5309:
4855:
5304:{\displaystyle {\begin{aligned}\sigma ^{\mu }\partial _{\mu }^{\prime }\psi _{\rm {R}}^{\prime }\left(x^{\prime }\right)&=\sigma ^{\mu }{\frac {\partial }{\partial x^{\prime \mu }}}\psi _{\rm {R}}^{\prime }\left(x^{\prime }\right)\\&=\sigma ^{\mu }{\frac {\partial x^{\nu }}{\partial x^{\prime \mu }}}{\frac {\partial }{\partial x^{\nu }}}R\psi _{\rm {R}}(x)\\&=\sigma ^{\mu }{\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }{\frac {\partial }{\partial x^{\nu }}}R\psi _{\rm {R}}(x)\\&=\sigma ^{\mu }{\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }\partial _{\nu }R\psi _{\rm {R}}(x)\end{aligned}}}
6076:
42:
5712:
4393:
8351:
3819:
6071:{\displaystyle {\begin{aligned}\sigma ^{\mu }{\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }\partial _{\nu }R\psi _{\rm {R}}(x)&=\sigma ^{\mu }{\left(\Lambda ^{-1{\mathsf {T}}}\right)_{\mu }}^{\nu }\partial _{\nu }R\psi _{\rm {R}}(x)\\&=\sigma _{\mu }{\Lambda ^{\mu }}_{\nu }\partial ^{\nu }R\psi _{\rm {R}}(x)\\&=\left(S^{-1}\right)^{\dagger }\sigma _{\mu }\partial ^{\mu }S^{-1}R\psi _{\rm {R}}(x)\end{aligned}}}
2962:
4175:
8077:
3597:
1587:
2402:
1750:
2715:
7897:
7748:
7050:
4388:{\displaystyle {\overline {\sigma }}^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi _{\rm {L}}(x)\mapsto {\overline {\sigma }}^{\mu }{\frac {\partial }{\partial x^{\prime \mu }}}\psi _{\rm {L}}^{\prime }\left(x^{\prime }\right)=S{\overline {\sigma }}^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi _{\rm {L}}(x)}
8346:{\displaystyle D_{\rm {R}}D_{\rm {L}}=\left(i\sigma ^{\mu }\partial _{\mu }+\eta m\omega K\right)\left(i{\overline {\sigma }}^{\mu }\partial _{\mu }+\zeta m\omega K\right)=-\left(\partial _{t}^{2}-{\vec {\nabla }}\cdot {\vec {\nabla }}+\eta \zeta ^{*}m^{2}\right)=-\left(\square +\eta \zeta ^{*}m^{2}\right)}
3259:
3814:{\displaystyle \sigma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi _{\rm {R}}(x)\mapsto \sigma ^{\mu }{\frac {\partial }{\partial x^{\prime \mu }}}\psi _{\rm {R}}^{\prime }\left(x^{\prime }\right)=\left(S^{-1}\right)^{\dagger }\sigma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}\psi _{\rm {R}}(x)}
10527:
The results presented here are identical to those of Aste (2010) equations 52 and 57, although the derivation performed here is completely different. The double-covering used here is also identical to Aste's equation 48, and to the current version (December 2020) of the
Knowledge article on
10432:
is again a pair of Weyl spinors, but this time arranged so that the left-handed spinor is the charge conjugate of the right-handed spinor. The result is a field with two less degrees of freedom than the Dirac spinor. It is unable to interact with the electromagnetic field, since it transforms as a
7556:
10015:, which can be taken to be a pair of Weyl spinors, one left-handed, and one right-handed. These are coupled together in such a way as to represent an electrically charged fermion field. The electric charge arises because the Dirac field transforms under the action of the complexified spin group
7199:. The Majorana equation is conventionally written as a four-component real equation, rather than a two-component complex equation; the above can be brought into four-component form (see that article for details). Similarly, the left-chiral Majorana equation (including an arbitrary phase factor
3436:
9607:
for spacetime; two different points on the fiber are related by a (Lorentz) boost/rotation, that is, by a local change of coordinates. The natural inhabitants of the spin structure are the Weyl spinors, in that the spin structure completely describes how the spinors behave under (Lorentz)
1430:
4533:
2221:
1598:
9872:
3131:
2066:
6885:
7332:
2957:{\displaystyle \left({\bar {\sigma }}^{\nu }p_{\nu }\right)\left(\sigma ^{\mu }p_{\mu }\right)\chi =\left(\sigma ^{\nu }p_{\nu }\right)\left({\bar {\sigma }}^{\mu }p_{\mu }\right)\chi =p_{\mu }p^{\mu }\chi =\left(E^{2}-{\vec {p}}\cdot {\vec {p}}\right)\chi =0}
8699:
5701:
2703:
6756:
6256:
4642:
3959:
7759:
10190:
8602:
5415:
4164:
7587:
2583:
7190:
1304:
eventually confirmed the existence of neutrino oscillations, and their non-zero mass. This discovery confirmed that Weyl's equation cannot completely describe the propagation of neutrinos, as the equations can only describe massless particles.
6896:
7416:
Thus, the
Majorana equation can be read as an equation that connects a spinor to its charge-conjugate form. The two distinct phases on the mass term are related to the two distinct eigenvalues of the charge conjugation operator; see
3142:
6552:
4844:
7430:
7981:
10511:, constructed much as the Majorana spinor, except with an additional minus sign between the charge-conjugate pair. This again renders it electrically neutral, but introduces a number of other quite surprising properties.
3348:
9303:
2472:
2202:
1582:{\displaystyle I_{2}{\frac {1}{c}}{\frac {\partial \psi }{\partial t}}+\sigma _{x}{\frac {\partial \psi }{\partial x}}+\sigma _{y}{\frac {\partial \psi }{\partial y}}+\sigma _{z}{\frac {\partial \psi }{\partial z}}=0}
9202:
4404:
6619:
2397:{\displaystyle \psi \left(\mathbf {r} ,t\right)={\begin{pmatrix}\psi _{1}\\\psi _{2}\\\end{pmatrix}}=\chi e^{-i(\mathbf {k} \cdot \mathbf {r} -\omega t)}=\chi e^{-i(\mathbf {p} \cdot \mathbf {r} -Et)/\hbar }}
4758:
2132:
5472:
1944:
1745:{\displaystyle \sigma ^{\mu }={\begin{pmatrix}\sigma ^{0}&\sigma ^{1}&\sigma ^{2}&\sigma ^{3}\end{pmatrix}}={\begin{pmatrix}I_{2}&\sigma _{x}&\sigma _{y}&\sigma _{z}\end{pmatrix}}}
9747:
3032:
1955:
1395:
10071:
9439:, there are only two such spinors possible, by convention labelled "left" and "right", as described above. A more formal, general presentation of Weyl spinors can be found in the article on the
6767:
5717:
4860:
3147:
7384:
4693:
6313:
10367:
5544:
3589:
6468:
6424:
5597:
10471:
3537:
7225:
6369:
4061:
10308:
10237:
3481:
8610:
5605:
9379:
7892:{\displaystyle \psi _{\rm {L}}\mapsto \psi _{\rm {L}}^{\prime }=\left(S^{\dagger }\right)^{-1}\psi _{\rm {L}}\qquad \psi _{\rm {R}}\mapsto \psi _{\rm {R}}^{\prime }=S\psi _{\rm {R}}}
4008:
2594:
8069:
8028:
6658:
6158:
4544:
3861:
9708:
9517:
10079:
9564:
9042:
8471:
7743:{\displaystyle D_{\rm {L}}\mapsto D_{\rm {L}}^{\prime }=SD_{\rm {L}}S^{\dagger }\qquad D_{\rm {R}}\mapsto D_{\rm {R}}^{\prime }=\left(S^{\dagger }\right)^{-1}D_{\rm {R}}S^{-1}}
9742:
8522:
7089:
5320:
4069:
10502:
10268:
10001:
9969:
8503:
8390:
7045:{\displaystyle m\omega \psi _{\rm {R}}^{*}(x)\mapsto m\omega \psi _{\rm {R}}^{\prime *}\left(x^{\prime }\right)=\left(S^{\dagger }\right)^{-1}m\omega \psi _{\rm {R}}^{*}(x)}
2483:
10423:
7096:
3849:
3340:
3318:
3020:
2995:
9433:
1858:
6121:
1269:
Neutrinos were experimentally observed in 1956 as particles with extremely small masses (and historically were even sometimes thought to be massless). The same year the
9913:
9075:
8726:
8431:
3293:
8933:
7414:
9109:
8966:
7217:
1813:
8998:
8878:
8804:
6646:
6322:
The Weyl equation is conventionally interpreted as describing a massless particle. However, with a slight alteration, one may obtain a two-component version of the
1878:
1787:
6150:
9594:
9467:
8904:
3254:{\displaystyle {\begin{aligned}\sigma ^{\mu }\partial _{\mu }\psi _{\rm {R}}&=0\\{\bar {\sigma }}^{\mu }\partial _{\mu }\psi _{\rm {L}}&=0\end{aligned}}}
9937:
9667:
8846:
7579:
1418:
7551:{\displaystyle D_{\rm {L}}=i{\overline {\sigma }}^{\mu }\partial _{\mu }+\zeta m\omega K\qquad D_{\rm {R}}=i\sigma ^{\mu }\partial _{\mu }+\eta m\omega K}
6479:
4769:
3431:{\displaystyle \mathbf {p} \cdot \mathbf {J} \left|\mathbf {p} ,\lambda \right\rangle =\lambda |\mathbf {p} |\left|\mathbf {p} ,\lambda \right\rangle }
7908:
9213:
2414:
2137:
9117:
1128:
4528:{\displaystyle \psi _{\rm {L}}(x)\mapsto \psi _{\rm {L}}^{\prime }\left(x^{\prime }\right)=\left(S^{\dagger }\right)^{-1}\psi _{\rm {L}}(x)~.}
10508:
9631:; the Clifford bundle is the space in which the spinors live. The general exploration of such structures and their relationships is termed
7055:
which is exactly the same
Lorentz covariance property noted earlier. Thus, the linear combination, using an arbitrary complex phase factor
2215:
solutions to the Weyl equation are referred to as the left and right handed Weyl spinors, each is with two components. Both have the form
211:
6560:
4701:
4538:
Proof: Neither of these transformation properties are in any way "obvious", and so deserve a careful derivation. Begin with the form
2082:
251:
9623:; this is effectively "the same thing" as the normal connection, just with spin indexes attached to it in a consistent fashion. The
5426:
1894:
8772:
9867:{\displaystyle \mathrm {End} (\mathbb {C} ^{N/2})\oplus \mathrm {End} (\mathbb {C} ^{N/2})=:\Delta _{n}^{+}\oplus \Delta _{n}^{-}}
3126:{\displaystyle |\mathbf {p} |=\hbar |\mathbf {k} |={\frac {\hbar \omega }{c}}\,\Rightarrow \,|\mathbf {k} |={\frac {\omega }{c}}}
2061:{\displaystyle {\bar {\sigma }}^{\mu }={\begin{pmatrix}I_{2}&-\sigma _{x}&-\sigma _{y}&-\sigma _{z}\end{pmatrix}}~.}
11097:
9446:
The abstract, general-relativistic form of the Weyl equation can be understood as follows: given a pseudo-Riemannian manifold
11086:
11065:
11021:
11002:
10894:
10869:
1333:
6880:{\displaystyle \psi _{\rm {R}}^{*}(x)\mapsto \psi _{\rm {R}}^{\prime *}\left(x^{\prime }\right)=S^{*}\psi _{\rm {R}}^{*}(x)}
1354:
10018:
986:
9644:
1211:
Mathematically, any Dirac fermion can be decomposed as two Weyl fermions of opposite chirality coupled by the mass term.
7340:
7337:
As noted earlier, the left and right chiral versions are related by a parity transformation. The skew complex conjugate
4650:
6264:
1121:
10313:
7327:{\displaystyle i{\overline {\sigma }}^{\mu }\partial _{\mu }\psi _{\rm {L}}(x)+\zeta m\omega \psi _{\rm {L}}^{*}(x)=0}
5480:
3542:
11042:
10921:
10561:
6432:
6381:
5553:
10436:
3500:
6332:
4169:
Thus, if the untransformed differential vanishes in one
Lorentz frame, then it also vanishes in another. Similarly
4024:
1756:
756:
8694:{\displaystyle {\mathcal {L}}=i\psi _{\rm {L}}^{\dagger }{\bar {\sigma }}^{\mu }\partial _{\mu }\psi _{\rm {L}}~.}
5696:{\displaystyle {\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }={\left(\Lambda ^{-1{\mathsf {T}}}\right)_{\mu }}^{\nu }}
54:
10277:
10198:
2698:{\displaystyle {\bar {\sigma }}^{\mu }p_{\mu }\chi =\left(I_{2}E+{\vec {\sigma }}\cdot {\vec {p}}\right)\chi =0}
1285:
in 1958. As experiments showed no signs of a neutrino mass, interest in the Weyl equation resurfaced. Thus, the
6751:{\displaystyle \psi _{\rm {R}}(x)\mapsto \psi _{\rm {R}}^{\prime }\left(x^{\prime }\right)=S\psi _{\rm {R}}(x)}
6251:{\displaystyle \psi _{\rm {L}}(x)\mapsto \psi _{\rm {L}}^{\prime }\left(x^{\prime }\right)=L\psi _{\rm {L}}(x)}
4637:{\displaystyle \psi _{\rm {R}}(x)\mapsto \psi _{\rm {R}}^{\prime }\left(x^{\prime }\right)=R\psi _{\rm {R}}(x)}
3954:{\displaystyle \psi _{\rm {R}}(x)\mapsto \psi _{\rm {R}}^{\prime }\left(x^{\prime }\right)=S\psi _{\rm {R}}(x)}
3444:
1232:
9315:
11255:
3967:
1114:
319:
168:
128:
10185:{\displaystyle \mathrm {Spin} ^{\mathbb {C} }(p,q)\cong \mathrm {Spin} (p,q)\times _{\mathbb {Z} _{2}}S^{1}}
8033:
7992:
2071:
The solutions of the right- and left-handed Weyl equations are different: they have right- and left-handed
856:
786:
158:
9672:
9476:
7581:
is a short-hand reminder to take the complex conjugate. Under
Lorentz transformations, these transform as
8744:. This is closely related to the solutions given above, and gives a natural geometric interpretation to
8597:{\displaystyle {\mathcal {L}}=i\psi _{\rm {R}}^{\dagger }\sigma ^{\mu }\partial _{\mu }\psi _{\rm {R}}~,}
5410:{\displaystyle \sigma _{\mu }{\Lambda ^{\mu }}_{\nu }=\left(S^{-1}\right)^{\dagger }\sigma _{\nu }S^{-1}}
4159:{\displaystyle \sigma _{\mu }{\Lambda ^{\mu }}_{\nu }=\left(S^{-1}\right)^{\dagger }\sigma _{\nu }S^{-1}}
3270:
2072:
1325:
1282:
314:
256:
17:
9529:
9007:
1169:. The Weyl fermions are one of the three possible types of elementary fermions, the other two being the
11275:
8826:
8765:
8440:
2578:{\displaystyle \sigma ^{\mu }p_{\mu }\chi =\left(I_{2}E-{\vec {\sigma }}\cdot {\vec {p}}\right)\chi =0}
1154:
163:
9713:
7058:
966:
10729:
10476:
10242:
9974:
9942:
9382:
8434:
7185:{\displaystyle i\sigma ^{\mu }\partial _{\mu }\psi _{\rm {R}}(x)+\eta m\omega \psi _{\rm {R}}^{*}(x)}
3296:
304:
9627:
can be defined in terms of the connection in an entirely conventional way. It acts naturally on the
8476:
8359:
5420:
a few indexes must be raised and lowered. This is easier said than done, as it invokes the identity
10372:
9524:
9309:
8818:. This is entirely analogous to how one might talk about a vector, and how it transforms under the
1197:
791:
421:
266:
231:
10473:
group. That is, it transforms as a spinor, but transversally, such that it is invariant under the
3827:
3323:
3301:
3003:
2978:
781:
9388:
1822:
1066:
536:
344:
339:
201:
6084:
9877:
9047:
7902:
just as above. Thus, the matched combinations of these are
Lorentz covariant, and one may take
3495:
936:
916:
416:
349:
261:
226:
10011:
There are three important special cases that can be constructed from Weyl spinors. One is the
9385:, thus allowing these abstractly defined formal structures to be interpreted as fermions. For
8711:
8395:
3278:
9604:
8909:
7393:
1235:. Hermann Weyl published his equation in 1929 as a simplified version of the Dirac equation.
826:
686:
382:
246:
108:
9084:
8941:
7202:
1792:
1196:
might be a Weyl fermion (it is now expected to be either a Dirac or a
Majorana fermion). In
11203:
10832:
10751:
10663:
10605:
9624:
9436:
8971:
8851:
8789:
6624:
6327:
4019:
3023:
1863:
1765:
1321:
1297:
1189:
1146:
1071:
851:
387:
241:
33:
10763:
6126:
153:
8:
11077:
9573:
9520:
8761:
4011:
3852:
1329:
1181:
1081:
411:
11207:
10836:
10755:
10667:
10609:
9449:
8886:
7195:
transforms in a covariant fashion; setting this to zero gives the complex two-component
1348:
The Weyl equation comes in two forms. The right-handed form can be written as follows:
796:
526:
11227:
11193:
10775:
10741:
10629:
10595:
9922:
9652:
9078:
8831:
8783:
are applied, the form of the equation itself does not change. However, the form of the
8757:
8514:
7564:
7418:
7387:
1403:
1281:, addressing Pauli's criticism. This was followed by the measurement of the neutrino's
1091:
1016:
946:
831:
816:
746:
711:
701:
661:
576:
466:
426:
216:
178:
133:
121:
98:
93:
10694:
10651:
11219:
11147:
11139:
11082:
11061:
11038:
11017:
10998:
10955:
10917:
10890:
10865:
10779:
10767:
10699:
10681:
10633:
10621:
10557:
9616:
8705:
7196:
6649:
6323:
3491:
1313:
1243:. However, three years before, Pauli had predicted the existence of a new elementary
1046:
976:
861:
706:
641:
611:
581:
441:
436:
281:
206:
196:
88:
8752:. This general setting has multiple strengths: it clarifies their interpretation as
6547:{\displaystyle \omega =i\sigma _{2}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}}
6426:
The symplectic group is defined as the set of all complex 2Ă2 matrices that satisfy
766:
11231:
11211:
11129:
10945:
10840:
10759:
10689:
10671:
10613:
10271:
8811:
8505:
These two
Majorana operators are thus "square roots" of the KleinâGordon operator.
6376:
6152:
Performing the same manipulations for the left-handed equation, one concludes that
5547:
1337:
1301:
1293:
1278:
1274:
1240:
1174:
1086:
1001:
981:
951:
901:
726:
721:
716:
601:
586:
516:
221:
148:
138:
73:
10583:
4839:{\displaystyle x^{\nu }={\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }x^{\prime \mu }}
11055:
11032:
10992:
10911:
10823:
10429:
9628:
9620:
8741:
1760:
1259:
1031:
1026:
996:
991:
961:
921:
891:
871:
776:
671:
666:
631:
616:
591:
571:
551:
481:
471:
392:
354:
271:
236:
68:
46:
41:
11245:
9597:
8881:
8819:
7976:{\displaystyle D_{\rm {L}}\psi _{\rm {L}}=0\qquad D_{\rm {R}}\psi _{\rm {R}}=0}
1816:
1421:
1317:
1309:
1286:
1263:
1236:
1220:
1205:
1185:
1056:
1006:
926:
911:
896:
876:
846:
841:
836:
821:
806:
771:
761:
691:
626:
596:
531:
496:
486:
461:
332:
299:
173:
11250:
811:
11269:
11143:
10959:
10845:
10818:
10771:
10685:
10625:
10529:
9632:
9612:
9308:
When properly examined in light of the
Clifford algebra, these are naturally
9298:{\displaystyle w_{j}^{*}={\frac {1}{\sqrt {2}}}\left(e_{2j}-ie_{2j+1}\right)}
8936:
8776:
4015:
1270:
1201:
1170:
1061:
1051:
1041:
1036:
956:
941:
741:
736:
566:
561:
521:
491:
456:
446:
377:
372:
10817:
Wu, C. S.; Ambler, E.; Hayward, R. W.; Hoppes, D. D.; Hudson, R. P. (1957).
11223:
11181:
11151:
10703:
10647:
10012:
9567:
9470:
9001:
1881:
1166:
1096:
1021:
906:
881:
866:
801:
681:
676:
556:
546:
541:
506:
501:
476:
431:
309:
276:
103:
83:
11256:
http://www.tfkp.physik.uni-erlangen.de/download/research/DW-derivation.pdf
10676:
3136:
The equation can be written in terms of left and right handed spinors as:
2467:{\displaystyle \chi ={\begin{pmatrix}\chi _{1}\\\chi _{2}\\\end{pmatrix}}}
2197:{\displaystyle {\bar {\sigma }}^{\mu }\partial _{\mu }\psi _{\rm {L}}=0~.}
2079:, respectively. It is convenient to indicate this explicitly, as follows:
10799:
9197:{\displaystyle w_{j}={\frac {1}{\sqrt {2}}}\left(e_{2j}+ie_{2j+1}\right)}
2998:
1076:
1011:
971:
931:
886:
751:
656:
651:
636:
621:
606:
11260:
11160:
10950:
10746:
9440:
9381:
This can be happily interpreted as the mathematical realization of the
8815:
6372:
2212:
1252:
1224:
731:
646:
511:
451:
143:
78:
10617:
8810:
entirely, the algebra of the spinors is described by a (complexified)
8740:
is also frequently used in a more general setting, as an element of a
11215:
11134:
11117:
8807:
2076:
8728:) as independent variables, the relevant Weyl equation is obtained.
11198:
8780:
8749:
2973:
1888:. The left-handed form of the Weyl equation is usually written as:
1248:
1228:
1193:
1158:
696:
10600:
9004:. Given this vector space, one can construct the Clifford algebra
6614:{\displaystyle \omega S^{*}=\left(S^{\dagger }\right)^{-1}\omega }
2968:
and concludes that the equations correspond to a particle that is
1289:
was built under the assumption that neutrinos were Weyl fermions.
9603:
Selecting a single point on the fiber corresponds to selecting a
8753:
1885:
1244:
1142:
4753:{\displaystyle x^{\prime \mu }={\Lambda ^{\mu }}_{\nu }x^{\nu }}
8822:, except that now, it has been adapted to the case of spinors.
8784:
8745:
2127:{\displaystyle \sigma ^{\mu }\partial _{\mu }\psi _{\rm {R}}=0}
5467:{\displaystyle \eta \Lambda ^{\mathsf {T}}\eta =\Lambda ^{-1}}
1939:{\displaystyle {\bar {\sigma }}^{\mu }\partial _{\mu }\psi =0}
11180:
Jia, Shuang; Xu, Su-Yang; Hasan, M. Zahid (25 October 2016).
9566:, and so one can identify the spin group fiber-wise with the
2477:
is a momentum-dependent two-component spinor which satisfies
11016:. Manchester Physics (2nd ed.). John Wiley & Sons.
4695:
to be determined. The
Lorentz transform, in coordinates, is
1296:
had proposed in 1957 the possibility of neutrino masses and
9473:
above it, with the spin group as the fiber. The spin group
8775:
under the action of the Lorentz group. This means that, as
2969:
10940:
Aste, Andreas (2010). "A direct road to Majorana fields".
6652:. The right handed field, as noted earlier, transforms as
5550:. The above identity is often used to define the elements
1255:, which eventually was described using the Weyl equation.
1239:
wrote in 1933 against Weyl's equation because it violated
8756:
in physics, and it shows precisely how to define spin in
3275:
The left and right components correspond to the helicity
10819:"Experimental Test of Parity Conservation in Beta Decay"
9596:
When this is done, the resulting structure is called a
8071:
are both Lorentz covariant. The product is explicitly
6890:
Applying the defining relationship, one concludes that
1204:
that behave as Weyl fermions, leading to the notion of
1188:
are Weyl fermions. Previous to the confirmation of the
27:
Relativistic wave equation describing massless fermions
6510:
5491:
3447:
2429:
2257:
1986:
1685:
1620:
10816:
10479:
10439:
10375:
10316:
10280:
10245:
10201:
10082:
10021:
9977:
9945:
9925:
9880:
9750:
9716:
9675:
9655:
9576:
9532:
9479:
9452:
9391:
9318:
9216:
9120:
9087:
9050:
9010:
8974:
8944:
8912:
8889:
8854:
8834:
8792:
8714:
8613:
8525:
8479:
8443:
8398:
8362:
8080:
8036:
7995:
7911:
7762:
7590:
7567:
7433:
7396:
7343:
7228:
7205:
7099:
7061:
6899:
6770:
6661:
6627:
6563:
6482:
6435:
6384:
6335:
6267:
6161:
6129:
6087:
5715:
5608:
5556:
5483:
5429:
5323:
4858:
4772:
4704:
4653:
4547:
4407:
4178:
4072:
4027:
3970:
3864:
3855:, provided that the right-handed field transforms as
3830:
3600:
3545:
3503:
3351:
3326:
3304:
3281:
3145:
3035:
3006:
2981:
2718:
2597:
2486:
2417:
2224:
2140:
2085:
1958:
1897:
1866:
1825:
1795:
1768:
1601:
1433:
1406:
1390:{\displaystyle \sigma ^{\mu }\partial _{\mu }\psi =0}
1357:
10903:
10066:{\displaystyle \mathrm {Spin} ^{\mathbb {C} }(p,q).}
7424:
Define a pair of operators, the Majorana operators,
4010:
is related to the Lorentz transform by means of the
10944:. Vol. 2010, no. 2. pp. 1776â1809.
10554:ITEP Lectures on Particle Physics and Field Theory
10496:
10465:
10417:
10361:
10302:
10262:
10231:
10184:
10065:
9995:
9963:
9931:
9907:
9866:
9736:
9702:
9661:
9588:
9558:
9511:
9461:
9427:
9373:
9297:
9196:
9103:
9069:
9036:
8992:
8960:
8927:
8898:
8872:
8840:
8798:
8720:
8693:
8596:
8497:
8465:
8425:
8384:
8345:
8063:
8022:
7986:as a pair of complex 2-spinor Majorana equations.
7975:
7891:
7742:
7573:
7550:
7408:
7379:{\displaystyle \omega \psi ^{*}=i\sigma ^{2}\psi }
7378:
7326:
7211:
7184:
7083:
7044:
6879:
6750:
6640:
6613:
6546:
6462:
6418:
6363:
6307:
6250:
6144:
6115:
6070:
5695:
5591:
5538:
5466:
5409:
5303:
4838:
4752:
4688:{\displaystyle R\in \mathrm {SL} (2,\mathbb {C} )}
4687:
4636:
4527:
4398:provided that the left-handed field transforms as
4387:
4158:
4055:
4002:
3953:
3843:
3813:
3583:
3531:
3475:
3430:
3334:
3312:
3287:
3253:
3125:
3014:
2989:
2956:
2697:
2577:
2466:
2396:
2196:
2126:
2060:
1938:
1872:
1852:
1807:
1781:
1744:
1581:
1412:
1389:
10793:
10791:
10789:
6761:and so the complex conjugate field transforms as
6308:{\displaystyle L=\left(S^{\dagger }\right)^{-1}.}
11267:
11182:"Weyl semimetals, Fermi arcs and chiral anomaly"
10362:{\displaystyle \mathrm {Spin} (p,q)\times S^{1}}
8814:. The spinors transform under the action of the
5539:{\displaystyle \eta ={\mbox{diag}}(+1,-1,-1,-1)}
3584:{\displaystyle \Lambda \in \mathrm {SO} (1,3)~.}
10909:
10656:Proceedings of the National Academy of Sciences
6463:{\displaystyle S^{\mathsf {T}}\omega S=\omega }
6419:{\displaystyle \mathrm {Sp} (2,\mathbb {C} )~.}
5592:{\displaystyle \Lambda \in \mathrm {SO} (1,3).}
1312:was demonstrated experimentally in crystalline
10786:
10723:
10721:
10719:
10717:
10715:
10713:
10466:{\displaystyle \mathrm {spin} ^{\mathbb {C} }}
10239:is the circle, and can be identified with the
9111:, one may construct a pair of Weyl spinors as
6557:The defining relationship can be rewritten as
3532:{\displaystyle x\mapsto x^{\prime }=\Lambda x}
9915:. A left-handed (respectively, right-handed)
8356:Verifying this requires keeping in mind that
6364:{\displaystyle \mathrm {SL} (2,\mathbb {C} )}
4056:{\displaystyle \mathrm {SL} (2,\mathbb {C} )}
1122:
11246:http://aesop.phys.utk.edu/qft/2004-5/2-2.pdf
10935:
10933:
10310:is just fancy notation denoting the product
9064:
9051:
11251:http://www.nbi.dk/~kleppe/random/ll/l2.html
10853:
10710:
10545:
6317:
3591:More precisely, the equations transform as
11179:
11158:
11030:
11011:
10975:Riemannian Geometry and Geometric Analysis
10887:The Cambridge Handbook of Physics Formulas
10797:
10577:
10575:
10573:
10303:{\displaystyle \times _{\mathbb {Z} _{2}}}
10232:{\displaystyle S^{1}\cong \mathrm {U} (1)}
9638:
8768:. This is informally sketched as follows.
3476:{\textstyle \lambda =\pm {\frac {1}{2}}~.}
2709:By direct manipulation, one obtains that
1129:
1115:
40:
11197:
11133:
10949:
10930:
10844:
10745:
10693:
10675:
10599:
10457:
10288:
10160:
10100:
10039:
9807:
9767:
9718:
9677:
6403:
6354:
4678:
4046:
3993:
3094:
3090:
1336:) also observed Weyl-like excitations in
1262:proposed that Weyl fermions may exist as
11098:"Weyl fermions are spotted at long last"
11095:
10878:
10640:
9374:{\displaystyle w_{j}w_{m}=-w_{m}w_{j}~.}
11159:Vishwanath, Ashvin (8 September 2015).
11074:
11053:
11031:Martin, Brian R.; Shaw, Graham (2013).
10990:
10913:An Introduction to Quantum Field Theory
10859:
10727:
10570:
10551:
8508:
4003:{\displaystyle S\in SL(2,\mathbb {C} )}
2206:
1400:Expanding this equation, and inserting
1328:) teams. Independently, the same year,
14:
11268:
11115:
10977:(3rd ed.). Springer Universitext.
10910:Peskin, M.E.; Schroeder, D.V. (1995).
10730:"The history of neutrino oscillations"
8064:{\displaystyle D_{\rm {R}}D_{\rm {L}}}
8023:{\displaystyle D_{\rm {L}}D_{\rm {R}}}
7753:whereas the Weyl spinors transform as
6442:
6081:One thus regains the original form if
5838:
5669:
5439:
1192:, it was considered possible that the
11261:http://www.weylmann.com/weyldirac.pdf
10864:. Addison Wesley, Prentice Hall Inc.
10810:
5314:In order to make use of the Weyl map
3485:
1334:Massachusetts Institute of Technology
10972:
10939:
10884:
10646:
10584:"Dirac, Majorana, and Weyl fermions"
9939:-dimensional space is an element of
9703:{\displaystyle \mathbb {C} l^{0}(n)}
9512:{\displaystyle \mathrm {Spin} (p,q)}
8513:The equations are obtained from the
3295:of the particles, the projection of
10581:
9645:classification of Clifford algebras
7421:and Majorana equation for details.
24:
10984:
10481:
10451:
10448:
10445:
10442:
10327:
10324:
10321:
10318:
10247:
10216:
10134:
10131:
10128:
10125:
10094:
10091:
10088:
10085:
10033:
10030:
10027:
10024:
9979:
9947:
9850:
9832:
9798:
9795:
9792:
9758:
9755:
9752:
9559:{\displaystyle \mathrm {SO} (p,q)}
9537:
9534:
9490:
9487:
9484:
9481:
9037:{\displaystyle \mathrm {Cl} (p,q)}
9015:
9012:
8679:
8664:
8633:
8616:
8582:
8567:
8545:
8528:
8258:
8243:
8223:
8182:
8127:
8099:
8087:
8055:
8043:
8014:
8002:
7961:
7949:
7930:
7918:
7883:
7866:
7860:
7845:
7832:
7790:
7784:
7769:
7721:
7679:
7673:
7658:
7635:
7618:
7612:
7597:
7524:
7501:
7470:
7440:
7298:
7265:
7250:
7162:
7129:
7114:
7022:
6970:
6953:
6947:
6912:
6857:
6829:
6812:
6806:
6777:
6733:
6712:
6698:
6692:
6668:
6389:
6386:
6340:
6337:
6233:
6212:
6198:
6192:
6168:
6049:
6018:
5954:
5936:
5919:
5881:
5863:
5826:
5786:
5768:
5738:
5657:
5617:
5567:
5564:
5557:
5452:
5434:
5337:
5282:
5264:
5234:
5191:
5166:
5162:
5131:
5088:
5063:
5059:
5046:
5038:
5023:
4991:
4977:
4971:
4954:
4946:
4942:
4914:
4900:
4894:
4883:
4874:
4828:
4794:
4724:
4710:
4664:
4661:
4619:
4598:
4584:
4578:
4554:
4504:
4458:
4444:
4438:
4414:
4370:
4348:
4344:
4310:
4296:
4290:
4273:
4265:
4261:
4224:
4202:
4198:
4086:
4032:
4029:
3936:
3915:
3901:
3895:
3871:
3796:
3774:
3770:
3718:
3704:
3698:
3681:
3673:
3669:
3639:
3617:
3613:
3556:
3553:
3546:
3523:
3515:
3231:
3216:
3176:
3161:
2176:
2161:
2112:
2097:
1918:
1564:
1556:
1531:
1523:
1498:
1490:
1465:
1457:
1369:
25:
11287:
11239:
11096:Johnston, Hamish (23 July 2015).
10798:Vishwanath, Ashvin (2015-09-08).
8748:as geometric objects living on a
8466:{\displaystyle \eta \zeta ^{*}=1}
3054:
2389:
11116:Ciudad, David (20 August 2015).
10994:Quantum Field Theory Demystified
10425:identified (a double covering).
10006:
9737:{\displaystyle \mathbb {C} l(n)}
9710:of the complex Clifford algebra
7084:{\displaystyle \eta =e^{i\phi }}
3413:
3398:
3371:
3361:
3353:
3328:
3306:
3101:
3063:
3042:
3008:
2983:
2972:. As a result, the magnitude of
2368:
2360:
2321:
2313:
2234:
11012:Martin, B.R.; Shaw, G. (2008).
10966:
10860:Pearson, E. Abers, ed. (2004).
10764:10.1088/0031-8949/2005/T121/001
10521:
10497:{\displaystyle \mathrm {U} (1)}
10433:scalar under the action of the
10263:{\displaystyle \mathrm {U} (1)}
9996:{\displaystyle \Delta _{n}^{-}}
9964:{\displaystyle \Delta _{n}^{+}}
8731:
8704:By treating the spinor and its
7942:
7838:
7651:
7494:
1316:(TaAs) by the collaboration of
10889:. Cambridge University Press.
10652:"Gravitation and the electron"
10507:The third special case is the
10491:
10485:
10412:
10394:
10388:
10376:
10343:
10331:
10257:
10251:
10226:
10220:
10150:
10138:
10118:
10106:
10073:This group has the structure
10057:
10045:
9825:
9802:
9785:
9762:
9731:
9725:
9697:
9691:
9553:
9541:
9506:
9494:
9422:
9410:
9404:
9392:
9031:
9019:
8987:
8975:
8867:
8855:
8651:
8498:{\displaystyle \eta =\zeta ~.}
8385:{\displaystyle \omega ^{2}=-1}
8261:
8246:
7851:
7775:
7664:
7603:
7315:
7309:
7277:
7271:
7179:
7173:
7141:
7135:
7039:
7033:
6932:
6929:
6923:
6874:
6868:
6797:
6794:
6788:
6745:
6739:
6683:
6680:
6674:
6407:
6393:
6358:
6344:
6245:
6239:
6183:
6180:
6174:
6061:
6055:
5966:
5960:
5893:
5887:
5798:
5792:
5583:
5571:
5533:
5497:
5294:
5288:
5203:
5197:
5100:
5094:
4682:
4668:
4631:
4625:
4569:
4566:
4560:
4516:
4510:
4429:
4426:
4420:
4382:
4376:
4239:
4236:
4230:
4050:
4036:
3997:
3983:
3948:
3942:
3886:
3883:
3877:
3808:
3802:
3654:
3651:
3645:
3572:
3560:
3507:
3403:
3393:
3203:
3106:
3096:
3091:
3068:
3058:
3047:
3037:
2934:
2919:
2836:
2731:
2675:
2660:
2605:
2555:
2540:
2381:
2356:
2334:
2309:
2148:
1966:
1905:
1233:relativistic quantum mechanics
1231:particles in the framework of
1227:, and was first used to model
1165:. The equation is named after
13:
1:
10539:
10418:{\displaystyle (s,u)=(-s,-u)}
8806:itself does change. Ignoring
1759:whose components are the 2Ă2
1300:, it was not until 1998 that
1200:, some materials can display
169:Spontaneous symmetry breaking
129:Symmetry in quantum mechanics
10556:. Vol. 1. p. 292.
8170:
7458:
7238:
4331:
4248:
4185:
3844:{\displaystyle S^{\dagger }}
3335:{\displaystyle \mathbf {p} }
3313:{\displaystyle \mathbf {J} }
3015:{\displaystyle \mathbf {k} }
2990:{\displaystyle \mathbf {p} }
7:
11161:"Where the Weyl things are"
10800:"Where the Weyl things are"
10588:American Journal of Physics
9428:{\displaystyle (p,q)=(1,3)}
3271:Helicity (particle physics)
3264:
1853:{\displaystyle \mu =1,2,3,}
1343:
1326:Chinese Academy of Sciences
10:
11292:
10504:action of the spin group.
9642:
8827:pseudo-Riemannian manifold
8766:pseudo-Riemannian manifold
6326:. This arises because the
6116:{\displaystyle S^{-1}R=1,}
3268:
1214:
1155:relativistic wave equation
164:Explicit symmetry breaking
11070:– via Google Books.
11057:Supersymmetry Demystified
11047:– via Google Books.
10926:– via Google Books.
10552:Shifman, Mikhail (1999).
9908:{\displaystyle N=2^{n/2}}
9383:Pauli exclusion principle
9070:{\displaystyle \{e_{i}\}}
7386:can be recognized as the
5599:One takes the transpose:
3320:onto the linear momentum
3297:angular momentum operator
1277:could be violated by the
1223:was published in 1928 by
320:BargmannâWigner equations
10846:10.1103/PhysRev.105.1413
10514:
9525:special orthogonal group
9312:, that is, one has that
8721:{\displaystyle \dagger }
8426:{\displaystyle Ki=-iK~.}
6318:Relationship to Majorana
3288:{\displaystyle \lambda }
2997:relates directly to the
1292:While Italian physicist
1198:condensed matter physics
1157:for describing massless
11075:Penrose, Roger (2007).
10582:Pal, Palash B. (2011).
9639:Mathematical definition
8928:{\displaystyle x\in M,}
8880:, one may consider its
8433:The RHS reduces to the
7409:{\displaystyle \psi ~.}
345:Electroweak interaction
340:Quantum electrodynamics
315:WheelerâDeWitt equation
202:Background field method
10885:Woan, G., ed. (2010).
10728:Bilenky, S.M. (2005).
10498:
10467:
10419:
10363:
10304:
10264:
10233:
10186:
10067:
9997:
9965:
9933:
9909:
9868:
9738:
9704:
9669:, the even subalgebra
9663:
9605:local coordinate frame
9590:
9560:
9513:
9463:
9429:
9375:
9299:
9198:
9105:
9104:{\displaystyle T_{x}M}
9071:
9038:
8994:
8962:
8961:{\displaystyle T_{x}M}
8929:
8900:
8874:
8842:
8800:
8760:, or, indeed, for any
8722:
8695:
8598:
8499:
8467:
8427:
8386:
8347:
8065:
8024:
7977:
7893:
7744:
7575:
7552:
7410:
7380:
7328:
7213:
7212:{\displaystyle \zeta }
7186:
7085:
7046:
6881:
6752:
6642:
6615:
6548:
6464:
6420:
6365:
6309:
6252:
6146:
6117:
6072:
5697:
5593:
5540:
5468:
5411:
5305:
4840:
4754:
4689:
4638:
4529:
4389:
4160:
4057:
4004:
3955:
3845:
3815:
3585:
3533:
3496:Lorentz transformation
3477:
3432:
3336:
3314:
3289:
3255:
3127:
3016:
2991:
2958:
2699:
2579:
2468:
2398:
2198:
2128:
2062:
1940:
1874:
1854:
1809:
1808:{\displaystyle \mu =0}
1783:
1746:
1583:
1414:
1391:
350:Quantum chromodynamics
227:Effective field theory
10973:Jost, Jurgen (2002).
10677:10.1073/pnas.15.4.323
10499:
10468:
10420:
10369:with opposite points
10364:
10305:
10265:
10234:
10187:
10068:
9998:
9966:
9934:
9910:
9869:
9739:
9705:
9664:
9591:
9561:
9514:
9464:
9430:
9376:
9300:
9199:
9106:
9072:
9039:
8995:
8993:{\displaystyle (p,q)}
8963:
8930:
8906:. At any given point
8901:
8875:
8873:{\displaystyle (p,q)}
8843:
8801:
8799:{\displaystyle \psi }
8771:The Weyl equation is
8723:
8696:
8599:
8500:
8468:
8435:KleinâGordon operator
8428:
8387:
8348:
8066:
8025:
7978:
7894:
7745:
7576:
7553:
7411:
7381:
7329:
7214:
7187:
7086:
7047:
6882:
6753:
6643:
6641:{\displaystyle S^{*}}
6616:
6549:
6465:
6421:
6366:
6310:
6253:
6147:
6118:
6073:
5698:
5594:
5541:
5469:
5412:
5306:
4841:
4755:
4690:
4639:
4530:
4390:
4161:
4058:
4005:
3956:
3846:
3816:
3586:
3534:
3478:
3433:
3337:
3315:
3290:
3256:
3128:
3017:
2992:
2959:
2700:
2580:
2469:
2399:
2199:
2129:
2063:
1941:
1875:
1873:{\displaystyle \psi }
1855:
1810:
1784:
1782:{\displaystyle I_{2}}
1747:
1584:
1415:
1392:
1298:neutrino oscillations
1266:in condensed matter.
1190:neutrino oscillations
305:KleinâGordon equation
247:LSZ reduction formula
11060:. USA: McGraw-Hill.
11054:LaBelle, P. (2010).
10997:. USA: McGraw-Hill.
10991:McMahon, D. (2008).
10477:
10437:
10373:
10314:
10278:
10243:
10199:
10080:
10019:
9975:
9943:
9923:
9878:
9748:
9714:
9673:
9653:
9625:covariant derivative
9615:, the analog of the
9574:
9530:
9477:
9450:
9437:Minkowski space-time
9389:
9316:
9214:
9118:
9085:
9048:
9008:
8972:
8942:
8910:
8887:
8852:
8832:
8790:
8712:
8611:
8523:
8515:Lagrangian densities
8509:Lagrangian densities
8477:
8441:
8396:
8360:
8078:
8034:
7993:
7909:
7760:
7588:
7565:
7431:
7394:
7341:
7226:
7203:
7097:
7059:
6897:
6768:
6659:
6625:
6561:
6480:
6433:
6382:
6333:
6328:special linear group
6265:
6159:
6145:{\displaystyle R=S.}
6127:
6085:
5713:
5606:
5554:
5481:
5427:
5321:
4856:
4770:
4702:
4651:
4545:
4405:
4176:
4070:
4025:
4020:special linear group
3968:
3862:
3828:
3598:
3543:
3501:
3445:
3349:
3324:
3302:
3279:
3143:
3033:
3024:de Broglie relations
3004:
2979:
2716:
2595:
2484:
2415:
2222:
2207:Plane wave solutions
2138:
2083:
1956:
1895:
1864:
1823:
1793:
1766:
1599:
1431:
1404:
1355:
1322:Princeton University
1182:elementary particles
1147:quantum field theory
388:Theory of everything
242:Lattice field theory
212:Correlation function
34:Quantum field theory
11208:2016NatMa..15.1140J
11118:"Massless yet real"
11078:The Road to Reality
10837:1957PhRv..105.1413W
10756:2005PhST..121...17B
10668:1929PNAS...15..323W
10610:2011AmJPh..79..485P
9992:
9960:
9917:complex Weyl spinor
9863:
9845:
9589:{\displaystyle M~.}
9231:
8825:Given an arbitrary
8762:Riemannian manifold
8643:
8555:
8236:
7870:
7794:
7683:
7622:
7308:
7172:
7032:
6960:
6922:
6867:
6819:
6787:
6702:
6202:
4981:
4904:
4887:
4588:
4448:
4300:
3905:
3853:Hermitian transpose
3708:
3490:Both equations are
1308:In 2015, the first
367:Incomplete theories
10951:10.3390/sym2041776
10916:. Addison-Wesley.
10494:
10463:
10415:
10359:
10300:
10260:
10229:
10182:
10063:
9993:
9978:
9961:
9946:
9929:
9905:
9864:
9849:
9831:
9734:
9700:
9659:
9608:boosts/rotations.
9586:
9556:
9509:
9462:{\displaystyle M,}
9459:
9425:
9371:
9295:
9217:
9194:
9101:
9079:vector space basis
9067:
9034:
8990:
8958:
8925:
8899:{\displaystyle TM}
8896:
8870:
8838:
8796:
8758:General Relativity
8718:
8691:
8627:
8594:
8539:
8495:
8463:
8423:
8382:
8343:
8222:
8061:
8020:
7973:
7889:
7854:
7778:
7740:
7667:
7606:
7571:
7548:
7419:charge conjugation
7406:
7376:
7324:
7292:
7209:
7182:
7156:
7081:
7042:
7016:
6941:
6906:
6877:
6851:
6800:
6771:
6748:
6686:
6638:
6611:
6544:
6538:
6460:
6416:
6361:
6305:
6248:
6186:
6142:
6113:
6068:
6066:
5693:
5589:
5546:is the flat-space
5536:
5495:
5464:
5407:
5301:
5299:
4965:
4888:
4873:
4836:
4763:or, equivalently,
4750:
4685:
4634:
4572:
4525:
4432:
4385:
4284:
4156:
4053:
4000:
3951:
3889:
3841:
3811:
3692:
3581:
3529:
3486:Lorentz invariance
3473:
3428:
3332:
3310:
3285:
3251:
3249:
3123:
3012:
2987:
2954:
2695:
2575:
2464:
2458:
2394:
2286:
2194:
2124:
2058:
2046:
1936:
1884:â one of the Weyl
1870:
1850:
1805:
1779:
1742:
1736:
1671:
1579:
1410:
1387:
1145:, particularly in
252:Partition function
179:Topological charge
99:General relativity
94:Special relativity
11276:Quantum mechanics
11192:(11): 1140â1144.
11088:978-0-679-77631-4
11081:. Vintage Books.
11067:978-0-07-163641-4
11023:978-0-470-03294-7
11004:978-0-07-154382-8
10896:978-0-521-57507-2
10871:978-0-13-146100-0
10862:Quantum Mechanics
10618:10.1119/1.3549729
9932:{\displaystyle n}
9744:is isomorphic to
9662:{\displaystyle n}
9617:metric connection
9582:
9469:one constructs a
9367:
9245:
9244:
9144:
9143:
8841:{\displaystyle M}
8687:
8654:
8590:
8491:
8419:
8264:
8249:
8173:
7574:{\displaystyle K}
7461:
7402:
7241:
7197:Majorana equation
6650:complex conjugate
6412:
6324:Majorana equation
5494:
5180:
5077:
5055:
4963:
4647:for some unknown
4521:
4362:
4334:
4282:
4251:
4216:
4188:
3788:
3690:
3631:
3577:
3492:Lorentz invariant
3469:
3465:
3206:
3121:
3088:
2937:
2922:
2839:
2734:
2678:
2663:
2608:
2558:
2543:
2190:
2151:
2054:
1969:
1908:
1571:
1538:
1505:
1472:
1452:
1413:{\displaystyle c}
1338:photonic crystals
1324:) and H. Ding's (
1314:tantalum arsenide
1251:, to explain the
1175:Majorana fermions
1161:particles called
1139:
1138:
232:Expectation value
207:BRST quantization
154:Poincaré symmetry
109:YangâMills theory
89:Quantum mechanics
16:(Redirected from
11283:
11235:
11216:10.1038/nmat4787
11201:
11186:Nature Materials
11176:
11174:
11172:
11155:
11137:
11135:10.1038/nmat4411
11122:Nature Materials
11112:
11110:
11108:
11092:
11071:
11048:
11037:(3rd ed.).
11034:Particle Physics
11027:
11014:Particle Physics
11008:
10979:
10978:
10970:
10964:
10963:
10953:
10937:
10928:
10927:
10907:
10901:
10900:
10882:
10876:
10875:
10857:
10851:
10850:
10848:
10831:(4): 1413â1415.
10814:
10808:
10807:
10795:
10784:
10783:
10749:
10725:
10708:
10707:
10697:
10679:
10644:
10638:
10637:
10603:
10579:
10568:
10567:
10549:
10533:
10525:
10503:
10501:
10500:
10495:
10484:
10472:
10470:
10469:
10464:
10462:
10461:
10460:
10454:
10424:
10422:
10421:
10416:
10368:
10366:
10365:
10360:
10358:
10357:
10330:
10309:
10307:
10306:
10301:
10299:
10298:
10297:
10296:
10291:
10272:electromagnetism
10269:
10267:
10266:
10261:
10250:
10238:
10236:
10235:
10230:
10219:
10211:
10210:
10191:
10189:
10188:
10183:
10181:
10180:
10171:
10170:
10169:
10168:
10163:
10137:
10105:
10104:
10103:
10097:
10072:
10070:
10069:
10064:
10044:
10043:
10042:
10036:
10002:
10000:
9999:
9994:
9991:
9986:
9970:
9968:
9967:
9962:
9959:
9954:
9938:
9936:
9935:
9930:
9914:
9912:
9911:
9906:
9904:
9903:
9899:
9873:
9871:
9870:
9865:
9862:
9857:
9844:
9839:
9824:
9823:
9819:
9810:
9801:
9784:
9783:
9779:
9770:
9761:
9743:
9741:
9740:
9735:
9721:
9709:
9707:
9706:
9701:
9690:
9689:
9680:
9668:
9666:
9665:
9660:
9595:
9593:
9592:
9587:
9580:
9565:
9563:
9562:
9557:
9540:
9518:
9516:
9515:
9510:
9493:
9468:
9466:
9465:
9460:
9434:
9432:
9431:
9426:
9380:
9378:
9377:
9372:
9365:
9364:
9363:
9354:
9353:
9338:
9337:
9328:
9327:
9304:
9302:
9301:
9296:
9294:
9290:
9289:
9288:
9264:
9263:
9246:
9240:
9236:
9230:
9225:
9203:
9201:
9200:
9195:
9193:
9189:
9188:
9187:
9163:
9162:
9145:
9139:
9135:
9130:
9129:
9110:
9108:
9107:
9102:
9097:
9096:
9076:
9074:
9073:
9068:
9063:
9062:
9043:
9041:
9040:
9035:
9018:
8999:
8997:
8996:
8991:
8967:
8965:
8964:
8959:
8954:
8953:
8934:
8932:
8931:
8926:
8905:
8903:
8902:
8897:
8879:
8877:
8876:
8871:
8847:
8845:
8844:
8839:
8812:Clifford algebra
8805:
8803:
8802:
8797:
8727:
8725:
8724:
8719:
8700:
8698:
8697:
8692:
8685:
8684:
8683:
8682:
8672:
8671:
8662:
8661:
8656:
8655:
8647:
8642:
8637:
8636:
8620:
8619:
8603:
8601:
8600:
8595:
8588:
8587:
8586:
8585:
8575:
8574:
8565:
8564:
8554:
8549:
8548:
8532:
8531:
8504:
8502:
8501:
8496:
8489:
8472:
8470:
8469:
8464:
8456:
8455:
8432:
8430:
8429:
8424:
8417:
8391:
8389:
8388:
8383:
8372:
8371:
8352:
8350:
8349:
8344:
8342:
8338:
8337:
8336:
8327:
8326:
8297:
8293:
8292:
8291:
8282:
8281:
8266:
8265:
8257:
8251:
8250:
8242:
8235:
8230:
8210:
8206:
8190:
8189:
8180:
8179:
8174:
8166:
8155:
8151:
8135:
8134:
8125:
8124:
8104:
8103:
8102:
8092:
8091:
8090:
8070:
8068:
8067:
8062:
8060:
8059:
8058:
8048:
8047:
8046:
8029:
8027:
8026:
8021:
8019:
8018:
8017:
8007:
8006:
8005:
7982:
7980:
7979:
7974:
7966:
7965:
7964:
7954:
7953:
7952:
7935:
7934:
7933:
7923:
7922:
7921:
7898:
7896:
7895:
7890:
7888:
7887:
7886:
7869:
7864:
7863:
7850:
7849:
7848:
7837:
7836:
7835:
7825:
7824:
7816:
7812:
7811:
7793:
7788:
7787:
7774:
7773:
7772:
7749:
7747:
7746:
7741:
7739:
7738:
7726:
7725:
7724:
7714:
7713:
7705:
7701:
7700:
7682:
7677:
7676:
7663:
7662:
7661:
7650:
7649:
7640:
7639:
7638:
7621:
7616:
7615:
7602:
7601:
7600:
7580:
7578:
7577:
7572:
7557:
7555:
7554:
7549:
7532:
7531:
7522:
7521:
7506:
7505:
7504:
7478:
7477:
7468:
7467:
7462:
7454:
7445:
7444:
7443:
7415:
7413:
7412:
7407:
7400:
7388:charge conjugate
7385:
7383:
7382:
7377:
7372:
7371:
7356:
7355:
7333:
7331:
7330:
7325:
7307:
7302:
7301:
7270:
7269:
7268:
7258:
7257:
7248:
7247:
7242:
7234:
7218:
7216:
7215:
7210:
7191:
7189:
7188:
7183:
7171:
7166:
7165:
7134:
7133:
7132:
7122:
7121:
7112:
7111:
7090:
7088:
7087:
7082:
7080:
7079:
7051:
7049:
7048:
7043:
7031:
7026:
7025:
7009:
7008:
7000:
6996:
6995:
6978:
6974:
6973:
6959:
6951:
6950:
6921:
6916:
6915:
6886:
6884:
6883:
6878:
6866:
6861:
6860:
6850:
6849:
6837:
6833:
6832:
6818:
6810:
6809:
6786:
6781:
6780:
6757:
6755:
6754:
6749:
6738:
6737:
6736:
6720:
6716:
6715:
6701:
6696:
6695:
6673:
6672:
6671:
6647:
6645:
6644:
6639:
6637:
6636:
6620:
6618:
6617:
6612:
6607:
6606:
6598:
6594:
6593:
6576:
6575:
6553:
6551:
6550:
6545:
6543:
6542:
6501:
6500:
6469:
6467:
6466:
6461:
6447:
6446:
6445:
6425:
6423:
6422:
6417:
6410:
6406:
6392:
6377:symplectic group
6370:
6368:
6367:
6362:
6357:
6343:
6314:
6312:
6311:
6306:
6301:
6300:
6292:
6288:
6287:
6257:
6255:
6254:
6249:
6238:
6237:
6236:
6220:
6216:
6215:
6201:
6196:
6195:
6173:
6172:
6171:
6151:
6149:
6148:
6143:
6122:
6120:
6119:
6114:
6100:
6099:
6077:
6075:
6074:
6069:
6067:
6054:
6053:
6052:
6039:
6038:
6026:
6025:
6016:
6015:
6006:
6005:
6000:
5996:
5995:
5972:
5959:
5958:
5957:
5944:
5943:
5934:
5933:
5928:
5927:
5926:
5915:
5914:
5899:
5886:
5885:
5884:
5871:
5870:
5861:
5860:
5855:
5854:
5853:
5848:
5844:
5843:
5842:
5841:
5817:
5816:
5791:
5790:
5789:
5776:
5775:
5766:
5765:
5760:
5759:
5758:
5753:
5749:
5748:
5729:
5728:
5702:
5700:
5699:
5694:
5692:
5691:
5686:
5685:
5684:
5679:
5675:
5674:
5673:
5672:
5645:
5644:
5639:
5638:
5637:
5632:
5628:
5627:
5598:
5596:
5595:
5590:
5570:
5548:Minkowski metric
5545:
5543:
5542:
5537:
5496:
5492:
5473:
5471:
5470:
5465:
5463:
5462:
5444:
5443:
5442:
5416:
5414:
5413:
5408:
5406:
5405:
5393:
5392:
5383:
5382:
5377:
5373:
5372:
5352:
5351:
5346:
5345:
5344:
5333:
5332:
5310:
5308:
5307:
5302:
5300:
5287:
5286:
5285:
5272:
5271:
5262:
5261:
5256:
5255:
5254:
5249:
5245:
5244:
5225:
5224:
5209:
5196:
5195:
5194:
5181:
5179:
5178:
5177:
5161:
5159:
5158:
5153:
5152:
5151:
5146:
5142:
5141:
5122:
5121:
5106:
5093:
5092:
5091:
5078:
5076:
5075:
5074:
5058:
5056:
5054:
5053:
5052:
5036:
5035:
5034:
5021:
5019:
5018:
5003:
4999:
4995:
4994:
4980:
4975:
4974:
4964:
4962:
4961:
4960:
4941:
4939:
4938:
4922:
4918:
4917:
4903:
4898:
4897:
4886:
4881:
4872:
4871:
4845:
4843:
4842:
4837:
4835:
4834:
4822:
4821:
4816:
4815:
4814:
4809:
4805:
4804:
4782:
4781:
4759:
4757:
4756:
4751:
4749:
4748:
4739:
4738:
4733:
4732:
4731:
4717:
4716:
4694:
4692:
4691:
4686:
4681:
4667:
4643:
4641:
4640:
4635:
4624:
4623:
4622:
4606:
4602:
4601:
4587:
4582:
4581:
4559:
4558:
4557:
4534:
4532:
4531:
4526:
4519:
4509:
4508:
4507:
4497:
4496:
4488:
4484:
4483:
4466:
4462:
4461:
4447:
4442:
4441:
4419:
4418:
4417:
4394:
4392:
4391:
4386:
4375:
4374:
4373:
4363:
4361:
4360:
4359:
4343:
4341:
4340:
4335:
4327:
4318:
4314:
4313:
4299:
4294:
4293:
4283:
4281:
4280:
4279:
4260:
4258:
4257:
4252:
4244:
4229:
4228:
4227:
4217:
4215:
4214:
4213:
4197:
4195:
4194:
4189:
4181:
4165:
4163:
4162:
4157:
4155:
4154:
4142:
4141:
4132:
4131:
4126:
4122:
4121:
4101:
4100:
4095:
4094:
4093:
4082:
4081:
4062:
4060:
4059:
4054:
4049:
4035:
4009:
4007:
4006:
4001:
3996:
3960:
3958:
3957:
3952:
3941:
3940:
3939:
3923:
3919:
3918:
3904:
3899:
3898:
3876:
3875:
3874:
3850:
3848:
3847:
3842:
3840:
3839:
3820:
3818:
3817:
3812:
3801:
3800:
3799:
3789:
3787:
3786:
3785:
3769:
3767:
3766:
3757:
3756:
3751:
3747:
3746:
3726:
3722:
3721:
3707:
3702:
3701:
3691:
3689:
3688:
3687:
3668:
3666:
3665:
3644:
3643:
3642:
3632:
3630:
3629:
3628:
3612:
3610:
3609:
3590:
3588:
3587:
3582:
3575:
3559:
3538:
3536:
3535:
3530:
3519:
3518:
3482:
3480:
3479:
3474:
3467:
3466:
3458:
3437:
3435:
3434:
3429:
3427:
3423:
3416:
3406:
3401:
3396:
3385:
3381:
3374:
3364:
3356:
3341:
3339:
3338:
3333:
3331:
3319:
3317:
3316:
3311:
3309:
3294:
3292:
3291:
3286:
3260:
3258:
3257:
3252:
3250:
3236:
3235:
3234:
3224:
3223:
3214:
3213:
3208:
3207:
3199:
3181:
3180:
3179:
3169:
3168:
3159:
3158:
3132:
3130:
3129:
3124:
3122:
3114:
3109:
3104:
3099:
3089:
3084:
3076:
3071:
3066:
3061:
3050:
3045:
3040:
3021:
3019:
3018:
3013:
3011:
2996:
2994:
2993:
2988:
2986:
2963:
2961:
2960:
2955:
2944:
2940:
2939:
2938:
2930:
2924:
2923:
2915:
2909:
2908:
2888:
2887:
2878:
2877:
2862:
2858:
2857:
2856:
2847:
2846:
2841:
2840:
2832:
2823:
2819:
2818:
2817:
2808:
2807:
2787:
2783:
2782:
2781:
2772:
2771:
2757:
2753:
2752:
2751:
2742:
2741:
2736:
2735:
2727:
2704:
2702:
2701:
2696:
2685:
2681:
2680:
2679:
2671:
2665:
2664:
2656:
2647:
2646:
2626:
2625:
2616:
2615:
2610:
2609:
2601:
2584:
2582:
2581:
2576:
2565:
2561:
2560:
2559:
2551:
2545:
2544:
2536:
2527:
2526:
2506:
2505:
2496:
2495:
2473:
2471:
2470:
2465:
2463:
2462:
2455:
2454:
2441:
2440:
2403:
2401:
2400:
2395:
2393:
2392:
2388:
2371:
2363:
2338:
2337:
2324:
2316:
2291:
2290:
2283:
2282:
2269:
2268:
2248:
2244:
2237:
2203:
2201:
2200:
2195:
2188:
2181:
2180:
2179:
2169:
2168:
2159:
2158:
2153:
2152:
2144:
2133:
2131:
2130:
2125:
2117:
2116:
2115:
2105:
2104:
2095:
2094:
2067:
2065:
2064:
2059:
2052:
2051:
2050:
2043:
2042:
2028:
2027:
2013:
2012:
1998:
1997:
1977:
1976:
1971:
1970:
1962:
1945:
1943:
1942:
1937:
1926:
1925:
1916:
1915:
1910:
1909:
1901:
1879:
1877:
1876:
1871:
1859:
1857:
1856:
1851:
1814:
1812:
1811:
1806:
1788:
1786:
1785:
1780:
1778:
1777:
1751:
1749:
1748:
1743:
1741:
1740:
1733:
1732:
1721:
1720:
1709:
1708:
1697:
1696:
1676:
1675:
1668:
1667:
1656:
1655:
1644:
1643:
1632:
1631:
1611:
1610:
1588:
1586:
1585:
1580:
1572:
1570:
1562:
1554:
1552:
1551:
1539:
1537:
1529:
1521:
1519:
1518:
1506:
1504:
1496:
1488:
1486:
1485:
1473:
1471:
1463:
1455:
1453:
1445:
1443:
1442:
1419:
1417:
1416:
1411:
1396:
1394:
1393:
1388:
1377:
1376:
1367:
1366:
1302:Super-Kamiokande
1294:Bruno Pontecorvo
1279:weak interaction
1131:
1124:
1117:
222:Effective action
149:Lorentz symmetry
74:Electromagnetism
44:
30:
29:
21:
11291:
11290:
11286:
11285:
11284:
11282:
11281:
11280:
11266:
11265:
11242:
11170:
11168:
11106:
11104:
11089:
11068:
11045:
11024:
11005:
10987:
10985:Further reading
10982:
10971:
10967:
10938:
10931:
10924:
10908:
10904:
10897:
10883:
10879:
10872:
10858:
10854:
10824:Physical Review
10815:
10811:
10796:
10787:
10734:Physica Scripta
10726:
10711:
10645:
10641:
10580:
10571:
10564:
10550:
10546:
10542:
10537:
10536:
10526:
10522:
10517:
10480:
10478:
10475:
10474:
10456:
10455:
10441:
10440:
10438:
10435:
10434:
10430:Majorana spinor
10374:
10371:
10370:
10353:
10349:
10317:
10315:
10312:
10311:
10292:
10287:
10286:
10285:
10281:
10279:
10276:
10275:
10246:
10244:
10241:
10240:
10215:
10206:
10202:
10200:
10197:
10196:
10176:
10172:
10164:
10159:
10158:
10157:
10153:
10124:
10099:
10098:
10084:
10083:
10081:
10078:
10077:
10038:
10037:
10023:
10022:
10020:
10017:
10016:
10009:
9987:
9982:
9976:
9973:
9972:
9971:(respectively,
9955:
9950:
9944:
9941:
9940:
9924:
9921:
9920:
9895:
9891:
9887:
9879:
9876:
9875:
9858:
9853:
9840:
9835:
9815:
9811:
9806:
9805:
9791:
9775:
9771:
9766:
9765:
9751:
9749:
9746:
9745:
9717:
9715:
9712:
9711:
9685:
9681:
9676:
9674:
9671:
9670:
9654:
9651:
9650:
9647:
9641:
9629:Clifford bundle
9621:spin connection
9575:
9572:
9571:
9533:
9531:
9528:
9527:
9480:
9478:
9475:
9474:
9451:
9448:
9447:
9390:
9387:
9386:
9359:
9355:
9349:
9345:
9333:
9329:
9323:
9319:
9317:
9314:
9313:
9275:
9271:
9256:
9252:
9251:
9247:
9235:
9226:
9221:
9215:
9212:
9211:
9174:
9170:
9155:
9151:
9150:
9146:
9134:
9125:
9121:
9119:
9116:
9115:
9092:
9088:
9086:
9083:
9082:
9058:
9054:
9049:
9046:
9045:
9011:
9009:
9006:
9005:
8973:
8970:
8969:
8949:
8945:
8943:
8940:
8939:
8911:
8908:
8907:
8888:
8885:
8884:
8853:
8850:
8849:
8833:
8830:
8829:
8791:
8788:
8787:
8742:Clifford module
8734:
8713:
8710:
8709:
8678:
8677:
8673:
8667:
8663:
8657:
8646:
8645:
8644:
8638:
8632:
8631:
8615:
8614:
8612:
8609:
8608:
8581:
8580:
8576:
8570:
8566:
8560:
8556:
8550:
8544:
8543:
8527:
8526:
8524:
8521:
8520:
8511:
8478:
8475:
8474:
8451:
8447:
8442:
8439:
8438:
8397:
8394:
8393:
8367:
8363:
8361:
8358:
8357:
8332:
8328:
8322:
8318:
8308:
8304:
8287:
8283:
8277:
8273:
8256:
8255:
8241:
8240:
8231:
8226:
8221:
8217:
8185:
8181:
8175:
8165:
8164:
8160:
8156:
8130:
8126:
8120:
8116:
8112:
8108:
8098:
8097:
8093:
8086:
8085:
8081:
8079:
8076:
8075:
8054:
8053:
8049:
8042:
8041:
8037:
8035:
8032:
8031:
8013:
8012:
8008:
8001:
8000:
7996:
7994:
7991:
7990:
7960:
7959:
7955:
7948:
7947:
7943:
7929:
7928:
7924:
7917:
7916:
7912:
7910:
7907:
7906:
7882:
7881:
7877:
7865:
7859:
7858:
7844:
7843:
7839:
7831:
7830:
7826:
7817:
7807:
7803:
7799:
7798:
7789:
7783:
7782:
7768:
7767:
7763:
7761:
7758:
7757:
7731:
7727:
7720:
7719:
7715:
7706:
7696:
7692:
7688:
7687:
7678:
7672:
7671:
7657:
7656:
7652:
7645:
7641:
7634:
7633:
7629:
7617:
7611:
7610:
7596:
7595:
7591:
7589:
7586:
7585:
7566:
7563:
7562:
7527:
7523:
7517:
7513:
7500:
7499:
7495:
7473:
7469:
7463:
7453:
7452:
7439:
7438:
7434:
7432:
7429:
7428:
7395:
7392:
7391:
7367:
7363:
7351:
7347:
7342:
7339:
7338:
7303:
7297:
7296:
7264:
7263:
7259:
7253:
7249:
7243:
7233:
7232:
7227:
7224:
7223:
7204:
7201:
7200:
7167:
7161:
7160:
7128:
7127:
7123:
7117:
7113:
7107:
7103:
7098:
7095:
7094:
7072:
7068:
7060:
7057:
7056:
7027:
7021:
7020:
7001:
6991:
6987:
6983:
6982:
6969:
6965:
6961:
6952:
6946:
6945:
6917:
6911:
6910:
6898:
6895:
6894:
6862:
6856:
6855:
6845:
6841:
6828:
6824:
6820:
6811:
6805:
6804:
6782:
6776:
6775:
6769:
6766:
6765:
6732:
6731:
6727:
6711:
6707:
6703:
6697:
6691:
6690:
6667:
6666:
6662:
6660:
6657:
6656:
6632:
6628:
6626:
6623:
6622:
6599:
6589:
6585:
6581:
6580:
6571:
6567:
6562:
6559:
6558:
6537:
6536:
6531:
6522:
6521:
6516:
6506:
6505:
6496:
6492:
6481:
6478:
6477:
6441:
6440:
6436:
6434:
6431:
6430:
6402:
6385:
6383:
6380:
6379:
6353:
6336:
6334:
6331:
6330:
6320:
6293:
6283:
6279:
6275:
6274:
6266:
6263:
6262:
6232:
6231:
6227:
6211:
6207:
6203:
6197:
6191:
6190:
6167:
6166:
6162:
6160:
6157:
6156:
6128:
6125:
6124:
6092:
6088:
6086:
6083:
6082:
6065:
6064:
6048:
6047:
6043:
6031:
6027:
6021:
6017:
6011:
6007:
6001:
5988:
5984:
5980:
5979:
5970:
5969:
5953:
5952:
5948:
5939:
5935:
5929:
5922:
5918:
5917:
5916:
5910:
5906:
5897:
5896:
5880:
5879:
5875:
5866:
5862:
5856:
5849:
5837:
5836:
5829:
5825:
5821:
5820:
5819:
5818:
5812:
5808:
5801:
5785:
5784:
5780:
5771:
5767:
5761:
5754:
5741:
5737:
5733:
5732:
5731:
5730:
5724:
5720:
5716:
5714:
5711:
5710:
5687:
5680:
5668:
5667:
5660:
5656:
5652:
5651:
5650:
5649:
5640:
5633:
5620:
5616:
5612:
5611:
5610:
5609:
5607:
5604:
5603:
5563:
5555:
5552:
5551:
5490:
5482:
5479:
5478:
5455:
5451:
5438:
5437:
5433:
5428:
5425:
5424:
5398:
5394:
5388:
5384:
5378:
5365:
5361:
5357:
5356:
5347:
5340:
5336:
5335:
5334:
5328:
5324:
5322:
5319:
5318:
5298:
5297:
5281:
5280:
5276:
5267:
5263:
5257:
5250:
5237:
5233:
5229:
5228:
5227:
5226:
5220:
5216:
5207:
5206:
5190:
5189:
5185:
5173:
5169:
5165:
5160:
5154:
5147:
5134:
5130:
5126:
5125:
5124:
5123:
5117:
5113:
5104:
5103:
5087:
5086:
5082:
5070:
5066:
5062:
5057:
5045:
5041:
5037:
5030:
5026:
5022:
5020:
5014:
5010:
5001:
5000:
4990:
4986:
4982:
4976:
4970:
4969:
4953:
4949:
4945:
4940:
4934:
4930:
4923:
4913:
4909:
4905:
4899:
4893:
4892:
4882:
4877:
4867:
4863:
4859:
4857:
4854:
4853:
4827:
4823:
4817:
4810:
4797:
4793:
4789:
4788:
4787:
4786:
4777:
4773:
4771:
4768:
4767:
4744:
4740:
4734:
4727:
4723:
4722:
4721:
4709:
4705:
4703:
4700:
4699:
4677:
4660:
4652:
4649:
4648:
4618:
4617:
4613:
4597:
4593:
4589:
4583:
4577:
4576:
4553:
4552:
4548:
4546:
4543:
4542:
4503:
4502:
4498:
4489:
4479:
4475:
4471:
4470:
4457:
4453:
4449:
4443:
4437:
4436:
4413:
4412:
4408:
4406:
4403:
4402:
4369:
4368:
4364:
4355:
4351:
4347:
4342:
4336:
4326:
4325:
4309:
4305:
4301:
4295:
4289:
4288:
4272:
4268:
4264:
4259:
4253:
4243:
4242:
4223:
4222:
4218:
4209:
4205:
4201:
4196:
4190:
4180:
4179:
4177:
4174:
4173:
4147:
4143:
4137:
4133:
4127:
4114:
4110:
4106:
4105:
4096:
4089:
4085:
4084:
4083:
4077:
4073:
4071:
4068:
4067:
4045:
4028:
4026:
4023:
4022:
4012:double covering
3992:
3969:
3966:
3965:
3935:
3934:
3930:
3914:
3910:
3906:
3900:
3894:
3893:
3870:
3869:
3865:
3863:
3860:
3859:
3835:
3831:
3829:
3826:
3825:
3795:
3794:
3790:
3781:
3777:
3773:
3768:
3762:
3758:
3752:
3739:
3735:
3731:
3730:
3717:
3713:
3709:
3703:
3697:
3696:
3680:
3676:
3672:
3667:
3661:
3657:
3638:
3637:
3633:
3624:
3620:
3616:
3611:
3605:
3601:
3599:
3596:
3595:
3552:
3544:
3541:
3540:
3514:
3510:
3502:
3499:
3498:
3488:
3457:
3446:
3443:
3442:
3412:
3411:
3407:
3402:
3397:
3392:
3370:
3369:
3365:
3360:
3352:
3350:
3347:
3346:
3327:
3325:
3322:
3321:
3305:
3303:
3300:
3299:
3280:
3277:
3276:
3273:
3267:
3248:
3247:
3237:
3230:
3229:
3225:
3219:
3215:
3209:
3198:
3197:
3196:
3193:
3192:
3182:
3175:
3174:
3170:
3164:
3160:
3154:
3150:
3146:
3144:
3141:
3140:
3113:
3105:
3100:
3095:
3077:
3075:
3067:
3062:
3057:
3046:
3041:
3036:
3034:
3031:
3030:
3007:
3005:
3002:
3001:
2982:
2980:
2977:
2976:
2929:
2928:
2914:
2913:
2904:
2900:
2899:
2895:
2883:
2879:
2873:
2869:
2852:
2848:
2842:
2831:
2830:
2829:
2828:
2824:
2813:
2809:
2803:
2799:
2798:
2794:
2777:
2773:
2767:
2763:
2762:
2758:
2747:
2743:
2737:
2726:
2725:
2724:
2723:
2719:
2717:
2714:
2713:
2670:
2669:
2655:
2654:
2642:
2638:
2637:
2633:
2621:
2617:
2611:
2600:
2599:
2598:
2596:
2593:
2592:
2550:
2549:
2535:
2534:
2522:
2518:
2517:
2513:
2501:
2497:
2491:
2487:
2485:
2482:
2481:
2457:
2456:
2450:
2446:
2443:
2442:
2436:
2432:
2425:
2424:
2416:
2413:
2412:
2384:
2367:
2359:
2349:
2345:
2320:
2312:
2302:
2298:
2285:
2284:
2278:
2274:
2271:
2270:
2264:
2260:
2253:
2252:
2233:
2232:
2228:
2223:
2220:
2219:
2209:
2175:
2174:
2170:
2164:
2160:
2154:
2143:
2142:
2141:
2139:
2136:
2135:
2111:
2110:
2106:
2100:
2096:
2090:
2086:
2084:
2081:
2080:
2045:
2044:
2038:
2034:
2029:
2023:
2019:
2014:
2008:
2004:
1999:
1993:
1989:
1982:
1981:
1972:
1961:
1960:
1959:
1957:
1954:
1953:
1921:
1917:
1911:
1900:
1899:
1898:
1896:
1893:
1892:
1865:
1862:
1861:
1824:
1821:
1820:
1794:
1791:
1790:
1773:
1769:
1767:
1764:
1763:
1761:identity matrix
1735:
1734:
1728:
1724:
1722:
1716:
1712:
1710:
1704:
1700:
1698:
1692:
1688:
1681:
1680:
1670:
1669:
1663:
1659:
1657:
1651:
1647:
1645:
1639:
1635:
1633:
1627:
1623:
1616:
1615:
1606:
1602:
1600:
1597:
1596:
1563:
1555:
1553:
1547:
1543:
1530:
1522:
1520:
1514:
1510:
1497:
1489:
1487:
1481:
1477:
1464:
1456:
1454:
1444:
1438:
1434:
1432:
1429:
1428:
1405:
1402:
1401:
1372:
1368:
1362:
1358:
1356:
1353:
1352:
1346:
1260:Conyers Herring
1217:
1206:Weyl semimetals
1135:
1106:
1105:
1104:
1102:
406:
398:
397:
393:Quantum gravity
368:
360:
359:
355:Higgs mechanism
335:
325:
324:
310:Proca equations
295:
287:
286:
272:Renormalization
237:Feynman diagram
192:
184:
183:
124:
114:
113:
64:
49:
47:Feynman diagram
28:
23:
22:
15:
12:
11:
5:
11289:
11279:
11278:
11264:
11263:
11258:
11253:
11248:
11241:
11240:External links
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10895:
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10806:. Vol. 8.
10785:
10747:hep-ph/0410090
10709:
10662:(4): 323â334.
10650:(1929-04-15).
10639:
10594:(5): 485â498.
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10274:. The product
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9724:
9720:
9699:
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9640:
9637:
9598:spin structure
9585:
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9332:
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9310:anti-commuting
9306:
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8980:
8977:
8957:
8952:
8948:
8924:
8921:
8918:
8915:
8895:
8892:
8882:tangent bundle
8869:
8866:
8863:
8860:
8857:
8837:
8820:rotation group
8795:
8733:
8730:
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8702:
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8690:
8681:
8676:
8670:
8666:
8660:
8653:
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8510:
8507:
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8450:
8446:
8437:provided that
8422:
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7847:
7842:
7834:
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7110:
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7078:
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7064:
7053:
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7024:
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7004:
6999:
6994:
6990:
6986:
6981:
6977:
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6968:
6964:
6958:
6955:
6949:
6944:
6940:
6937:
6934:
6931:
6928:
6925:
6920:
6914:
6909:
6905:
6902:
6888:
6887:
6876:
6873:
6870:
6865:
6859:
6854:
6848:
6844:
6840:
6836:
6831:
6827:
6823:
6817:
6814:
6808:
6803:
6799:
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6774:
6759:
6758:
6747:
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6735:
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6714:
6710:
6706:
6700:
6694:
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6679:
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6665:
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6631:
6610:
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6597:
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6566:
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6360:
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6339:
6319:
6316:
6304:
6299:
6296:
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6286:
6282:
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6270:
6259:
6258:
6247:
6244:
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6235:
6230:
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6210:
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6185:
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6179:
6176:
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6165:
6141:
6138:
6135:
6132:
6112:
6109:
6106:
6103:
6098:
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6079:
6078:
6063:
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6057:
6051:
6046:
6042:
6037:
6034:
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6024:
6020:
6014:
6010:
6004:
5999:
5994:
5991:
5987:
5983:
5978:
5975:
5973:
5971:
5968:
5965:
5962:
5956:
5951:
5947:
5942:
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5932:
5925:
5921:
5913:
5909:
5905:
5902:
5900:
5898:
5895:
5892:
5889:
5883:
5878:
5874:
5869:
5865:
5859:
5852:
5847:
5840:
5835:
5832:
5828:
5824:
5815:
5811:
5807:
5804:
5802:
5800:
5797:
5794:
5788:
5783:
5779:
5774:
5770:
5764:
5757:
5752:
5747:
5744:
5740:
5736:
5727:
5723:
5719:
5718:
5704:
5703:
5690:
5683:
5678:
5671:
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5663:
5659:
5655:
5648:
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5636:
5631:
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5535:
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5526:
5523:
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5511:
5508:
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5502:
5499:
5489:
5486:
5475:
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5454:
5450:
5447:
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5436:
5432:
5418:
5417:
5404:
5401:
5397:
5391:
5387:
5381:
5376:
5371:
5368:
5364:
5360:
5355:
5350:
5343:
5339:
5331:
5327:
5312:
5311:
5296:
5293:
5290:
5284:
5279:
5275:
5270:
5266:
5260:
5253:
5248:
5243:
5240:
5236:
5232:
5223:
5219:
5215:
5212:
5210:
5208:
5205:
5202:
5199:
5193:
5188:
5184:
5176:
5172:
5168:
5164:
5157:
5150:
5145:
5140:
5137:
5133:
5129:
5120:
5116:
5112:
5109:
5107:
5105:
5102:
5099:
5096:
5090:
5085:
5081:
5073:
5069:
5065:
5061:
5051:
5048:
5044:
5040:
5033:
5029:
5025:
5017:
5013:
5009:
5006:
5004:
5002:
4998:
4993:
4989:
4985:
4979:
4973:
4968:
4959:
4956:
4952:
4948:
4944:
4937:
4933:
4929:
4926:
4924:
4921:
4916:
4912:
4908:
4902:
4896:
4891:
4885:
4880:
4876:
4870:
4866:
4862:
4861:
4849:This leads to
4847:
4846:
4833:
4830:
4826:
4820:
4813:
4808:
4803:
4800:
4796:
4792:
4785:
4780:
4776:
4761:
4760:
4747:
4743:
4737:
4730:
4726:
4720:
4715:
4712:
4708:
4684:
4680:
4676:
4673:
4670:
4666:
4663:
4659:
4656:
4645:
4644:
4633:
4630:
4627:
4621:
4616:
4612:
4609:
4605:
4600:
4596:
4592:
4586:
4580:
4575:
4571:
4568:
4565:
4562:
4556:
4551:
4536:
4535:
4524:
4518:
4515:
4512:
4506:
4501:
4495:
4492:
4487:
4482:
4478:
4474:
4469:
4465:
4460:
4456:
4452:
4446:
4440:
4435:
4431:
4428:
4425:
4422:
4416:
4411:
4396:
4395:
4384:
4381:
4378:
4372:
4367:
4358:
4354:
4350:
4346:
4339:
4333:
4330:
4324:
4321:
4317:
4312:
4308:
4304:
4298:
4292:
4287:
4278:
4275:
4271:
4267:
4263:
4256:
4250:
4247:
4241:
4238:
4235:
4232:
4226:
4221:
4212:
4208:
4204:
4200:
4193:
4187:
4184:
4167:
4166:
4153:
4150:
4146:
4140:
4136:
4130:
4125:
4120:
4117:
4113:
4109:
4104:
4099:
4092:
4088:
4080:
4076:
4052:
4048:
4044:
4041:
4038:
4034:
4031:
3999:
3995:
3991:
3988:
3985:
3982:
3979:
3976:
3973:
3962:
3961:
3950:
3947:
3944:
3938:
3933:
3929:
3926:
3922:
3917:
3913:
3909:
3903:
3897:
3892:
3888:
3885:
3882:
3879:
3873:
3868:
3838:
3834:
3822:
3821:
3810:
3807:
3804:
3798:
3793:
3784:
3780:
3776:
3772:
3765:
3761:
3755:
3750:
3745:
3742:
3738:
3734:
3729:
3725:
3720:
3716:
3712:
3706:
3700:
3695:
3686:
3683:
3679:
3675:
3671:
3664:
3660:
3656:
3653:
3650:
3647:
3641:
3636:
3627:
3623:
3619:
3615:
3608:
3604:
3580:
3574:
3571:
3568:
3565:
3562:
3558:
3555:
3551:
3548:
3528:
3525:
3522:
3517:
3513:
3509:
3506:
3487:
3484:
3472:
3464:
3461:
3456:
3453:
3450:
3439:
3438:
3426:
3422:
3419:
3415:
3410:
3405:
3400:
3395:
3391:
3388:
3384:
3380:
3377:
3373:
3368:
3363:
3359:
3355:
3330:
3308:
3284:
3269:Main article:
3266:
3263:
3262:
3261:
3246:
3243:
3240:
3238:
3233:
3228:
3222:
3218:
3212:
3205:
3202:
3195:
3194:
3191:
3188:
3185:
3183:
3178:
3173:
3167:
3163:
3157:
3153:
3149:
3148:
3134:
3133:
3120:
3117:
3112:
3108:
3103:
3098:
3093:
3087:
3083:
3080:
3074:
3070:
3065:
3060:
3056:
3053:
3049:
3044:
3039:
3010:
2985:
2966:
2965:
2953:
2950:
2947:
2943:
2936:
2933:
2927:
2921:
2918:
2912:
2907:
2903:
2898:
2894:
2891:
2886:
2882:
2876:
2872:
2868:
2865:
2861:
2855:
2851:
2845:
2838:
2835:
2827:
2822:
2816:
2812:
2806:
2802:
2797:
2793:
2790:
2786:
2780:
2776:
2770:
2766:
2761:
2756:
2750:
2746:
2740:
2733:
2730:
2722:
2707:
2706:
2694:
2691:
2688:
2684:
2677:
2674:
2668:
2662:
2659:
2653:
2650:
2645:
2641:
2636:
2632:
2629:
2624:
2620:
2614:
2607:
2604:
2586:
2585:
2574:
2571:
2568:
2564:
2557:
2554:
2548:
2542:
2539:
2533:
2530:
2525:
2521:
2516:
2512:
2509:
2504:
2500:
2494:
2490:
2475:
2474:
2461:
2453:
2449:
2445:
2444:
2439:
2435:
2431:
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2406:
2405:
2391:
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2377:
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2370:
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2336:
2333:
2330:
2327:
2323:
2319:
2315:
2311:
2308:
2305:
2301:
2297:
2294:
2289:
2281:
2277:
2273:
2272:
2267:
2263:
2259:
2258:
2256:
2251:
2247:
2243:
2240:
2236:
2231:
2227:
2208:
2205:
2193:
2187:
2184:
2178:
2173:
2167:
2163:
2157:
2150:
2147:
2123:
2120:
2114:
2109:
2103:
2099:
2093:
2089:
2069:
2068:
2057:
2049:
2041:
2037:
2033:
2030:
2026:
2022:
2018:
2015:
2011:
2007:
2003:
2000:
1996:
1992:
1988:
1987:
1985:
1980:
1975:
1968:
1965:
1947:
1946:
1935:
1932:
1929:
1924:
1920:
1914:
1907:
1904:
1869:
1849:
1846:
1843:
1840:
1837:
1834:
1831:
1828:
1817:Pauli matrices
1804:
1801:
1798:
1776:
1772:
1753:
1752:
1739:
1731:
1727:
1723:
1719:
1715:
1711:
1707:
1703:
1699:
1695:
1691:
1687:
1686:
1684:
1679:
1674:
1666:
1662:
1658:
1654:
1650:
1646:
1642:
1638:
1634:
1630:
1626:
1622:
1621:
1619:
1614:
1609:
1605:
1590:
1589:
1578:
1575:
1569:
1566:
1561:
1558:
1550:
1546:
1542:
1536:
1533:
1528:
1525:
1517:
1513:
1509:
1503:
1500:
1495:
1492:
1484:
1480:
1476:
1470:
1467:
1462:
1459:
1451:
1448:
1441:
1437:
1422:speed of light
1409:
1398:
1397:
1386:
1383:
1380:
1375:
1371:
1365:
1361:
1345:
1342:
1310:Weyl semimetal
1287:Standard Model
1264:quasiparticles
1237:Wolfgang Pauli
1221:Dirac equation
1216:
1213:
1202:quasiparticles
1186:Standard Model
1137:
1136:
1134:
1133:
1126:
1119:
1111:
1108:
1107:
1100:
1099:
1094:
1089:
1084:
1079:
1074:
1069:
1064:
1059:
1054:
1049:
1044:
1039:
1034:
1029:
1024:
1019:
1014:
1009:
1004:
999:
994:
989:
984:
979:
974:
969:
964:
959:
954:
949:
944:
939:
934:
929:
924:
919:
914:
909:
904:
899:
894:
889:
884:
879:
874:
869:
864:
859:
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:
624:
619:
614:
609:
604:
599:
594:
589:
584:
579:
574:
569:
564:
559:
554:
549:
544:
539:
534:
529:
524:
519:
514:
509:
504:
499:
494:
489:
484:
479:
474:
469:
464:
459:
454:
449:
444:
439:
434:
429:
424:
419:
414:
408:
407:
404:
403:
400:
399:
396:
395:
390:
385:
380:
375:
369:
366:
365:
362:
361:
358:
357:
352:
347:
342:
336:
333:Standard Model
331:
330:
327:
326:
323:
322:
317:
312:
307:
302:
300:Dirac equation
296:
293:
292:
289:
288:
285:
284:
282:Wick's theorem
279:
274:
269:
267:Regularization
264:
259:
254:
249:
244:
239:
234:
229:
224:
219:
214:
209:
204:
199:
193:
190:
189:
186:
185:
182:
181:
176:
174:Noether charge
171:
166:
161:
159:Gauge symmetry
156:
151:
146:
141:
136:
131:
125:
120:
119:
116:
115:
112:
111:
106:
101:
96:
91:
86:
81:
76:
71:
65:
62:
61:
58:
57:
51:
50:
45:
37:
36:
26:
9:
6:
4:
3:
2:
11288:
11277:
11274:
11273:
11271:
11262:
11259:
11257:
11254:
11252:
11249:
11247:
11244:
11243:
11233:
11229:
11225:
11221:
11217:
11213:
11209:
11205:
11200:
11195:
11191:
11187:
11183:
11178:
11167:. Vol. 8
11166:
11162:
11157:
11153:
11149:
11145:
11141:
11136:
11131:
11127:
11123:
11119:
11114:
11103:
11102:Physics World
11099:
11094:
11090:
11084:
11080:
11079:
11073:
11069:
11063:
11059:
11058:
11052:
11046:
11044:9781118681664
11040:
11036:
11035:
11029:
11028:
11025:
11019:
11015:
11010:
11006:
11000:
10996:
10995:
10989:
10988:
10976:
10969:
10961:
10957:
10952:
10947:
10943:
10936:
10934:
10925:
10923:0-201-50397-2
10919:
10915:
10914:
10906:
10898:
10892:
10888:
10881:
10873:
10867:
10863:
10856:
10847:
10842:
10838:
10834:
10830:
10826:
10825:
10820:
10813:
10805:
10801:
10794:
10792:
10790:
10781:
10777:
10773:
10769:
10765:
10761:
10757:
10753:
10748:
10743:
10739:
10735:
10731:
10724:
10722:
10720:
10718:
10716:
10714:
10705:
10701:
10696:
10691:
10687:
10683:
10678:
10673:
10669:
10665:
10661:
10657:
10653:
10649:
10648:Weyl, Hermann
10643:
10635:
10631:
10627:
10623:
10619:
10615:
10611:
10607:
10602:
10597:
10593:
10589:
10585:
10578:
10576:
10574:
10565:
10563:9789810239480
10559:
10555:
10548:
10544:
10531:
10530:Lorentz group
10524:
10520:
10512:
10510:
10505:
10488:
10431:
10426:
10409:
10406:
10403:
10400:
10397:
10391:
10385:
10382:
10379:
10354:
10350:
10346:
10340:
10337:
10334:
10293:
10282:
10273:
10254:
10223:
10212:
10207:
10203:
10177:
10173:
10165:
10154:
10147:
10144:
10141:
10121:
10115:
10112:
10109:
10076:
10075:
10074:
10060:
10054:
10051:
10048:
10014:
10007:Special cases
10004:
9988:
9983:
9956:
9951:
9926:
9918:
9900:
9896:
9892:
9888:
9884:
9881:
9859:
9854:
9846:
9841:
9836:
9828:
9820:
9816:
9812:
9788:
9780:
9776:
9772:
9728:
9722:
9694:
9686:
9682:
9656:
9646:
9636:
9634:
9633:spin geometry
9630:
9626:
9622:
9618:
9614:
9613:spin manifold
9609:
9606:
9601:
9599:
9583:
9577:
9569:
9550:
9547:
9544:
9526:
9522:
9503:
9500:
9497:
9472:
9456:
9453:
9444:
9442:
9438:
9419:
9416:
9413:
9407:
9401:
9398:
9395:
9384:
9368:
9360:
9356:
9350:
9346:
9342:
9339:
9334:
9330:
9324:
9320:
9311:
9291:
9285:
9282:
9279:
9276:
9272:
9268:
9265:
9260:
9257:
9253:
9248:
9241:
9237:
9232:
9227:
9222:
9218:
9210:
9209:
9208:
9190:
9184:
9181:
9178:
9175:
9171:
9167:
9164:
9159:
9156:
9152:
9147:
9140:
9136:
9131:
9126:
9122:
9114:
9113:
9112:
9098:
9093:
9089:
9080:
9059:
9055:
9028:
9025:
9022:
9003:
8984:
8981:
8978:
8955:
8950:
8946:
8938:
8937:tangent space
8922:
8919:
8916:
8913:
8893:
8890:
8883:
8864:
8861:
8858:
8848:of dimension
8835:
8828:
8823:
8821:
8817:
8813:
8809:
8793:
8786:
8782:
8778:
8774:
8769:
8767:
8763:
8759:
8755:
8751:
8747:
8743:
8739:
8729:
8715:
8707:
8688:
8674:
8668:
8658:
8648:
8639:
8628:
8624:
8621:
8607:
8606:
8591:
8577:
8571:
8561:
8557:
8551:
8540:
8536:
8533:
8519:
8518:
8517:
8516:
8506:
8492:
8486:
8483:
8480:
8460:
8457:
8452:
8448:
8444:
8436:
8420:
8414:
8411:
8408:
8405:
8402:
8399:
8379:
8376:
8373:
8368:
8364:
8339:
8333:
8329:
8323:
8319:
8315:
8312:
8309:
8305:
8301:
8298:
8294:
8288:
8284:
8278:
8274:
8270:
8267:
8252:
8237:
8232:
8227:
8218:
8214:
8211:
8207:
8203:
8200:
8197:
8194:
8191:
8186:
8176:
8167:
8161:
8157:
8152:
8148:
8145:
8142:
8139:
8136:
8131:
8121:
8117:
8113:
8109:
8105:
8094:
8082:
8074:
8073:
8072:
8050:
8038:
8009:
7997:
7989:The products
7987:
7970:
7967:
7956:
7944:
7939:
7936:
7925:
7913:
7905:
7904:
7903:
7878:
7874:
7871:
7855:
7840:
7827:
7821:
7818:
7813:
7808:
7804:
7800:
7795:
7779:
7764:
7756:
7755:
7754:
7735:
7732:
7728:
7716:
7710:
7707:
7702:
7697:
7693:
7689:
7684:
7668:
7653:
7646:
7642:
7630:
7626:
7623:
7607:
7592:
7584:
7583:
7582:
7568:
7545:
7542:
7539:
7536:
7533:
7528:
7518:
7514:
7510:
7507:
7496:
7491:
7488:
7485:
7482:
7479:
7474:
7464:
7455:
7449:
7446:
7435:
7427:
7426:
7425:
7422:
7420:
7403:
7397:
7389:
7373:
7368:
7364:
7360:
7357:
7352:
7348:
7344:
7321:
7318:
7312:
7304:
7293:
7289:
7286:
7283:
7280:
7274:
7260:
7254:
7244:
7235:
7229:
7222:
7221:
7220:
7206:
7198:
7176:
7168:
7157:
7153:
7150:
7147:
7144:
7138:
7124:
7118:
7108:
7104:
7100:
7093:
7092:
7091:
7076:
7073:
7069:
7065:
7062:
7036:
7028:
7017:
7013:
7010:
7005:
7002:
6997:
6992:
6988:
6984:
6979:
6975:
6966:
6962:
6956:
6942:
6938:
6935:
6926:
6918:
6907:
6903:
6900:
6893:
6892:
6891:
6871:
6863:
6852:
6846:
6842:
6838:
6834:
6825:
6821:
6815:
6801:
6791:
6783:
6772:
6764:
6763:
6762:
6742:
6728:
6724:
6721:
6717:
6708:
6704:
6687:
6677:
6663:
6655:
6654:
6653:
6651:
6633:
6629:
6608:
6603:
6600:
6595:
6590:
6586:
6582:
6577:
6572:
6568:
6564:
6539:
6533:
6528:
6525:
6518:
6513:
6507:
6502:
6497:
6493:
6489:
6486:
6483:
6476:
6475:
6474:
6457:
6454:
6451:
6448:
6437:
6429:
6428:
6427:
6413:
6399:
6396:
6378:
6374:
6350:
6347:
6329:
6325:
6315:
6302:
6297:
6294:
6289:
6284:
6280:
6276:
6271:
6268:
6242:
6228:
6224:
6221:
6217:
6208:
6204:
6187:
6177:
6163:
6155:
6154:
6153:
6139:
6136:
6133:
6130:
6110:
6107:
6104:
6101:
6096:
6093:
6089:
6058:
6044:
6040:
6035:
6032:
6028:
6022:
6012:
6008:
6002:
5997:
5992:
5989:
5985:
5981:
5976:
5974:
5963:
5949:
5945:
5940:
5930:
5923:
5911:
5907:
5903:
5901:
5890:
5876:
5872:
5867:
5857:
5850:
5845:
5833:
5830:
5822:
5813:
5809:
5805:
5803:
5795:
5781:
5777:
5772:
5762:
5755:
5750:
5745:
5742:
5734:
5725:
5721:
5709:
5708:
5707:
5688:
5681:
5676:
5664:
5661:
5653:
5646:
5641:
5634:
5629:
5624:
5621:
5613:
5602:
5601:
5600:
5586:
5580:
5577:
5574:
5560:
5549:
5530:
5527:
5524:
5521:
5518:
5515:
5512:
5509:
5506:
5503:
5500:
5487:
5484:
5459:
5456:
5448:
5445:
5430:
5423:
5422:
5421:
5402:
5399:
5395:
5389:
5385:
5379:
5374:
5369:
5366:
5362:
5358:
5353:
5348:
5341:
5329:
5325:
5317:
5316:
5315:
5291:
5277:
5273:
5268:
5258:
5251:
5246:
5241:
5238:
5230:
5221:
5217:
5213:
5211:
5200:
5186:
5182:
5174:
5170:
5155:
5148:
5143:
5138:
5135:
5127:
5118:
5114:
5110:
5108:
5097:
5083:
5079:
5071:
5067:
5049:
5042:
5031:
5027:
5015:
5011:
5007:
5005:
4996:
4987:
4983:
4966:
4957:
4950:
4935:
4931:
4927:
4925:
4919:
4910:
4906:
4889:
4878:
4868:
4864:
4852:
4851:
4850:
4831:
4824:
4818:
4811:
4806:
4801:
4798:
4790:
4783:
4778:
4774:
4766:
4765:
4764:
4745:
4741:
4735:
4728:
4718:
4713:
4706:
4698:
4697:
4696:
4674:
4671:
4657:
4654:
4628:
4614:
4610:
4607:
4603:
4594:
4590:
4573:
4563:
4549:
4541:
4540:
4539:
4522:
4513:
4499:
4493:
4490:
4485:
4480:
4476:
4472:
4467:
4463:
4454:
4450:
4433:
4423:
4409:
4401:
4400:
4399:
4379:
4365:
4356:
4352:
4337:
4328:
4322:
4319:
4315:
4306:
4302:
4285:
4276:
4269:
4254:
4245:
4233:
4219:
4210:
4206:
4191:
4182:
4172:
4171:
4170:
4151:
4148:
4144:
4138:
4134:
4128:
4123:
4118:
4115:
4111:
4107:
4102:
4097:
4090:
4078:
4074:
4066:
4065:
4064:
4042:
4039:
4021:
4017:
4016:Lorentz group
4013:
3989:
3986:
3980:
3977:
3974:
3971:
3945:
3931:
3927:
3924:
3920:
3911:
3907:
3890:
3880:
3866:
3858:
3857:
3856:
3854:
3836:
3832:
3805:
3791:
3782:
3778:
3763:
3759:
3753:
3748:
3743:
3740:
3736:
3732:
3727:
3723:
3714:
3710:
3693:
3684:
3677:
3662:
3658:
3648:
3634:
3625:
3621:
3606:
3602:
3594:
3593:
3592:
3578:
3569:
3566:
3563:
3549:
3526:
3520:
3511:
3504:
3497:
3493:
3483:
3470:
3462:
3459:
3454:
3451:
3448:
3424:
3420:
3417:
3408:
3389:
3386:
3382:
3378:
3375:
3366:
3357:
3345:
3344:
3343:
3298:
3282:
3272:
3244:
3241:
3239:
3226:
3220:
3210:
3200:
3189:
3186:
3184:
3171:
3165:
3155:
3151:
3139:
3138:
3137:
3118:
3115:
3110:
3085:
3081:
3078:
3072:
3051:
3029:
3028:
3027:
3025:
3000:
2975:
2971:
2951:
2948:
2945:
2941:
2931:
2925:
2916:
2910:
2905:
2901:
2896:
2892:
2889:
2884:
2880:
2874:
2870:
2866:
2863:
2859:
2853:
2849:
2843:
2833:
2825:
2820:
2814:
2810:
2804:
2800:
2795:
2791:
2788:
2784:
2778:
2774:
2768:
2764:
2759:
2754:
2748:
2744:
2738:
2728:
2720:
2712:
2711:
2710:
2692:
2689:
2686:
2682:
2672:
2666:
2657:
2651:
2648:
2643:
2639:
2634:
2630:
2627:
2622:
2618:
2612:
2602:
2591:
2590:
2589:
2572:
2569:
2566:
2562:
2552:
2546:
2537:
2531:
2528:
2523:
2519:
2514:
2510:
2507:
2502:
2498:
2492:
2488:
2480:
2479:
2478:
2459:
2451:
2447:
2437:
2433:
2426:
2421:
2418:
2411:
2410:
2409:
2385:
2378:
2375:
2372:
2364:
2353:
2350:
2346:
2342:
2339:
2331:
2328:
2325:
2317:
2306:
2303:
2299:
2295:
2292:
2287:
2279:
2275:
2265:
2261:
2254:
2249:
2245:
2241:
2238:
2229:
2225:
2218:
2217:
2216:
2214:
2204:
2191:
2185:
2182:
2171:
2165:
2155:
2145:
2121:
2118:
2107:
2101:
2091:
2087:
2078:
2074:
2055:
2047:
2039:
2035:
2031:
2024:
2020:
2016:
2009:
2005:
2001:
1994:
1990:
1983:
1978:
1973:
1963:
1952:
1951:
1950:
1933:
1930:
1927:
1922:
1912:
1902:
1891:
1890:
1889:
1887:
1883:
1867:
1847:
1844:
1841:
1838:
1835:
1832:
1829:
1826:
1818:
1802:
1799:
1796:
1774:
1770:
1762:
1758:
1737:
1729:
1725:
1717:
1713:
1705:
1701:
1693:
1689:
1682:
1677:
1672:
1664:
1660:
1652:
1648:
1640:
1636:
1628:
1624:
1617:
1612:
1607:
1603:
1595:
1594:
1593:
1576:
1573:
1567:
1559:
1548:
1544:
1540:
1534:
1526:
1515:
1511:
1507:
1501:
1493:
1482:
1478:
1474:
1468:
1460:
1449:
1446:
1439:
1435:
1427:
1426:
1425:
1424:, it becomes
1423:
1407:
1384:
1381:
1378:
1373:
1363:
1359:
1351:
1350:
1349:
1341:
1339:
1335:
1331:
1327:
1323:
1319:
1315:
1311:
1306:
1303:
1299:
1295:
1290:
1288:
1284:
1280:
1276:
1272:
1271:Wu experiment
1267:
1265:
1261:
1256:
1254:
1250:
1246:
1242:
1238:
1234:
1230:
1226:
1222:
1212:
1209:
1207:
1203:
1199:
1195:
1191:
1187:
1183:
1178:
1176:
1172:
1168:
1164:
1163:Weyl fermions
1160:
1156:
1152:
1151:Weyl equation
1148:
1144:
1132:
1127:
1125:
1120:
1118:
1113:
1112:
1110:
1109:
1103:
1098:
1095:
1093:
1090:
1088:
1085:
1083:
1080:
1078:
1075:
1073:
1072:Zamolodchikov
1070:
1068:
1067:Zamolodchikov
1065:
1063:
1060:
1058:
1055:
1053:
1050:
1048:
1045:
1043:
1040:
1038:
1035:
1033:
1030:
1028:
1025:
1023:
1020:
1018:
1015:
1013:
1010:
1008:
1005:
1003:
1000:
998:
995:
993:
990:
988:
985:
983:
980:
978:
975:
973:
970:
968:
965:
963:
960:
958:
955:
953:
950:
948:
945:
943:
940:
938:
935:
933:
930:
928:
925:
923:
920:
918:
915:
913:
910:
908:
905:
903:
900:
898:
895:
893:
890:
888:
885:
883:
880:
878:
875:
873:
870:
868:
865:
863:
860:
858:
855:
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:
623:
620:
618:
615:
613:
610:
608:
605:
603:
600:
598:
595:
593:
590:
588:
585:
583:
580:
578:
575:
573:
570:
568:
565:
563:
560:
558:
555:
553:
550:
548:
545:
543:
540:
538:
535:
533:
530:
528:
525:
523:
520:
518:
515:
513:
510:
508:
505:
503:
500:
498:
495:
493:
490:
488:
485:
483:
480:
478:
475:
473:
470:
468:
465:
463:
460:
458:
455:
453:
450:
448:
445:
443:
440:
438:
435:
433:
430:
428:
425:
423:
420:
418:
415:
413:
410:
409:
402:
401:
394:
391:
389:
386:
384:
381:
379:
378:Supersymmetry
376:
374:
373:String theory
371:
370:
364:
363:
356:
353:
351:
348:
346:
343:
341:
338:
337:
334:
329:
328:
321:
318:
316:
313:
311:
308:
306:
303:
301:
298:
297:
291:
290:
283:
280:
278:
275:
273:
270:
268:
265:
263:
260:
258:
255:
253:
250:
248:
245:
243:
240:
238:
235:
233:
230:
228:
225:
223:
220:
218:
215:
213:
210:
208:
205:
203:
200:
198:
195:
194:
188:
187:
180:
177:
175:
172:
170:
167:
165:
162:
160:
157:
155:
152:
150:
147:
145:
142:
140:
137:
135:
132:
130:
127:
126:
123:
118:
117:
110:
107:
105:
102:
100:
97:
95:
92:
90:
87:
85:
82:
80:
77:
75:
72:
70:
67:
66:
60:
59:
56:
53:
52:
48:
43:
39:
38:
35:
32:
31:
19:
11189:
11185:
11169:. Retrieved
11164:
11125:
11121:
11105:. Retrieved
11101:
11076:
11056:
11033:
11013:
10993:
10974:
10968:
10941:
10912:
10905:
10886:
10880:
10861:
10855:
10828:
10822:
10812:
10803:
10737:
10733:
10659:
10655:
10642:
10591:
10587:
10553:
10547:
10523:
10506:
10427:
10194:
10013:Dirac spinor
10010:
9916:
9648:
9610:
9602:
9568:frame bundle
9521:double cover
9471:fiber bundle
9445:
9435:dimensional
9307:
9206:
9002:vector space
9000:dimensional
8824:
8770:
8737:
8735:
8732:Weyl spinors
8708:(denoted by
8703:
8512:
8355:
7988:
7985:
7901:
7752:
7560:
7423:
7336:
7194:
7054:
6889:
6760:
6556:
6472:
6321:
6260:
6080:
5705:
5476:
5419:
5313:
4848:
4762:
4646:
4537:
4397:
4168:
3963:
3823:
3489:
3440:
3274:
3135:
2967:
2708:
2587:
2476:
2407:
2210:
2070:
1948:
1882:wavefunction
1754:
1591:
1399:
1347:
1307:
1291:
1273:showed that
1268:
1257:
1218:
1210:
1180:None of the
1179:
1167:Hermann Weyl
1162:
1150:
1140:
1101:
947:Stueckelberg
687:Jona-Lasinio
277:Vacuum state
262:Quantization
104:Gauge theory
84:Strong force
69:Field theory
11171:22 November
11165:APS Physics
11107:22 November
10804:APS Physics
10509:ELKO spinor
8738:Weyl spinor
3964:The matrix
2999:wave-vector
2075:, and thus
1330:M. SoljaÄiÄ
1087:Zinn-Justin
937:Sommerfield
862:Pomeranchuk
832:Osterwalder
827:Oppenheimer
757:ĆopuszaĆski
582:Fredenhagen
383:Technicolor
18:Weyl spinor
11199:1612.00416
11128:(9): 863.
10540:References
9643:See also:
9441:spin group
9044:on it. If
8816:spin group
8473:, that is
6373:isomorphic
3494:under the
2213:plane-wave
1318:M.Z. Hasan
1253:beta decay
1225:Paul Dirac
1082:Zimmermann
977:Vainshtein
722:Kontsevich
667:Iliopoulos
642:Heisenberg
467:Bogoliubov
405:Scientists
257:Propagator
144:T-symmetry
139:P-symmetry
134:C-symmetry
122:Symmetries
79:Weak force
63:Background
11144:1476-1122
10960:2073-8994
10780:119341278
10772:0031-8949
10740:: 17â22.
10686:0027-8424
10634:118685467
10626:0002-9505
10601:1006.1718
10407:−
10398:−
10347:×
10283:×
10213:≅
10155:×
10122:≅
9989:−
9980:Δ
9948:Δ
9860:−
9851:Δ
9847:⊕
9833:Δ
9789:⊕
9649:For even
9343:−
9266:−
9228:∗
8917:∈
8808:spacetime
8794:ψ
8781:rotations
8773:invariant
8736:The term
8716:†
8706:conjugate
8675:ψ
8669:μ
8665:∂
8659:μ
8652:¯
8649:σ
8640:†
8629:ψ
8578:ψ
8572:μ
8568:∂
8562:μ
8558:σ
8552:†
8541:ψ
8487:ζ
8481:η
8453:∗
8449:ζ
8445:η
8409:−
8392:and that
8377:−
8365:ω
8324:∗
8320:ζ
8316:η
8310:◻
8302:−
8279:∗
8275:ζ
8271:η
8262:→
8259:∇
8253:⋅
8247:→
8244:∇
8238:−
8224:∂
8215:−
8201:ω
8195:ζ
8187:μ
8183:∂
8177:μ
8171:¯
8168:σ
8146:ω
8140:η
8132:μ
8128:∂
8122:μ
8118:σ
7957:ψ
7926:ψ
7879:ψ
7867:′
7856:ψ
7852:↦
7841:ψ
7828:ψ
7819:−
7809:†
7791:′
7780:ψ
7776:↦
7765:ψ
7733:−
7708:−
7698:†
7680:′
7665:↦
7647:†
7619:′
7604:↦
7543:ω
7537:η
7529:μ
7525:∂
7519:μ
7515:σ
7489:ω
7483:ζ
7475:μ
7471:∂
7465:μ
7459:¯
7456:σ
7398:ψ
7374:ψ
7365:σ
7353:∗
7349:ψ
7345:ω
7305:∗
7294:ψ
7290:ω
7284:ζ
7261:ψ
7255:μ
7251:∂
7245:μ
7239:¯
7236:σ
7207:ζ
7169:∗
7158:ψ
7154:ω
7148:η
7125:ψ
7119:μ
7115:∂
7109:μ
7105:σ
7077:ϕ
7063:η
7029:∗
7018:ψ
7014:ω
7003:−
6993:†
6971:′
6957:∗
6954:′
6943:ψ
6939:ω
6933:↦
6919:∗
6908:ψ
6904:ω
6864:∗
6853:ψ
6847:∗
6830:′
6816:∗
6813:′
6802:ψ
6798:↦
6784:∗
6773:ψ
6729:ψ
6713:′
6699:′
6688:ψ
6684:↦
6664:ψ
6634:∗
6609:ω
6601:−
6591:†
6573:∗
6565:ω
6526:−
6494:σ
6484:ω
6458:ω
6449:ω
6295:−
6285:†
6229:ψ
6213:′
6199:′
6188:ψ
6184:↦
6164:ψ
6123:that is,
6094:−
6045:ψ
6033:−
6023:μ
6019:∂
6013:μ
6009:σ
6003:†
5990:−
5950:ψ
5941:ν
5937:∂
5931:ν
5924:μ
5920:Λ
5912:μ
5908:σ
5877:ψ
5868:ν
5864:∂
5858:ν
5851:μ
5831:−
5827:Λ
5814:μ
5810:σ
5782:ψ
5773:ν
5769:∂
5763:μ
5756:ν
5743:−
5739:Λ
5726:μ
5722:σ
5706:to write
5689:ν
5682:μ
5662:−
5658:Λ
5642:μ
5635:ν
5622:−
5618:Λ
5561:∈
5558:Λ
5528:−
5519:−
5510:−
5485:η
5457:−
5453:Λ
5446:η
5435:Λ
5431:η
5400:−
5390:ν
5386:σ
5380:†
5367:−
5349:ν
5342:μ
5338:Λ
5330:μ
5326:σ
5278:ψ
5269:ν
5265:∂
5259:μ
5252:ν
5239:−
5235:Λ
5222:μ
5218:σ
5187:ψ
5175:ν
5167:∂
5163:∂
5156:μ
5149:ν
5136:−
5132:Λ
5119:μ
5115:σ
5084:ψ
5072:ν
5064:∂
5060:∂
5050:μ
5047:′
5039:∂
5032:ν
5024:∂
5016:μ
5012:σ
4992:′
4978:′
4967:ψ
4958:μ
4955:′
4947:∂
4943:∂
4936:μ
4932:σ
4915:′
4901:′
4890:ψ
4884:′
4879:μ
4875:∂
4869:μ
4865:σ
4832:μ
4829:′
4819:μ
4812:ν
4799:−
4795:Λ
4779:ν
4746:ν
4736:ν
4729:μ
4725:Λ
4714:μ
4711:′
4658:∈
4615:ψ
4599:′
4585:′
4574:ψ
4570:↦
4550:ψ
4500:ψ
4491:−
4481:†
4459:′
4445:′
4434:ψ
4430:↦
4410:ψ
4366:ψ
4357:μ
4349:∂
4345:∂
4338:μ
4332:¯
4329:σ
4311:′
4297:′
4286:ψ
4277:μ
4274:′
4266:∂
4262:∂
4255:μ
4249:¯
4246:σ
4240:↦
4220:ψ
4211:μ
4203:∂
4199:∂
4192:μ
4186:¯
4183:σ
4149:−
4139:ν
4135:σ
4129:†
4116:−
4098:ν
4091:μ
4087:Λ
4079:μ
4075:σ
4063:given by
3975:∈
3932:ψ
3916:′
3902:′
3891:ψ
3887:↦
3867:ψ
3837:†
3792:ψ
3783:μ
3775:∂
3771:∂
3764:μ
3760:σ
3754:†
3741:−
3719:′
3705:′
3694:ψ
3685:μ
3682:′
3674:∂
3670:∂
3663:μ
3659:σ
3655:↦
3635:ψ
3626:μ
3618:∂
3614:∂
3607:μ
3603:σ
3550:∈
3547:Λ
3524:Λ
3516:′
3508:↦
3455:±
3449:λ
3421:λ
3390:λ
3379:λ
3358:⋅
3283:λ
3227:ψ
3221:μ
3217:∂
3211:μ
3204:¯
3201:σ
3172:ψ
3166:μ
3162:∂
3156:μ
3152:σ
3116:ω
3092:⇒
3082:ω
3079:ℏ
3055:ℏ
2946:χ
2935:→
2926:⋅
2920:→
2911:−
2890:χ
2885:μ
2875:μ
2864:χ
2854:μ
2844:μ
2837:¯
2834:σ
2815:ν
2805:ν
2801:σ
2789:χ
2779:μ
2769:μ
2765:σ
2749:ν
2739:ν
2732:¯
2729:σ
2687:χ
2676:→
2667:⋅
2661:→
2658:σ
2628:χ
2623:μ
2613:μ
2606:¯
2603:σ
2567:χ
2556:→
2547:⋅
2541:→
2538:σ
2532:−
2508:χ
2503:μ
2493:μ
2489:σ
2448:χ
2434:χ
2419:χ
2390:ℏ
2373:−
2365:⋅
2351:−
2343:χ
2329:ω
2326:−
2318:⋅
2304:−
2296:χ
2276:ψ
2262:ψ
2226:ψ
2172:ψ
2166:μ
2162:∂
2156:μ
2149:¯
2146:σ
2108:ψ
2102:μ
2098:∂
2092:μ
2088:σ
2077:chirality
2036:σ
2032:−
2021:σ
2017:−
2006:σ
2002:−
1974:μ
1967:¯
1964:σ
1928:ψ
1923:μ
1919:∂
1913:μ
1906:¯
1903:σ
1868:ψ
1827:μ
1797:μ
1726:σ
1714:σ
1702:σ
1661:σ
1649:σ
1637:σ
1625:σ
1608:μ
1604:σ
1565:∂
1560:ψ
1557:∂
1545:σ
1532:∂
1527:ψ
1524:∂
1512:σ
1499:∂
1494:ψ
1491:∂
1479:σ
1466:∂
1461:ψ
1458:∂
1379:ψ
1374:μ
1370:∂
1364:μ
1360:σ
1258:In 1937,
1017:Wetterich
1002:Weisskopf
952:Sudarshan
902:Schwinger
817:Nishijima
782:Maldacena
747:Leutwyler
712:Kinoshita
612:Goldstone
602:Gell-Mann
517:Doplicher
294:Equations
11270:Category
11224:27777402
11152:26288972
10942:Symmetry
10704:16587474
9874:, where
9611:Given a
8754:fermions
8750:manifold
7390:form of
3425:⟩
3383:⟩
3265:Helicity
2974:momentum
2970:massless
2073:helicity
1815:and the
1420:for the
1344:Equation
1283:helicity
1249:neutrino
1229:spin-1/2
1194:neutrino
1173:and the
1159:spin-1/2
1032:Wightman
997:Weinberg
987:Virasoro
967:Tomonaga
962:Thirring
957:Symanzik
917:Semenoff
892:Schrader
857:Polyakov
777:Majorana
717:Klebanov
672:Ivanenko
662:'t Hooft
632:Guralnik
577:Fröhlich
572:Fritzsch
567:Frampton
482:Buchholz
427:Bargmann
417:Anderson
217:Crossing
11232:1115349
11204:Bibcode
10833:Bibcode
10752:Bibcode
10664:Bibcode
10606:Bibcode
9619:is the
9523:of the
8746:spinors
6648:is the
6375:to the
4018:by the
4014:of the
3851:is the
3022:by the
1886:spinors
1880:is the
1245:fermion
1215:History
1184:in the
1143:physics
1042:Wilczek
1007:Wentzel
982:Veltman
927:Shirkov
922:Shifman
912:Seiberg
897:Schwarz
877:Rubakov
802:Naimark
752:Lipatov
742:Lehmann
707:Kendall
597:Gelfand
592:Glashow
552:Feynman
532:Faddeev
527:Englert
497:Coleman
487:Cachazo
472:Brodsky
457:Bjorken
447:Berezin
437:Belavin
197:Anomaly
55:History
11230:
11222:
11150:
11142:
11085:
11064:
11041:
11020:
11001:
10958:
10920:
10893:
10868:
10778:
10770:
10702:
10695:522457
10692:
10684:
10632:
10624:
10560:
10195:where
9581:
9366:
9077:are a
8785:spinor
8777:boosts
8686:
8589:
8490:
8418:
7561:where
7401:
6621:where
6473:where
6411:
5477:where
4520:
3824:where
3576:
3539:where
3468:
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2189:
2053:
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1332:team (
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1037:Wigner
972:Tyutin
932:Skyrme
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492:Callan
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11228:S2CID
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7219:) is
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537:Fadin
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11039:ISBN
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762:Low
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