18247:−
18238:−
18105:→
18044:→
18032:−
17942:−
17930:→
17920:−
17836:α
17800:α
17792:−
17774:−
17764:∞
17755:∫
17666:α
17663:−
17639:−
17627:−
17617:∞
17608:∫
17595:−
17508:−
17494:∞
17485:∫
17472:−
17403:−
17391:−
17376:α
17373:−
17348:−
17288:−
17237:∫
17226:α
17223:−
17198:−
17138:τ
17131:α
17128:−
17114:˙
17089:∫
17034:∫
17013:−
16956:ε
16927:→
16911:−
16905:−
16897:−
16844:Gaussian
16796:ε
16768:→
16756:−
16745:−
16547:−
16525:∝
16453:∗
16444:ψ
16431:ψ
16317:ψ
16292:∫
16239:ψ
16229:∫
16207:ψ
16085:∫
16078:⟩
16066:∣
16054:⟨
16008:to point
15918:−
15903:δ
15881:φ
15872:∂
15856:φ
15717:Wetterich
15702:Weisskopf
15652:Sudarshan
15602:Schwinger
15517:Nishijima
15482:Maldacena
15447:Leutwyler
15412:Kinoshita
15312:Goldstone
15302:Gell-Mann
15217:Doplicher
14994:Equations
14720:classical
14689:β
14683:−
14671:
14621:⟩
14612:α
14600:ℏ
14584:−
14573:⟩
14564:α
14453:ℏ
14450:≫
14337:˙
14285:∫
14238:ℏ
14206:∫
14162:^
14117:ψ
14111:^
14102:−
14081:ψ
14072:∂
14068:∂
14063:ℏ
14025:ψ
13984:μ
13950:ψ
13943:ℏ
13905:∫
13902:−
13894:∫
13873:ψ
13778:μ
13723:μ
13627:˙
13611:∫
13598:−
13553:∫
13526:Euclidean
13446:˙
13415:∫
13403:˙
13228:ψ
13222:ℏ
13208:˙
13150:−
13119:∫
13087:ψ
13058:ℏ
13047:^
13035:−
13008:ℏ
12997:^
12982:−
12951:−
12881:ϕ
12870:ℏ
12854:ϕ
12841:∫
12831:ϕ
12820:ℏ
12804:ϕ
12792:ϕ
12779:∫
12770:⟩
12764:⟨
12720:˙
12684:∫
12679:ℏ
12525:∫
12518:⟩
12479:^
12468:ℏ
12460:−
12446:^
12409:−
12390:^
12379:ℏ
12371:−
12357:^
12320:−
12308:^
12297:ℏ
12289:−
12266:⟨
12222:˙
12186:∫
12181:ℏ
12071:∫
12047:⟩
12010:^
11999:ℏ
11991:−
11977:^
11949:−
11937:^
11926:ℏ
11918:−
11895:⟨
11817:−
11805:˙
11786:˙
11748:∫
11668:μ
11665:∫
11553:˙
11544:∂
11536:∂
11502:Itō lemma
11456:−
11421:˙
11343:−
11279:ε
11249:−
11243:ε
11178:ε
11175:ε
11170:≈
11165:ε
11135:−
11129:ε
11092:−
11051:ε
11035:−
11029:ε
11008:ε
10935:ε
10919:−
10913:ε
10813:ε
10808:≈
10793:−
10787:ε
10736:ε
10720:−
10714:ε
10597:∫
10490:ψ
10466:δ
10458:δ
10441:ψ
10356:ε
10342:∂
10334:∂
10328:−
10319:˙
10310:∂
10302:∂
10276:∫
10268:−
10244:∫
10225:ψ
10221:∫
10210:, gives:
10155:˙
10152:ε
10140:˙
10131:∂
10123:∂
10114:ε
10105:∂
10097:∂
10091:∫
10062:∫
10043:ψ
10039:∫
9993:˙
9990:ε
9978:˙
9966:ε
9922:∫
9903:ψ
9899:∫
9793:˙
9743:∫
9724:ψ
9720:∫
9673:ψ
9653:−
9644:∇
9623:⋅
9608:∂
9603:ψ
9600:∂
9552:ε
9531:−
9498:ε
9492:−
9469:ψ
9466:∫
9463:≈
9457:ε
9439:ψ
9392:ε
9375:˙
9341:ε
9335:−
9311:close to
9229:−
9217:˙
9188:ε
9173:∫
9150:ε
9114:∫
9095:ψ
9090:∞
9085:∞
9082:−
9078:∫
9068:ε
9050:ψ
8971:∫
8899:ω
8896:ℏ
8826:…
8798:for
8762:ω
8756:−
8740:ω
8731:−
8713:ω
8707:−
8636:⋅
8623:ω
8617:−
8589:ℏ
8586:π
8581:ω
8444:ω
8435:−
8427:−
8414:ω
8408:−
8374:−
8363:ω
8354:−
8346:−
8333:ω
8324:−
8256:ℏ
8248:ω
8239:−
8230:
8212:−
8198:ω
8189:−
8181:−
8159:ω
8153:−
8125:ℏ
8122:π
8117:ω
7942:∗
7915:ψ
7889:ψ
7877:ℏ
7854:−
7845:
7837:∞
7822:∑
7796:ω
7793:
7762:−
7756:ω
7753:
7705:ω
7685:ℏ
7671:
7644:ω
7641:
7635:ℏ
7629:π
7621:ω
7475:−
7459:ω
7456:
7450:ℏ
7444:π
7436:ω
7423:ℏ
7379:−
7363:ω
7360:
7339:−
7323:ω
7313:ℏ
7193:π
7185:π
7182:
7141:−
7128:∞
7113:∏
7059:−
7043:ℏ
7037:π
6979:−
6957:π
6936:−
6920:ω
6909:−
6895:∞
6880:∏
6870:ℏ
6812:−
6790:ω
6786:−
6770:−
6745:π
6700:ℏ
6682:
6663:∫
6652:π
6641:∞
6626:∏
6619:ℏ
6479:−
6457:ω
6453:−
6437:−
6412:π
6356:∞
6341:∑
6261:−
6245:ω
6242:
6211:−
6195:−
6179:ω
6176:
6124:ω
6070:ω
6051:−
6039:˙
5984:∫
5938:∫
5871:−
5855:ω
5852:
5831:−
5822:ω
5819:
5781:−
5765:ω
5762:
5748:−
5732:ω
5729:
5588:ω
5569:−
5557:˙
5475:ψ
5460:∇
5454:−
5442:ψ
5432:∂
5428:∂
5318:∫
5299:ψ
5295:∫
5270:−
5243:ψ
5239:∫
5218:ψ
5170:∇
5074:−
5055:∝
5040:−
4975:∇
4885:−
4873:∫
4806:−
4794:−
4786:∝
4771:−
4705:−
4676:~
4623:ε
4617:−
4595:ε
4588:~
4546:ε
4522:ε
4515:~
4484:~
4425:ε
4417:∗
4414:⋯
4411:∗
4406:ε
4398:∗
4393:ε
4370:−
4267:ε
4252:ε
4236:−
4230:ε
4195:−
4187:
4175:∏
4130:∫
4034:˙
4011:∫
4007:−
3999:
3952:∫
3933:−
3865:˙
3853:∫
3818:to point
3758:˙
3717:∫
3705:˙
3502:ψ
3490:˙
3421:∫
3389:ψ
3254:ε
3245:−
3234:−
3206:~
3165:∑
3161:ε
3156:ℏ
3143:
3102:−
3084:˙
3051:−
3036:˙
2999:Actually
2832:⋯
2733:∫
2727:ℏ
2714:
2706:∞
2698:∞
2695:−
2691:∫
2687:⋯
2682:∞
2674:∞
2671:−
2667:∫
2606:−
2584:Δ
2578:ε
2465:−
2451:⋯
2287:curlicues
1986:and time
1959:ε
1860:ε
1781:ε
1775:−
1670:−
1605:ε
1602:−
1580:−
1574:ε
1491:at fixed
1463:∂
1455:∂
1443:˙
1405:ε
1402:−
1396:˙
1384:ε
1375:ε
1334:˙
1296:˙
1287:∂
1279:∂
1241:ε
1238:−
1216:−
1210:ε
1170:ε
1016:cosmology
979:with the
958:unitarity
938:canonical
852:Zeilinger
697:Ehrenfest
426:Equations
103:⟩
100:Ψ
89:^
77:⟩
74:Ψ
51:ℏ
26305:Infinity
26259:(nephew)
26250:(sister)
25984:Category
25778:Timeline
25530:Ensemble
25510:Bayesian
25403:Collapse
25275:Glossary
25258:Timeline
25154:(2023).
25132:(2004).
25066:16577107
25020:(1928).
24997:(1998).
24931:(1979).
24802:(2004).
24780:(2010).
24579:(2010).
24454:(2002).
24409:(1979).
24397:: 64–72.
24382:(1933).
24194:(1971).
23688:See also
23550:models.
23548:spinfoam
22864:⟩
22847:⟨
22477:⟩
22440:⟨
21829:⟩
21784:⟨
21445:on shell
21306:, where
21051:⟩
20989:⟨
20501:operator
20476:are its
20451:⟨
20200:⟩
20158:⟨
20053:⟩
20007:⟨
19734:on-shell
19713:⟩
19688:⟨
19674:⟩
19658:⟨
19622:⟩
19572:⟨
19558:⟩
19526:⟨
19277:Big Bang
18986: P(
18771:⟩
18767:⟨
16012:in time
15732:Wightman
15697:Weinberg
15687:Virasoro
15667:Tomonaga
15662:Thirring
15657:Symanzik
15617:Semenoff
15592:Schrader
15557:Polyakov
15477:Majorana
15417:Klebanov
15372:Ivanenko
15362:'t Hooft
15332:Guralnik
15277:Fröhlich
15272:Fritzsch
15267:Frampton
15182:Buchholz
15127:Bargmann
15117:Anderson
14917:Crossing
12161:′
12023:′
11956:′
10500:⟩
10436:⟨
2965:⟩
2937:⟨
2556:, where
2394:at time
2380:at time
2320:for the
2256:, where
2194:—
2041:, where
1751:at time
1065:using a
966:deriving
962:S-matrix
777:Millikan
702:Einstein
687:Davisson
642:Blackett
627:Aharonov
495:Ensemble
475:Bayesian
380:Overview
261:Collapse
241:Symmetry
132:Glossary
26268:Related
26129:"
25937:Related
25916:History
25655:Science
25487:Rydberg
25253:History
25103:Zee, A.
25057:1085402
25034:Bibcode
24967:Bibcode
24683:Bibcode
24526:Bibcode
24477:Bibcode
24426:Bibcode
24364:0309456
24344:Bibcode
24281:Bibcode
24239:Bibcode
24100:Bibcode
23732:Remarks
23540:Hawking
23507:In one
23009:Caveats
20706:, then
20484:is its
20478:moments
19780:defined
19502:of any
19374:If the
18993:
18980:)
18969:
16684:and in
15742:Wilczek
15707:Wentzel
15682:Veltman
15627:Shirkov
15622:Shifman
15612:Seiberg
15597:Schwarz
15577:Rubakov
15502:Naimark
15452:Lipatov
15442:Lehmann
15407:Kendall
15297:Gelfand
15292:Glashow
15252:Feynman
15232:Faddeev
15227:Englert
15197:Coleman
15187:Cachazo
15172:Brodsky
15157:Bjorken
15147:Berezin
15137:Belavin
14897:Anomaly
14755:History
14537:
14514:
14510:
14496:
14374:is the
11862:√
10756:√
8947:
8931:
8023:
8011:
7992:
7980:
4464:
4448:
3364:
3337:
3287:in the
2982:and at
2920:terms.
2268:of the
2260:is the
2096:s, say
1906:basis.
817:Simmons
807:Rydberg
772:Moseley
752:Kramers
742:Hilbert
727:Glauber
722:Feynman
707:Everett
677:Compton
448:Rydberg
137:History
26317:
26257:
26248:
26239:Family
26230:(2005)
26228:
26219:(1999)
26217:
26208:(1999)
26206:
26197:(1997)
26195:
26186:(1988)
26184:
26175:(1985)
26173:
26164:(1985)
26162:
26153:(1965)
26151:
26142:(1964)
26140:
26131:(1959)
26033:Career
25630:Popper
25162:
25140:
25117:
25092:
25064:
25054:
25005:
24985:
24939:
24917:
24897:
24874:
24846:
24812:
24788:
24766:
24747:
24728:
24711:
24701:
24651:
24632:
24609:
24587:
24557:
24495:
24370:
24362:
24320:
24299:
24211:
22102:where
21916:where
21664:Then,
21546:where
21409:about
20480:, and
20462:) ...
20225:where
19782:to be
18740:action
18708:called
18638:direct
17563:where
15797:Zumino
15762:Yukawa
15752:Witten
15747:Wilson
15737:Wigner
15672:Tyutin
15632:Skyrme
15582:Ruelle
15552:Plefka
15547:Peskin
15537:Parisi
15497:Møller
15487:Migdal
15472:Maiani
15467:Lüders
15432:Landau
15427:Kuraev
15402:Källén
15392:Jordan
15377:Jackiw
15317:Gribov
15207:DeWitt
15202:Dashen
15192:Callan
15162:Bleuer
15132:Becchi
15122:Anselm
14395:, and
14376:action
13679:where
13284:where
9878:(0) =
9862:where
8902:
7947:
7079:
6284:
5508:, and
5412:does:
3544:where
2951:|
2864:where
2371:zigzag
2262:action
1652:matrix
1426:where
1349:holds
1150:action
987:. The
847:Zeeman
842:Wigner
792:Planck
762:Landau
747:Jordan
398:Matrix
328:Popper
26119:Works
25540:Local
25482:Pauli
25472:Dirac
24983:S2CID
24955:(PDF)
24825:arXiv
24709:S2CID
24664:arXiv
24514:(PDF)
24414:(PDF)
24387:(PDF)
24368:S2CID
24297:S2CID
24271:arXiv
23987:(PDF)
23940:arXiv
23831:arXiv
23806:(PDF)
23799:(PDF)
23415:, or
23205:, or
21568:is a
18642:field
15792:Zuber
15642:Stora
15607:Segal
15587:Salam
15572:Proca
15567:Popov
15542:Pauli
15522:Oehme
15512:Neveu
15507:Nambu
15492:Mills
15382:Jaffe
15357:Hagen
15352:Higgs
15327:Gupta
15322:Gross
15307:Glimm
15287:Furry
15257:Fierz
15247:Fermi
15242:Fayet
15237:Fadin
15222:Dyson
15212:Dirac
15177:Brout
15152:Bethe
15112:Adler
14891:Tools
13818:with
11296:Then
9886:) = 0
9702:and
8487:into
8028:(1 +
7997:(1 −
5401:then
3595:with
2299:phase
991:is a
802:Raman
787:Pauli
782:Onnes
717:Fermi
692:Debye
682:Dirac
647:Bloch
637:Bethe
505:Local
443:Pauli
433:Dirac
231:State
25160:ISBN
25138:ISBN
25115:ISBN
25090:ISBN
25062:PMID
25003:ISBN
24937:ISBN
24915:ISBN
24895:ISBN
24872:ISBN
24844:ISBN
24810:ISBN
24786:ISBN
24764:ISBN
24745:ISBN
24726:ISBN
24699:ISBN
24649:ISBN
24630:ISBN
24607:ISBN
24585:ISBN
24555:ISBN
24493:ISBN
24318:ISBN
24209:ISBN
23546:and
21580:and
21578:BRST
20698:and
20516:for
18967:) =
18322:→ +∞
18315:→ −∞
16981:and
15989:fail
15943:and
15757:Yang
15727:Wick
15722:Weyl
15712:Wess
15692:Ward
15397:Jost
15347:Hepp
15337:Haag
15262:Fock
15142:Bell
11644:here
11195:Let
10569:and
10529:and
9852:) +
9844:) =
9824:Now
9016:and
8005:cos
8003:and
7974:sin
7510:Let
5653:) +
5642:) =
2486:<
2473:<
2454:<
2448:<
2435:<
2243:The
2232:The
2180:L dt
2069:, …
1913:and
1514:and
1259:and
1022:and
1014:and
893:The
837:Wien
832:Weyl
797:Rabi
767:Laue
757:Lamb
712:Fock
667:Bose
662:Born
657:Bohr
652:Bohm
632:Bell
26315:QED
25052:PMC
25042:doi
24975:doi
24691:doi
24534:doi
24485:doi
24434:doi
24422:84B
24352:doi
24289:doi
24247:doi
24235:149
24108:doi
23660:eff
23598:exp
22833:= 0
22246:not
21562:= 0
21214:or
20491:If
20438:QFT
20078:or
19751:If
19301:as
18756:of
18734:In
18702:in
18680:of
18667:One
17568:(Τ)
17448:to
15949:are
15777:Zee
15462:Low
15437:Lee
14645:= 0
14390:= t
14383:= 0
12943:by
11646:).
11504:in
8227:exp
7842:exp
7790:sin
7750:cos
7668:exp
7638:sin
7453:sin
7357:sin
7179:sin
6679:exp
6239:sin
6173:cos
5849:sin
5816:sin
5759:sin
5726:sin
4184:exp
3996:exp
3834:= 1
3827:= 1
3315:...
3300:to
3140:exp
2928:→ ∞
2711:exp
2565:+ 1
2553:− 1
2535:+ 1
2387:to
2282:all
2010:(0)
1991:+ 2
1921:to
1819:or
1723:in
948:or
905:of
26347::
25088:,
25060:.
25050:.
25040:.
25030:14
25028:.
25024:.
24981:.
24973:.
24961:.
24957:.
24707:.
24697:.
24689:.
24575:;
24532:.
24522:20
24520:.
24516:.
24491:.
24483:.
24432:.
24420:.
24416:.
24405:;
24393:.
24389:.
24366:.
24360:MR
24358:.
24350:.
24340:28
24338:.
24312:.
24295:.
24287:.
24279:.
24267:36
24265:.
24245:.
24233:.
24225:;
24131:.
24106:.
24094:.
24090:.
23989:.
23884:.
23854:.
23821:^
23519:.
23374:,
23269:,
23164:,
22993:0.
22871:0.
22810:0.
22484:0.
22314:0.
21584:.
21381:ln
21279:0.
20506:,
20488:.
19748:.
19371:.
19272:in
19185:in
19080:in
18972:P(
18959:P(
18947:P(
18764:,
18710:a
18611:=
18202:NR
18175:=
18158:=
17977::
17710:−
17575::
17328::
17172:,
17168:−
16697:;
16596:−
16409:−
16397:,
16016::
15991:.
15983:.
15959:.
14668:Tr
14649:iβ
14544:.
14512:=
14482:ħβ
14480:=
14478:it
13862::
13855:.
11393:be
10835:xẋ
10580::
10028::
9024:.
8009:=
8007:ωT
7978:=
7976:ωT
7522:−
7515:=
7101:,
6308:δS
5677:δS
5675:+
5668:=
5655:δx
4910:1.
4318:+
3836:)
3829:,
3367:.
3354:−1
3346:+
3325:x̃
3323:,
3317:dx
3310:dx
2996:.
2958:,
2944:,
2907:dx
2895:=
2873:,
2652:.
2545:−
2078:,
2062:,
2055:,
1933:.
1891:.
1823:.
1756:.
1495:.
1358:+
1113:.
1098:,
1003:.
917:.
26125:"
26018:e
26011:t
26004:v
25226:e
25219:t
25212:v
25168:.
25146:.
25123:.
25068:.
25044::
25036::
25011:.
24989:.
24977::
24969::
24963:4
24945:.
24923:.
24903:.
24880:.
24852:.
24833:.
24827::
24818:.
24794:.
24772:.
24753:.
24734:.
24715:.
24693::
24685::
24672:.
24666::
24657:.
24638:.
24615:.
24593:.
24563:.
24540:.
24536::
24528::
24501:.
24487::
24479::
24446:.
24436::
24428::
24395:3
24374:.
24354::
24346::
24326:.
24303:.
24291::
24283::
24273::
24253:.
24249::
24241::
24217:.
24142:.
24116:.
24110::
24102::
24096:4
23993:.
23961:.
23948:.
23942::
23894:.
23866:.
23839:.
23833::
23815:.
23748:.
23669:o
23666:A
23657:S
23638:)
23626:f
23623:f
23620:e
23615:S
23605:(
23592:o
23587:A
23583:=
23567:Γ
23487:)
23478:q
23466:p
23460:+
23451:p
23439:q
23433:(
23428:2
23425:1
23397:q
23385:p
23356:p
23344:q
23321:p
23318:q
23292:q
23280:p
23251:p
23239:q
23216:p
23213:q
23191:2
23184:i
23178:+
23175:p
23172:q
23150:2
23143:i
23134:p
23131:q
23111:)
23102:q
23090:p
23084:+
23075:p
23063:q
23057:(
23052:2
23049:1
22990:=
22987:]
22984:J
22981:[
22978:Z
22974:]
22967:J
22955:i
22948:[
22944:]
22941:)
22938:x
22935:(
22929:[
22926:Q
22923:)
22920:x
22917:(
22914:J
22910:x
22905:d
22901:d
22868:=
22860:]
22857:F
22854:[
22851:Q
22831:f
22807:=
22804:]
22801:J
22798:[
22795:Z
22791:]
22784:J
22772:i
22765:[
22761:]
22758:)
22755:x
22752:(
22746:[
22743:Q
22740:)
22737:x
22734:(
22731:J
22728:+
22725:]
22722:J
22719:[
22716:Z
22712:]
22705:J
22693:i
22686:[
22682:)
22679:x
22676:(
22667:j
22653:=
22650:]
22647:J
22644:[
22641:Z
22637:]
22630:J
22618:i
22611:[
22607:]
22604:)
22601:x
22598:(
22592:[
22589:Q
22586:)
22583:x
22580:(
22577:J
22574:+
22571:]
22568:J
22565:[
22562:Z
22558:]
22551:J
22539:i
22532:[
22528:]
22525:S
22522:[
22519:)
22516:x
22513:(
22510:q
22481:=
22473:)
22470:x
22467:(
22458:j
22444:F
22436:i
22433:+
22427:]
22424:F
22421:[
22418:)
22415:x
22412:(
22409:q
22403:=
22397:]
22394:S
22391:[
22388:)
22385:x
22382:(
22379:q
22376:F
22370:i
22367:+
22361:]
22358:F
22355:[
22352:)
22349:x
22346:(
22343:q
22311:=
22308:]
22302:[
22299:]
22296:F
22293:[
22290:)
22287:x
22284:(
22281:q
22272:D
22254:Q
22242:)
22240:x
22238:(
22236:q
22231:φ
22213:]
22207:[
22204:Q
22201:)
22198:x
22195:(
22190:L
22182:)
22166:(
22151:)
22148:x
22145:(
22136:f
22132:=
22129:)
22126:x
22123:(
22114:j
22086:)
22083:x
22080:(
22071:j
22057:=
22054:]
22051:S
22048:[
22045:)
22042:x
22039:(
22036:q
22009:]
22006:)
22003:y
22000:(
21994:[
21991:Q
21988:)
21985:y
21979:X
21976:(
21971:)
21968:d
21965:(
21957:=
21954:]
21951:)
21948:y
21945:(
21939:[
21936:)
21933:x
21930:(
21927:q
21901:)
21898:x
21895:(
21892:q
21888:x
21883:d
21879:d
21872:=
21869:Q
21856:Q
21836:0
21833:=
21819:s
21815:d
21805:f
21799:V
21788:F
21780:i
21777:+
21771:]
21768:F
21765:[
21762:Q
21733:,
21730:0
21727:=
21724:]
21718:[
21714:]
21708:S
21705:i
21701:e
21697:F
21693:[
21689:Q
21680:D
21653:F
21636:0
21633:=
21630:]
21624:[
21621:]
21618:F
21615:[
21612:Q
21603:D
21566:Q
21560:f
21552:φ
21548:f
21544:f
21527:)
21524:x
21521:(
21512:f
21498:=
21495:]
21492:)
21489:x
21486:(
21481:L
21476:[
21473:Q
21411:J
21390:.
21387:M
21378:i
21370:S
21365:=
21355:S
21338:S
21326:R
21316:φ
21313:D
21308:M
21303:φ
21300:D
21297:M
21276:=
21273:Z
21268:i
21264:J
21260:+
21257:Z
21254:]
21248:i
21242:[
21237:i
21234:,
21228:S
21199:,
21196:0
21193:=
21190:]
21187:J
21184:[
21181:Z
21178:)
21175:x
21172:(
21169:J
21166:+
21163:]
21160:J
21157:[
21154:Z
21150:]
21143:J
21131:i
21124:[
21117:)
21114:x
21111:(
21098:S
21064:0
21061:=
21056:J
21047:)
21044:x
21041:(
21038:J
21035:+
21032:]
21026:[
21020:)
21017:x
21014:(
21001:S
20958:.
20955:]
20952:J
20949:[
20946:G
20940:)
20935:n
20931:x
20927:(
20924:J
20905:n
20901:k
20895:n
20891:x
20879:n
20875:k
20858:)
20853:1
20849:x
20845:(
20842:J
20823:1
20819:k
20813:1
20809:x
20797:1
20793:k
20780:n
20776:)
20772:i
20766:(
20763:=
20760:]
20757:J
20754:[
20751:G
20747:]
20740:J
20728:i
20721:[
20717:F
20704:J
20700:G
20683:,
20680:)
20675:n
20671:x
20667:(
20654:n
20650:k
20644:n
20640:x
20628:n
20624:k
20610:)
20605:1
20601:x
20597:(
20584:1
20580:k
20574:1
20570:x
20558:1
20554:k
20543:=
20540:]
20534:[
20531:F
20518:φ
20514:K
20509:F
20504:K
20497:φ
20493:F
20482:Z
20470:n
20468:x
20466:(
20464:φ
20460:1
20457:x
20455:(
20453:φ
20433:e
20430:φ
20427:D
20409:.
20400:)
20390:,
20387:J
20381:+
20378:]
20372:[
20367:S
20361:(
20357:i
20353:e
20344:D
20331:)
20321:,
20318:J
20312:+
20309:]
20303:[
20298:S
20292:(
20288:i
20284:e
20280:]
20274:[
20271:F
20263:D
20252:=
20247:J
20239:F
20210:,
20205:J
20192:n
20188:i
20172:1
20168:i
20153:]
20150:J
20147:[
20144:Z
20139:n
20135:i
20131:=
20128:]
20125:J
20122:[
20115:n
20111:i
20102:1
20098:i
20094:,
20090:Z
20063:,
20058:J
20049:)
20044:n
20040:x
20036:(
20027:)
20022:1
20018:x
20014:(
20001:]
19998:J
19995:[
19992:Z
19986:n
19982:i
19978:=
19975:]
19972:J
19969:[
19963:)
19958:n
19954:x
19950:(
19947:J
19938:)
19933:1
19929:x
19925:(
19922:J
19914:Z
19909:n
19875:.
19869:)
19859:,
19856:J
19850:+
19847:]
19841:[
19836:S
19830:(
19826:i
19822:e
19813:D
19805:=
19802:]
19799:J
19796:[
19793:Z
19776:Z
19753:J
19744:i
19742:,
19739:S
19717:.
19707:i
19704:,
19698:S
19692:F
19684:i
19678:=
19669:i
19666:,
19662:F
19640:F
19607:]
19601:[
19596:S
19585:]
19579:[
19576:F
19568:i
19562:=
19544:]
19538:[
19535:F
19508:φ
19496:H
19477:]
19471:[
19468:H
19461:e
19434:,
19429:]
19423:[
19418:S
19413:i
19409:e
19383:ϕ
19380:D
19348:0
19345:=
19331:]
19325:[
19320:S
19269:O
19251:,
19243:2
19238:|
19234:]
19228:[
19225:F
19220:]
19214:[
19209:S
19204:i
19200:e
19196:]
19190:[
19181:O
19172:D
19163:|
19156:A
19150:F
19138:2
19133:|
19129:]
19123:[
19120:F
19115:]
19109:[
19104:S
19099:i
19095:e
19091:]
19085:[
19076:O
19067:D
19058:|
19051:B
19045:A
19039:F
19028:=
19025:)
19022:A
19016:B
19013:(
19007:P
18990:)
18988:A
18983:/
18978:B
18976:∩
18974:A
18965:A
18963:|
18961:B
18955:)
18953:A
18951:|
18949:B
18939:?
18937:B
18933:A
18917:ϕ
18914:D
18912:∫
18895:.
18887:]
18881:[
18876:S
18871:i
18867:e
18858:D
18846:]
18840:[
18835:S
18830:i
18826:e
18822:]
18816:[
18813:F
18805:D
18794:=
18788:F
18769:F
18762:F
18747:S
18716:i
18678:)
18676:x
18674:(
18672:ϕ
18663:)
18661:x
18659:(
18657:ϕ
18651:S
18613:m
18609:α
18604:ε
18587:,
18578:i
18575:+
18570:2
18566:m
18557:2
18553:p
18548:i
18543:=
18540:)
18537:p
18534:(
18531:K
18517:0
18514:p
18496:.
18487:i
18477:2
18473:m
18469:+
18464:2
18454:p
18440:0
18436:p
18431:i
18426:+
18417:i
18414:+
18407:2
18403:m
18399:+
18394:2
18384:p
18370:0
18366:p
18361:i
18356:=
18353:)
18350:p
18347:(
18344:K
18330:0
18327:p
18320:t
18313:t
18284:.
18275:m
18272:2
18266:2
18256:p
18244:)
18241:m
18233:0
18229:p
18225:(
18221:i
18216:=
18213:)
18210:p
18207:(
18198:K
18194:m
18191:2
18177:m
18173:0
18170:p
18165:p
18160:m
18156:0
18153:p
18135:.
18125:2
18121:m
18117:+
18112:2
18102:p
18093:+
18088:0
18084:p
18079:i
18074:+
18064:2
18060:m
18056:+
18051:2
18041:p
18027:0
18023:p
18018:i
18013:=
18010:)
18007:p
18004:(
18001:K
17996:0
17992:p
17988:2
17958:.
17950:2
17946:m
17937:2
17927:p
17915:2
17910:0
17906:p
17901:i
17896:=
17893:)
17890:p
17887:(
17884:K
17870:i
17868:−
17863:0
17860:p
17842:,
17833:+
17828:2
17824:p
17819:1
17814:=
17810:T
17806:d
17796:T
17787:2
17783:p
17778:T
17770:e
17759:0
17751:=
17748:)
17745:p
17742:(
17739:K
17726:p
17722:Τ
17718:Τ
17714:)
17712:y
17708:x
17706:(
17685:.
17681:T
17677:d
17670:T
17657:T
17650:2
17646:)
17642:y
17636:x
17633:(
17623:e
17612:0
17604:=
17601:)
17598:y
17592:x
17589:(
17586:K
17573:α
17566:W
17548:,
17544:T
17540:d
17536:)
17532:T
17528:(
17525:W
17522:)
17518:T
17514:,
17511:y
17505:x
17502:(
17499:K
17489:0
17481:=
17478:)
17475:y
17469:x
17466:(
17463:K
17450:y
17446:x
17429:.
17421:T
17414:2
17410:)
17406:y
17400:x
17397:(
17387:e
17380:T
17369:e
17365:=
17362:)
17358:T
17354:,
17351:y
17345:x
17342:(
17339:K
17326:Τ
17318:Τ
17313:Τ
17296:.
17291:L
17284:e
17278:y
17275:=
17272:)
17268:T
17264:(
17261:x
17256:x
17253:=
17250:)
17247:0
17244:(
17241:x
17230:T
17219:e
17215:=
17212:)
17208:T
17204:,
17201:y
17195:x
17192:(
17189:K
17176:)
17174:τ
17170:y
17166:x
17164:(
17162:K
17143:.
17135:d
17121:2
17111:x
17099:T
17093:0
17085:i
17081:e
17075:y
17072:=
17069:)
17065:T
17061:(
17058:x
17053:x
17050:=
17047:)
17044:0
17041:(
17038:x
17030:=
17027:)
17023:T
17019:,
17016:y
17010:x
17007:(
17004:K
16987:E
16983:ε
16979:E
16962:.
16953:i
16950:+
16944:m
16941:2
16934:2
16924:p
16917:i
16908:E
16900:i
16891:=
16888:)
16885:E
16882:,
16879:p
16876:(
16871:B
16867:G
16852:t
16850:−
16846:t
16838:G
16834:K
16830:E
16826:E
16819:p
16802:,
16793:i
16790:+
16784:m
16781:2
16775:2
16765:p
16753:E
16748:i
16739:=
16736:)
16733:E
16730:,
16727:p
16724:(
16719:F
16715:G
16701:)
16699:T
16695:p
16693:(
16691:K
16686:p
16669:,
16662:m
16659:2
16652:2
16648:p
16644:T
16641:i
16635:e
16631:=
16628:)
16625:T
16622:;
16619:p
16616:(
16613:K
16600:)
16598:y
16594:x
16592:(
16575:.
16568:T
16565:2
16558:2
16554:)
16550:y
16544:x
16541:(
16538:m
16535:i
16529:e
16522:)
16519:T
16516:;
16513:y
16510:,
16507:x
16504:(
16501:K
16488:m
16471:.
16468:)
16465:T
16462:;
16459:(
16456:K
16448:0
16440:=
16435:T
16413:)
16411:y
16407:x
16405:(
16401:)
16399:y
16395:x
16393:(
16391:K
16373:.
16370:x
16367:D
16361:]
16358:x
16355:[
16352:S
16349:i
16345:e
16341:)
16338:)
16335:0
16332:(
16329:x
16326:(
16321:0
16311:y
16308:=
16305:)
16302:T
16299:(
16296:x
16288:=
16285:x
16282:d
16278:)
16275:T
16272:;
16269:y
16266:,
16263:x
16260:(
16257:K
16254:)
16251:x
16248:(
16243:0
16233:x
16225:=
16222:)
16219:y
16216:(
16211:T
16193:)
16191:x
16189:(
16187:0
16184:ψ
16179:x
16158:.
16155:x
16152:D
16146:]
16143:x
16140:[
16137:S
16134:i
16130:e
16124:y
16121:=
16118:)
16115:T
16112:(
16109:x
16104:x
16101:=
16098:)
16095:0
16092:(
16089:x
16081:=
16075:0
16072:;
16069:x
16063:T
16060:;
16057:y
16051:=
16048:)
16045:T
16042:;
16039:y
16036:,
16033:x
16030:(
16027:K
16014:T
16010:y
16006:x
15945:y
15941:x
15924:)
15921:y
15915:x
15912:(
15907:3
15899:i
15896:=
15893:]
15890:)
15887:y
15884:(
15876:t
15868:,
15865:)
15862:x
15859:(
15853:[
15830:e
15823:t
15816:v
14694:]
14686:H
14679:e
14675:[
14665:=
14662:Z
14643:t
14638:α
14618:0
14615:;
14608:|
14596:t
14593:H
14590:i
14580:e
14576:=
14570:t
14567:;
14560:|
14533:ħ
14529:/
14525:t
14522:B
14519:k
14517:i
14506:T
14502:/
14499:1
14447:]
14443:x
14439:[
14434:S
14411:x
14405:D
14392:f
14388:t
14381:t
14359:t
14356:d
14352:]
14349:)
14346:t
14343:(
14333:x
14326:,
14323:)
14320:t
14317:(
14313:x
14309:[
14306:L
14299:f
14295:t
14289:0
14281:=
14278:]
14274:x
14270:[
14265:S
14253:x
14247:D
14234:]
14230:x
14226:[
14221:S
14216:i
14210:e
14203:=
14200:Z
14159:H
14144:.
14132:)
14129:t
14126:,
14123:x
14120:(
14108:H
14099:=
14096:)
14093:t
14090:,
14087:x
14084:(
14075:t
14040:)
14037:t
14034:,
14031:x
14028:(
14013:,
14001:)
13997:x
13993:(
13988:x
13980:d
13976:)
13973:)
13970:t
13967:(
13963:x
13959:(
13954:0
13939:/
13935:t
13932:d
13928:)
13925:)
13922:t
13919:(
13915:x
13911:(
13908:V
13898:e
13891:=
13888:)
13885:t
13882:,
13879:x
13876:(
13843:x
13840:=
13837:)
13834:0
13831:(
13827:x
13805:x
13782:x
13757:x
13727:x
13698:)
13694:x
13690:(
13687:f
13662:x
13656:D
13648:t
13645:d
13640:2
13635:|
13623:x
13615:|
13606:2
13603:m
13594:e
13590:)
13586:x
13582:(
13579:f
13574:x
13571:=
13568:)
13565:0
13562:(
13558:x
13547:Z
13544:1
13520:.
13508:t
13505:d
13500:]
13496:)
13493:)
13490:t
13487:(
13483:x
13479:(
13476:V
13473:+
13468:2
13463:|
13458:)
13455:t
13452:(
13442:x
13434:|
13428:2
13425:m
13419:[
13412:=
13409:)
13399:x
13392:,
13388:x
13384:(
13378:n
13375:a
13372:e
13369:d
13366:i
13363:l
13360:c
13357:u
13354:E
13349:S
13322:n
13319:a
13316:e
13313:d
13310:i
13307:l
13304:c
13301:u
13298:E
13293:S
13280:,
13266:x
13260:D
13254:)
13251:)
13248:t
13245:(
13241:x
13237:(
13232:0
13218:/
13214:)
13204:x
13197:,
13193:x
13189:(
13183:n
13180:a
13177:e
13174:d
13171:i
13168:l
13165:c
13162:u
13159:E
13154:S
13146:e
13140:x
13137:=
13134:)
13131:0
13128:(
13124:x
13113:Z
13110:1
13105:=
13102:)
13099:t
13096:,
13093:x
13090:(
13054:/
13044:H
13038:t
13031:e
13004:/
12994:H
12988:t
12985:i
12978:e
12957:t
12954:i
12931:t
12901:.
12884:]
12878:[
12875:S
12867:i
12861:e
12857:]
12851:[
12846:D
12834:]
12828:[
12825:S
12817:i
12811:e
12807:)
12801:(
12798:F
12795:]
12789:[
12784:D
12773:=
12767:F
12749:,
12735:)
12732:)
12729:t
12726:(
12717:x
12711:,
12708:)
12705:t
12702:(
12699:x
12696:(
12693:L
12690:t
12687:d
12676:i
12670:e
12666:)
12663:)
12658:2
12654:t
12650:(
12647:x
12644:(
12639:2
12635:F
12631:)
12628:)
12623:1
12619:t
12615:(
12612:x
12609:(
12604:1
12600:F
12596:]
12593:x
12590:[
12585:D
12576:f
12572:x
12568:=
12565:)
12562:t
12559:(
12556:x
12549:i
12545:x
12541:=
12538:)
12535:0
12532:(
12529:x
12521:=
12513:i
12509:x
12504:|
12498:)
12493:2
12489:t
12485:(
12476:H
12465:i
12456:e
12452:)
12443:x
12437:(
12432:2
12428:F
12422:)
12417:2
12413:t
12404:1
12400:t
12396:(
12387:H
12376:i
12367:e
12363:)
12354:x
12348:(
12343:1
12339:F
12333:)
12328:1
12324:t
12317:t
12314:(
12305:H
12294:i
12285:e
12280:|
12274:f
12270:x
12251:.
12237:)
12234:)
12231:t
12228:(
12219:x
12213:,
12210:)
12207:t
12204:(
12201:x
12198:(
12195:L
12192:t
12189:d
12178:i
12172:e
12168:)
12165:)
12158:t
12154:(
12151:x
12148:(
12145:F
12142:]
12139:x
12136:[
12131:D
12122:f
12118:x
12114:=
12111:)
12108:t
12105:(
12102:x
12095:i
12091:x
12087:=
12084:)
12081:0
12078:(
12075:x
12042:i
12038:x
12033:|
12027:)
12020:t
12016:(
12007:H
11996:i
11987:e
11983:)
11974:x
11968:(
11965:F
11960:)
11953:t
11946:t
11943:(
11934:H
11923:i
11914:e
11909:|
11903:f
11899:x
11876:x
11873:D
11865:g
11844:,
11841:t
11838:d
11833:)
11829:)
11826:x
11823:(
11820:V
11812:j
11802:x
11793:i
11783:x
11774:j
11771:i
11767:g
11761:2
11758:m
11752:(
11745:=
11742:S
11713:.
11710:x
11705:D
11697:]
11694:x
11691:[
11688:S
11685:i
11681:e
11677:]
11674:x
11671:[
11614:i
11611:=
11608:]
11605:p
11602:,
11599:x
11596:[
11570:1
11567:=
11563:]
11550:x
11539:S
11530:,
11527:x
11523:[
11485:1
11482:=
11479:x
11473:t
11470:d
11465:x
11462:d
11450:t
11447:d
11442:x
11439:d
11433:x
11430:=
11427:]
11418:x
11412:,
11409:x
11406:[
11385:L
11381:S
11377:f
11360:,
11354:2
11350:)
11346:1
11340:)
11337:t
11334:(
11331:f
11328:(
11325:=
11320:L
11305:)
11303:t
11301:(
11299:f
11273:2
11267:)
11261:)
11258:t
11255:(
11252:x
11246:)
11240:+
11237:t
11234:(
11231:x
11226:(
11218:=
11215:)
11212:t
11209:(
11206:f
11159:2
11153:)
11147:)
11144:t
11141:(
11138:x
11132:)
11126:+
11123:t
11120:(
11117:x
11112:(
11104:=
11101:]
11098:1
11095:[
11089:]
11086:2
11083:[
11070:ε
11047:)
11044:t
11041:(
11038:x
11032:)
11026:+
11023:t
11020:(
11017:x
11011:)
11005:+
11002:t
10999:(
10996:x
10993:=
10987:t
10984:d
10979:x
10976:d
10970:x
10967:=
10964:]
10961:2
10958:[
10931:)
10928:t
10925:(
10922:x
10916:)
10910:+
10907:t
10904:(
10901:x
10895:)
10892:t
10889:(
10886:x
10883:=
10877:t
10874:d
10869:x
10866:d
10860:x
10857:=
10854:]
10851:1
10848:[
10805:)
10802:t
10799:(
10796:x
10790:)
10784:+
10781:t
10778:(
10775:x
10759:t
10732:)
10729:t
10726:(
10723:x
10717:)
10711:+
10708:t
10705:(
10702:x
10696:=
10690:t
10687:d
10682:x
10679:d
10663:)
10661:t
10659:(
10657:x
10639:t
10636:d
10630:2
10625:)
10619:t
10616:d
10611:x
10608:d
10602:(
10594:=
10591:S
10578:i
10571:p
10567:x
10555:ħ
10551:ħ
10543:ħ
10531:x
10527:ẋ
10507:0
10504:=
10494:0
10485:|
10481:)
10478:t
10475:(
10469:x
10461:S
10451:|
10445:0
10422:)
10420:t
10418:(
10416:ε
10397:u
10394:D
10388:S
10385:i
10381:e
10376:)
10372:t
10369:d
10365:)
10362:t
10359:(
10352:)
10345:u
10337:S
10316:u
10305:S
10293:t
10290:d
10286:d
10280:(
10272:(
10263:x
10260:=
10257:)
10254:0
10251:(
10248:u
10240:)
10237:x
10234:(
10229:0
10208:t
10190:u
10187:D
10181:S
10178:i
10174:e
10169:)
10165:t
10162:d
10137:u
10126:S
10117:+
10108:u
10100:S
10087:(
10081:x
10078:=
10075:)
10072:0
10069:(
10066:u
10058:)
10055:x
10052:(
10047:0
10026:ε
10008:u
10005:D
9999:)
9984:+
9975:u
9969:,
9963:+
9960:u
9957:(
9954:S
9951:i
9947:e
9941:x
9938:=
9935:)
9932:0
9929:(
9926:u
9918:)
9915:x
9912:(
9907:0
9884:T
9882:(
9880:ε
9876:ε
9871:)
9869:t
9867:(
9865:ε
9860:)
9858:t
9856:(
9854:ε
9850:t
9848:(
9846:u
9842:t
9840:(
9838:x
9833:)
9831:t
9829:(
9827:x
9808:x
9805:D
9799:)
9790:x
9784:,
9781:x
9778:(
9775:S
9772:i
9768:e
9762:x
9759:=
9756:)
9753:0
9750:(
9747:x
9739:)
9736:x
9733:(
9728:0
9704:ẋ
9700:x
9669:)
9665:)
9662:x
9659:(
9656:V
9648:2
9637:2
9634:1
9627:(
9620:i
9617:=
9611:t
9584:ψ
9567:.
9563:x
9560:d
9549:2
9542:2
9538:)
9534:y
9528:x
9525:(
9522:i
9516:e
9510:)
9507:x
9504:(
9501:V
9495:i
9488:e
9484:)
9481:t
9478:;
9475:x
9472:(
9460:)
9454:+
9451:t
9448:;
9445:y
9442:(
9426:ε
9422:i
9418:)
9416:x
9414:(
9412:ψ
9387:2
9382:2
9372:x
9363:i
9359:e
9353:)
9350:x
9347:(
9344:V
9338:i
9331:e
9313:x
9309:y
9305:ẋ
9288:)
9285:1
9282:(
9278:x
9275:d
9271:)
9268:t
9265:(
9262:x
9259:D
9254:t
9251:d
9246:)
9241:)
9238:x
9235:(
9232:V
9224:2
9214:x
9205:2
9202:1
9195:(
9185:+
9182:t
9177:t
9169:i
9165:e
9159:y
9156:=
9153:)
9147:+
9144:t
9141:(
9138:x
9133:x
9130:=
9127:)
9124:t
9121:(
9118:x
9110:)
9107:t
9104:;
9101:x
9098:(
9074:=
9071:)
9065:+
9062:t
9059:;
9056:y
9053:(
8994:)
8991:t
8988:(
8985:r
8980:t
8977:d
8968:=
8965:s
8952:t
8943:r
8939:/
8935:e
8905:.
8892:)
8885:2
8882:1
8876:+
8873:n
8869:(
8865:=
8860:n
8856:E
8829:.
8823:,
8820:2
8817:,
8814:1
8811:,
8808:0
8805:=
8802:n
8790:)
8786:n
8783:+
8778:2
8775:1
8769:(
8765:T
8759:i
8752:e
8748:=
8743:T
8737:n
8734:i
8727:e
8720:2
8716:T
8710:i
8701:e
8686:e
8680:e
8675:)
8673:T
8671:(
8669:R
8651:.
8648:)
8645:T
8642:(
8639:R
8630:2
8626:T
8620:i
8611:e
8604:2
8601:1
8595:)
8578:m
8572:(
8567:=
8564:)
8559:i
8555:t
8551:,
8546:i
8542:x
8538:;
8533:f
8529:t
8525:,
8520:f
8516:x
8512:(
8509:K
8496:)
8494:T
8492:(
8490:R
8484:e
8466:.
8461:)
8456:)
8447:T
8441:i
8438:2
8431:e
8424:1
8417:T
8411:i
8404:e
8398:f
8394:x
8388:i
8384:x
8380:4
8366:T
8360:i
8357:2
8350:e
8343:1
8336:T
8330:i
8327:2
8320:e
8316:+
8313:1
8306:)
8300:2
8295:f
8291:x
8287:+
8282:2
8277:i
8273:x
8268:(
8263:(
8253:2
8245:m
8235:(
8220:2
8217:1
8207:)
8201:T
8195:i
8192:2
8185:e
8178:1
8174:(
8166:2
8162:T
8156:i
8147:e
8140:2
8137:1
8131:)
8114:m
8108:(
8103:=
8100:)
8095:i
8091:t
8087:,
8082:i
8078:x
8074:;
8069:f
8065:t
8061:,
8056:f
8052:x
8048:(
8045:K
8032:)
8030:e
8026:e
8020:2
8017:/
8014:1
8001:)
7999:e
7995:e
7989:2
7986:/
7983:1
7972:i
7950:.
7938:)
7932:i
7928:x
7924:(
7919:n
7911:)
7906:f
7902:x
7898:(
7893:n
7883:)
7873:T
7868:n
7864:E
7860:i
7850:(
7832:0
7829:=
7826:n
7818:=
7806:)
7799:T
7783:f
7779:x
7773:i
7769:x
7765:2
7759:T
7747:)
7742:2
7737:f
7733:x
7729:+
7724:2
7719:i
7715:x
7711:(
7702:m
7696:2
7693:1
7682:i
7676:(
7662:2
7659:1
7653:)
7647:T
7632:i
7626:2
7618:m
7612:(
7607:=
7600:)
7595:i
7591:t
7587:,
7582:i
7578:x
7574:;
7569:f
7565:t
7561:,
7556:f
7552:x
7548:(
7545:K
7526:i
7524:t
7519:f
7517:t
7513:T
7495:.
7488:)
7483:i
7479:t
7470:f
7466:t
7462:(
7447:i
7441:2
7433:m
7417:c
7413:S
7409:i
7403:e
7399:=
7392:)
7387:i
7383:t
7374:f
7370:t
7366:(
7352:)
7347:i
7343:t
7334:f
7330:t
7326:(
7307:c
7303:S
7299:i
7293:e
7289:Q
7286:=
7283:)
7278:i
7274:t
7270:,
7265:i
7261:x
7257:;
7252:f
7248:t
7244:,
7239:f
7235:x
7231:(
7228:K
7202:,
7196:x
7188:x
7173:=
7169:)
7161:2
7157:j
7151:2
7147:x
7138:1
7134:(
7123:1
7120:=
7117:j
7082:.
7072:)
7067:i
7063:t
7054:f
7050:t
7046:(
7040:i
7034:2
7030:m
7024:=
7021:Q
6987:2
6984:1
6974:)
6968:2
6963:)
6954:j
6949:)
6944:i
6940:t
6931:f
6927:t
6923:(
6914:(
6906:1
6902:(
6890:1
6887:=
6884:j
6876:Q
6864:c
6860:S
6856:i
6850:e
6846:=
6834:)
6829:)
6825:)
6820:i
6816:t
6807:f
6803:t
6799:(
6794:2
6778:i
6774:t
6765:f
6761:t
6753:2
6749:)
6742:j
6739:(
6732:(
6726:2
6723:m
6716:2
6711:j
6707:a
6697:2
6693:i
6687:(
6674:j
6670:a
6666:d
6657:2
6649:j
6636:1
6633:=
6630:j
6613:c
6609:S
6605:i
6599:e
6595:Q
6592:=
6585:)
6580:i
6576:t
6572:,
6567:i
6563:x
6559:;
6554:f
6550:t
6546:,
6541:f
6537:x
6533:(
6530:K
6500:.
6496:)
6492:)
6487:i
6483:t
6474:f
6470:t
6466:(
6461:2
6445:i
6441:t
6432:f
6428:t
6420:2
6416:)
6409:n
6406:(
6399:(
6393:2
6390:m
6383:2
6378:n
6374:a
6367:2
6364:1
6351:1
6348:=
6345:n
6337:+
6332:c
6328:S
6324:=
6321:S
6287:.
6280:)
6274:)
6269:i
6265:t
6256:f
6252:t
6248:(
6232:f
6228:x
6222:i
6218:x
6214:2
6208:)
6203:i
6199:t
6190:f
6186:t
6182:(
6170:)
6165:2
6160:f
6156:x
6152:+
6147:2
6142:i
6138:x
6134:(
6128:(
6121:m
6116:2
6113:1
6108:=
6098:t
6095:d
6090:)
6084:2
6080:x
6074:2
6066:m
6060:2
6057:1
6046:2
6036:x
6029:m
6023:2
6020:1
6013:(
6005:f
6001:t
5993:i
5989:t
5980:=
5977:t
5974:d
5968:L
5959:f
5955:t
5947:i
5943:t
5934:=
5925:c
5921:S
5890:.
5884:)
5879:i
5875:t
5866:f
5862:t
5858:(
5844:)
5839:i
5835:t
5828:t
5825:(
5808:f
5804:x
5800:+
5794:)
5789:i
5785:t
5776:f
5772:t
5768:(
5754:)
5751:t
5743:f
5739:t
5735:(
5718:i
5714:x
5710:=
5707:)
5704:t
5701:(
5696:c
5692:x
5673:c
5670:S
5666:S
5661:)
5659:t
5657:(
5651:t
5649:(
5647:c
5644:x
5640:t
5638:(
5636:x
5631:)
5629:t
5627:(
5625:x
5607:.
5602:2
5598:x
5592:2
5584:m
5578:2
5575:1
5564:2
5554:x
5547:m
5541:2
5538:1
5532:=
5527:L
5484:.
5479:t
5469:2
5464:2
5451:=
5446:t
5435:t
5423:i
5410:K
5405:t
5403:ψ
5386:,
5383:x
5378:D
5370:S
5367:i
5363:e
5357:y
5354:=
5351:)
5348:t
5345:(
5342:x
5337:x
5334:=
5331:)
5328:0
5325:(
5322:x
5314:)
5311:x
5308:(
5303:0
5292:=
5289:x
5286:d
5282:)
5279:t
5276:;
5273:y
5267:x
5264:(
5261:K
5258:)
5255:x
5252:(
5247:0
5236:=
5233:)
5230:y
5227:(
5222:t
5204:K
5187:.
5184:K
5179:2
5174:2
5164:i
5161:=
5158:)
5155:T
5152:;
5149:x
5146:(
5143:K
5137:t
5134:d
5130:d
5102:,
5095:T
5092:2
5085:2
5081:)
5077:y
5071:x
5068:(
5065:i
5059:e
5052:)
5049:T
5046:;
5043:y
5037:x
5034:(
5031:K
5014:ε
5009:i
4992:.
4989:K
4984:2
4979:2
4969:=
4966:)
4963:T
4960:;
4957:x
4954:(
4951:K
4945:t
4942:d
4938:d
4907:=
4904:y
4901:d
4897:)
4894:T
4891:;
4888:y
4882:x
4879:(
4876:K
4835:.
4827:T
4824:2
4817:2
4813:)
4809:y
4803:x
4800:(
4790:e
4783:)
4780:T
4777:;
4774:y
4768:x
4765:(
4762:K
4749:K
4732:.
4725:2
4719:2
4715:p
4711:T
4701:e
4697:=
4694:)
4691:T
4688:;
4685:p
4682:(
4673:K
4644:,
4637:2
4631:2
4627:p
4613:e
4609:=
4606:)
4603:p
4600:(
4585:G
4568:G
4551:.
4542:/
4538:T
4534:)
4530:p
4527:(
4512:G
4505:=
4502:)
4499:T
4496:;
4493:p
4490:(
4481:K
4460:ε
4456:/
4452:T
4430:,
4421:G
4402:G
4389:G
4385:=
4382:)
4379:T
4376:;
4373:y
4367:x
4364:(
4361:K
4347:ε
4345:G
4337:ε
4333:)
4331:t
4329:(
4327:x
4322:)
4320:ε
4316:t
4314:(
4312:x
4307:ε
4303:D
4286:,
4283:x
4278:D
4271:)
4262:2
4257:)
4248:)
4245:t
4242:(
4239:x
4233:)
4227:+
4224:t
4221:(
4218:x
4212:(
4204:2
4201:1
4191:(
4179:t
4169:y
4166:=
4163:)
4160:T
4157:(
4154:x
4149:x
4146:=
4143:)
4140:0
4137:(
4134:x
4126:=
4123:)
4120:T
4117:;
4114:y
4111:,
4108:x
4105:(
4102:K
4076:.
4073:x
4068:D
4061:)
4057:t
4053:d
4046:2
4041:2
4031:x
4020:T
4015:0
4003:(
3991:y
3988:=
3985:)
3982:T
3979:(
3976:x
3971:x
3968:=
3965:)
3962:0
3959:(
3956:x
3948:=
3945:)
3942:T
3939:;
3936:y
3930:x
3927:(
3924:K
3911:i
3891:,
3888:t
3884:d
3877:2
3872:2
3862:x
3850:=
3847:S
3832:ħ
3825:m
3820:y
3816:x
3805:.
3773:)
3770:)
3767:t
3764:(
3754:x
3747:,
3744:)
3741:t
3738:(
3734:x
3730:(
3727:L
3723:t
3720:d
3714:=
3711:]
3701:x
3694:,
3690:x
3686:[
3683:S
3660:S
3640:Z
3620:x
3617:=
3614:)
3611:0
3608:(
3604:x
3582:x
3560:x
3554:D
3528:)
3525:)
3522:t
3519:(
3515:x
3511:(
3506:0
3496:]
3486:x
3479:,
3475:x
3471:[
3468:S
3465:i
3461:e
3455:x
3449:D
3442:x
3439:=
3436:)
3433:0
3430:(
3426:x
3415:Z
3412:1
3407:=
3404:)
3401:t
3398:,
3395:x
3392:(
3361:2
3358:/
3352:j
3348:x
3343:j
3341:x
3332:j
3327:j
3319:n
3313:1
3304:n
3302:x
3297:1
3294:x
3271:)
3266:)
3262:j
3259:,
3248:1
3242:j
3238:x
3229:j
3225:x
3218:,
3213:j
3203:x
3195:(
3191:L
3186:1
3183:+
3180:n
3175:1
3172:=
3169:j
3153:i
3147:(
3114:)
3111:x
3108:(
3105:V
3097:2
3092:|
3081:x
3074:|
3070:m
3065:2
3062:1
3057:=
3054:V
3048:T
3045:=
3042:)
3033:x
3027:,
3024:x
3021:(
3018:L
3001:L
2993:b
2991:x
2986:b
2984:t
2979:a
2977:x
2972:a
2970:t
2962:a
2960:t
2955:a
2953:x
2948:b
2946:t
2941:b
2939:x
2926:n
2918:n
2914:j
2909:j
2903:)
2901:t
2899:(
2897:ẋ
2893:v
2888:)
2886:t
2884:(
2882:x
2877:)
2875:v
2871:x
2869:(
2867:L
2849:,
2844:n
2840:x
2836:d
2826:0
2822:x
2818:d
2813:)
2809:t
2806:d
2800:)
2795:)
2792:t
2789:(
2786:v
2783:,
2780:)
2777:t
2774:(
2771:x
2766:(
2761:L
2754:b
2750:t
2742:a
2738:t
2724:i
2718:(
2703:+
2679:+
2633:.
2627:1
2624:+
2621:n
2614:a
2610:t
2601:b
2597:t
2590:=
2587:t
2581:=
2563:n
2559:j
2551:j
2547:t
2542:j
2540:t
2533:n
2513:b
2509:t
2505:=
2500:1
2497:+
2494:n
2490:t
2481:n
2477:t
2468:1
2462:n
2458:t
2443:1
2439:t
2430:0
2426:t
2422:=
2417:a
2413:t
2398:b
2396:t
2391:b
2389:x
2384:a
2382:t
2377:a
2375:x
2258:S
2253:e
2206:ħ
2189:h
2185:q
2173:T
2166:∫
2160:F
2156:q
2152:F
2147:k
2145:q
2141:F
2137:F
2132:k
2130:q
2125:k
2123:q
2118:e
2113:h
2111:/
2109:F
2105:h
2100:k
2098:q
2094:q
2090:h
2084:t
2080:q
2075:m
2071:q
2067:2
2064:q
2060:1
2057:q
2052:T
2048:q
2043:F
2038:e
2021:)
2019:t
2017:(
2015:q
2008:q
2002:S
1998:H
1993:ε
1989:t
1984:t
1967:,
1962:S
1956:i
1952:e
1938:S
1927:H
1923:q
1919:p
1915:q
1911:p
1904:p
1900:q
1896:H
1887:q
1868:,
1863:)
1857:+
1854:t
1851:(
1848:q
1845:p
1842:i
1838:e
1804:,
1799:)
1796:q
1793:,
1790:p
1787:(
1784:H
1778:i
1771:e
1753:t
1749:p
1743:)
1741:t
1739:(
1737:p
1732:)
1730:t
1728:(
1726:q
1717:)
1715:t
1713:(
1711:q
1693:,
1688:)
1685:t
1682:(
1679:q
1676:p
1673:i
1666:e
1635:.
1628:]
1623:)
1620:q
1617:,
1614:p
1611:(
1608:H
1597:)
1592:)
1589:t
1586:(
1583:q
1577:)
1571:+
1568:t
1565:(
1562:q
1557:(
1552:p
1547:[
1542:i
1538:e
1524:q
1520:p
1516:q
1512:p
1508:p
1504:q
1493:q
1489:p
1472:,
1466:p
1458:H
1449:=
1440:q
1411:,
1408:H
1393:q
1387:p
1381:=
1378:L
1362:)
1360:ε
1356:t
1354:(
1352:q
1331:q
1305:,
1293:q
1282:L
1273:=
1270:p
1244:L
1233:)
1228:)
1225:t
1222:(
1219:q
1213:)
1207:+
1204:t
1201:(
1198:q
1193:(
1188:)
1185:t
1182:(
1179:p
1176:=
1173:H
1102:i
1100:−
882:e
875:t
868:v
96:|
86:H
80:=
70:|
63:t
60:d
56:d
48:i
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.