5298:
4844:
5293:{\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}}}
6065:
31:
5701:
4382:
8340:
3808:
6060:{\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}}}
2951:
4164:
8066:
3586:
1576:
2391:
1739:
2704:
7886:
7737:
7039:
4377:{\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)}
8335:{\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)}
3248:
3803:{\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)}
10516:
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
10421:
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
7545:
10004:, 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
7188:. 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
3425:
9596:
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)
1419:
4522:
2210:
1587:
9861:
3120:
2055:
6874:
7321:
2946:{\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}
8688:
5690:
2692:
6745:
6245:
4631:
3948:
7748:
10179:
8591:
5404:
4153:
7576:
2572:
7179:
1293:
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.
6885:
7405:
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
3131:
6541:
4833:
7419:
7970:
10500:, 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.
3337:
9292:
2461:
2191:
1571:{\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}
9191:
4393:
6608:
2386:{\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 }}
4747:
2121:
5461:
1933:
1734:{\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}}}
9736:
3021:
1944:
1384:
10060:
9428:, 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
6756:
5706:
4849:
3136:
7373:
4682:
6302:
10356:
5533:
3578:
6457:
6413:
5586:
10460:
3526:
7214:
6358:
4050:
10297:
10226:
3470:
8599:
5594:
9368:
7881:{\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}}}
3997:
2583:
8058:
8017:
6647:
6147:
4533:
3850:
9697:
9506:
10068:
9553:
9031:
8460:
7732:{\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}}
9731:
8511:
7078:
5309:
4058:
10491:
10257:
9990:
9958:
8492:
8379:
7034:{\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)}
2472:
10412:
7085:
3838:
3329:
3307:
3009:
2984:
9422:
1847:
6110:
1258:
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
9902:
9064:
8715:
8420:
3282:
8922:
7403:
9098:
8955:
7206:
1802:
8987:
8867:
8793:
6635:
6311:
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
1867:
1776:
6139:
9583:
9456:
8893:
3243:{\displaystyle {\begin{aligned}\sigma ^{\mu }\partial _{\mu }\psi _{\rm {R}}&=0\\{\bar {\sigma }}^{\mu }\partial _{\mu }\psi _{\rm {L}}&=0\end{aligned}}}
9926:
9656:
8835:
7568:
1407:
7540:{\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}
6468:
4758:
3420:{\displaystyle \mathbf {p} \cdot \mathbf {J} \left|\mathbf {p} ,\lambda \right\rangle =\lambda |\mathbf {p} |\left|\mathbf {p} ,\lambda \right\rangle }
7897:
9202:
2403:
2126:
9106:
1117:
4517:{\displaystyle \psi _{\rm {L}}(x)\mapsto \psi _{\rm {L}}^{\prime }\left(x^{\prime }\right)=\left(S^{\dagger }\right)^{-1}\psi _{\rm {L}}(x)~.}
10497:
9620:; the Clifford bundle is the space in which the spinors live. The general exploration of such structures and their relationships is termed
7044:
which is exactly the same
Lorentz covariance property noted earlier. Thus, the linear combination, using an arbitrary complex phase factor
2204:
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
200:
6549:
4690:
4527:
Proof: Neither of these transformation properties are in any way "obvious", and so deserve a careful derivation. Begin with the form
2071:
240:
9612:; this is effectively "the same thing" as the normal connection, just with spin indexes attached to it in a consistent fashion. The
5415:
1883:
8761:
9856:{\displaystyle \mathrm {End} (\mathbb {C} ^{N/2})\oplus \mathrm {End} (\mathbb {C} ^{N/2})=:\Delta _{n}^{+}\oplus \Delta _{n}^{-}}
3115:{\displaystyle |\mathbf {p} |=\hbar |\mathbf {k} |={\frac {\hbar \omega }{c}}\,\Rightarrow \,|\mathbf {k} |={\frac {\omega }{c}}}
2050:{\displaystyle {\bar {\sigma }}^{\mu }={\begin{pmatrix}I_{2}&-\sigma _{x}&-\sigma _{y}&-\sigma _{z}\end{pmatrix}}~.}
11086:
9435:
The abstract, general-relativistic form of the Weyl equation can be understood as follows: given a pseudo-Riemannian manifold
11075:
11054:
11010:
10991:
10883:
10858:
1322:
6869:{\displaystyle \psi _{\rm {R}}^{*}(x)\mapsto \psi _{\rm {R}}^{\prime *}\left(x^{\prime }\right)=S^{*}\psi _{\rm {R}}^{*}(x)}
1343:
10007:
975:
9633:
1200:
Mathematically, any Dirac fermion can be decomposed as two Weyl fermions of opposite chirality coupled by the mass term.
7329:
7326:
As noted earlier, the left and right chiral versions are related by a parity transformation. The skew complex conjugate
4639:
6253:
1110:
10302:
7316:{\displaystyle i{\overline {\sigma }}^{\mu }\partial _{\mu }\psi _{\rm {L}}(x)+\zeta m\omega \psi _{\rm {L}}^{*}(x)=0}
5469:
3531:
11031:
10910:
10550:
6421:
6370:
5542:
10425:
3489:
6321:
4158:
Thus, if the untransformed differential vanishes in one
Lorentz frame, then it also vanishes in another. Similarly
4013:
1745:
745:
8683:{\displaystyle {\mathcal {L}}=i\psi _{\rm {L}}^{\dagger }{\bar {\sigma }}^{\mu }\partial _{\mu }\psi _{\rm {L}}~.}
5685:{\displaystyle {\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }={\left(\Lambda ^{-1{\mathsf {T}}}\right)_{\mu }}^{\nu }}
43:
10266:
10187:
2687:{\displaystyle {\bar {\sigma }}^{\mu }p_{\mu }\chi =\left(I_{2}E+{\vec {\sigma }}\cdot {\vec {p}}\right)\chi =0}
1274:
in 1958. As experiments showed no signs of a neutrino mass, interest in the Weyl equation resurfaced. Thus, the
6740:{\displaystyle \psi _{\rm {R}}(x)\mapsto \psi _{\rm {R}}^{\prime }\left(x^{\prime }\right)=S\psi _{\rm {R}}(x)}
6240:{\displaystyle \psi _{\rm {L}}(x)\mapsto \psi _{\rm {L}}^{\prime }\left(x^{\prime }\right)=L\psi _{\rm {L}}(x)}
4626:{\displaystyle \psi _{\rm {R}}(x)\mapsto \psi _{\rm {R}}^{\prime }\left(x^{\prime }\right)=R\psi _{\rm {R}}(x)}
3943:{\displaystyle \psi _{\rm {R}}(x)\mapsto \psi _{\rm {R}}^{\prime }\left(x^{\prime }\right)=S\psi _{\rm {R}}(x)}
3433:
1221:
9304:
11244:
3956:
1103:
308:
157:
117:
10174:{\displaystyle \mathrm {Spin} ^{\mathbb {C} }(p,q)\cong \mathrm {Spin} (p,q)\times _{\mathbb {Z} _{2}}S^{1}}
8022:
7981:
2060:
The solutions of the right- and left-handed Weyl equations are different: they have right- and left-handed
845:
775:
147:
9661:
9465:
7570:
is a short-hand reminder to take the complex conjugate. Under
Lorentz transformations, these transform as
8733:. This is closely related to the solutions given above, and gives a natural geometric interpretation to
8586:{\displaystyle {\mathcal {L}}=i\psi _{\rm {R}}^{\dagger }\sigma ^{\mu }\partial _{\mu }\psi _{\rm {R}}~,}
5399:{\displaystyle \sigma _{\mu }{\Lambda ^{\mu }}_{\nu }=\left(S^{-1}\right)^{\dagger }\sigma _{\nu }S^{-1}}
4148:{\displaystyle \sigma _{\mu }{\Lambda ^{\mu }}_{\nu }=\left(S^{-1}\right)^{\dagger }\sigma _{\nu }S^{-1}}
3259:
2061:
1314:
1271:
303:
245:
9518:
8996:
1158:. The Weyl fermions are one of the three possible types of elementary fermions, the other two being the
11264:
8815:
8754:
8429:
2567:{\displaystyle \sigma ^{\mu }p_{\mu }\chi =\left(I_{2}E-{\vec {\sigma }}\cdot {\vec {p}}\right)\chi =0}
1143:
152:
9702:
7047:
955:
10718:
10465:
10231:
9963:
9931:
9371:
8423:
7174:{\displaystyle i\sigma ^{\mu }\partial _{\mu }\psi _{\rm {R}}(x)+\eta m\omega \psi _{\rm {R}}^{*}(x)}
3285:
293:
9616:
can be defined in terms of the connection in an entirely conventional way. It acts naturally on the
8465:
8348:
5409:
a few indexes must be raised and lowered. This is easier said than done, as it invokes the identity
10361:
9513:
9298:
8807:. This is entirely analogous to how one might talk about a vector, and how it transforms under the
1186:
780:
410:
255:
220:
10462:
group. That is, it transforms as a spinor, but transversally, such that it is invariant under the
3816:
3312:
3290:
2992:
2967:
770:
9377:
1811:
1055:
525:
333:
328:
190:
6073:
9866:
9036:
7891:
just as above. Thus, the matched combinations of these are
Lorentz covariant, and one may take
3484:
925:
905:
405:
338:
250:
215:
10000:
There are three important special cases that can be constructed from Weyl spinors. One is the
9374:, thus allowing these abstractly defined formal structures to be interpreted as fermions. For
8700:
8384:
3267:
9593:
8898:
7382:
1224:. Hermann Weyl published his equation in 1929 as a simplified version of the Dirac equation.
815:
675:
371:
235:
97:
9073:
8930:
7191:
1781:
1185:
might be a Weyl fermion (it is now expected to be either a Dirac or a
Majorana fermion). In
11192:
10821:
10740:
10652:
10594:
9613:
9425:
8960:
8840:
8778:
6613:
6316:
4008:
3012:
1852:
1754:
1310:
1286:
1178:
1135:
1060:
840:
376:
230:
22:
10752:
6115:
142:
8:
11066:
9562:
9509:
8750:
4000:
3841:
1318:
1170:
1070:
400:
11196:
10825:
10744:
10656:
10598:
9438:
8875:
7184:
transforms in a covariant fashion; setting this to zero gives the complex two-component
1337:
The Weyl equation comes in two forms. The right-handed form can be written as follows:
785:
515:
11216:
11182:
10764:
10730:
10618:
10584:
9911:
9641:
9067:
8820:
8772:
are applied, the form of the equation itself does not change. However, the form of the
8746:
8503:
7553:
7407:
7376:
1392:
1270:, addressing Pauli's criticism. This was followed by the measurement of the neutrino's
1080:
1005:
935:
820:
805:
735:
700:
690:
650:
565:
455:
415:
205:
167:
122:
110:
87:
82:
10683:
10640:
11208:
11136:
11128:
11071:
11050:
11027:
11006:
10987:
10944:
10906:
10879:
10854:
10768:
10756:
10688:
10670:
10622:
10610:
10546:
9605:
8694:
7185:
6638:
6312:
3480:
1302:
1232:. However, three years before, Pauli had predicted the existence of a new elementary
1035:
965:
850:
695:
630:
600:
570:
430:
425:
270:
195:
185:
77:
8741:. This general setting has multiple strengths: it clarifies their interpretation as
6536:{\displaystyle \omega =i\sigma _{2}={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}}
6415:
The symplectic group is defined as the set of all complex 2Ă2 matrices that satisfy
755:
11220:
11200:
11118:
10934:
10829:
10748:
10678:
10660:
10602:
10260:
8800:
8494:
These two
Majorana operators are thus "square roots" of the KleinâGordon operator.
6365:
6141:
Performing the same manipulations for the left-handed equation, one concludes that
5536:
1326:
1290:
1282:
1267:
1263:
1229:
1163:
1075:
990:
970:
940:
890:
715:
710:
705:
590:
575:
505:
210:
137:
127:
62:
10572:
4828:{\displaystyle x^{\nu }={\left(\Lambda ^{-1}\right)^{\nu }}_{\mu }x^{\prime \mu }}
11044:
11021:
10981:
10900:
10812:
10418:
9617:
9609:
8730:
1749:
1248:
1020:
1015:
985:
980:
950:
910:
880:
860:
765:
660:
655:
620:
605:
580:
560:
540:
470:
460:
381:
343:
260:
225:
57:
35:
30:
11234:
9586:
8870:
8808:
7965:{\displaystyle D_{\rm {L}}\psi _{\rm {L}}=0\qquad D_{\rm {R}}\psi _{\rm {R}}=0}
1805:
1410:
1306:
1298:
1275:
1252:
1225:
1209:
1194:
1174:
1045:
995:
915:
900:
885:
865:
835:
830:
825:
810:
795:
760:
750:
680:
615:
585:
520:
485:
475:
450:
321:
288:
162:
11239:
800:
11258:
11132:
10948:
10834:
10807:
10760:
10674:
10614:
10518:
9621:
9601:
9297:
When properly examined in light of the
Clifford algebra, these are naturally
9287:{\displaystyle w_{j}^{*}={\frac {1}{\sqrt {2}}}\left(e_{2j}-ie_{2j+1}\right)}
8925:
8765:
4004:
1259:
1190:
1159:
1050:
1040:
1030:
1025:
945:
930:
730:
725:
555:
550:
510:
480:
445:
435:
366:
361:
10806:
Wu, C. S.; Ambler, E.; Hayward, R. W.; Hoppes, D. D.; Hudson, R. P. (1957).
11212:
11170:
11140:
10692:
10636:
10001:
9556:
9459:
8990:
1870:
1155:
1085:
1010:
895:
870:
855:
790:
670:
665:
545:
535:
530:
495:
490:
465:
420:
298:
265:
92:
72:
11245:
http://www.tfkp.physik.uni-erlangen.de/download/research/DW-derivation.pdf
10665:
3125:
The equation can be written in terms of left and right handed spinors as:
2456:{\displaystyle \chi ={\begin{pmatrix}\chi _{1}\\\chi _{2}\\\end{pmatrix}}}
2186:{\displaystyle {\bar {\sigma }}^{\mu }\partial _{\mu }\psi _{\rm {L}}=0~.}
2068:, respectively. It is convenient to indicate this explicitly, as follows:
10788:
9186:{\displaystyle w_{j}={\frac {1}{\sqrt {2}}}\left(e_{2j}+ie_{2j+1}\right)}
2987:
1065:
1000:
960:
920:
875:
740:
645:
640:
625:
610:
595:
11249:
11149:
10939:
10735:
9429:
9370:
This can be happily interpreted as the mathematical realization of the
8804:
6361:
2201:
1241:
1213:
720:
635:
500:
440:
132:
67:
10606:
8799:
entirely, the algebra of the spinors is described by a (complexified)
8729:
is also frequently used in a more general setting, as an element of a
11204:
11123:
11106:
8796:
2065:
8717:) as independent variables, the relevant Weyl equation is obtained.
11187:
8769:
8738:
2962:
1877:. The left-handed form of the Weyl equation is usually written as:
1237:
1217:
1182:
1147:
685:
10589:
8993:. Given this vector space, one can construct the Clifford algebra
6603:{\displaystyle \omega S^{*}=\left(S^{\dagger }\right)^{-1}\omega }
2957:
and concludes that the equations correspond to a particle that is
1278:
was built under the assumption that neutrinos were Weyl fermions.
9592:
Selecting a single point on the fiber corresponds to selecting a
8742:
1874:
1233:
1131:
4742:{\displaystyle x^{\prime \mu }={\Lambda ^{\mu }}_{\nu }x^{\nu }}
8811:, except that now, it has been adapted to the case of spinors.
8773:
8734:
2116:{\displaystyle \sigma ^{\mu }\partial _{\mu }\psi _{\rm {R}}=0}
5456:{\displaystyle \eta \Lambda ^{\mathsf {T}}\eta =\Lambda ^{-1}}
1928:{\displaystyle {\bar {\sigma }}^{\mu }\partial _{\mu }\psi =0}
11169:
Jia, Shuang; Xu, Su-Yang; Hasan, M. Zahid (25 October 2016).
9555:, and so one can identify the spin group fiber-wise with the
2466:
is a momentum-dependent two-component spinor which satisfies
11005:. Manchester Physics (2nd ed.). John Wiley & Sons.
4684:
to be determined. The
Lorentz transform, in coordinates, is
1285:
had proposed in 1957 the possibility of neutrino masses and
9462:
above it, with the spin group as the fiber. The spin group
8764:
under the action of the Lorentz group. This means that, as
2958:
10929:
Aste, Andreas (2010). "A direct road to Majorana fields".
6641:. The right handed field, as noted earlier, transforms as
5539:. The above identity is often used to define the elements
1244:, which eventually was described using the Weyl equation.
1228:
wrote in 1933 against Weyl's equation because it violated
8745:
in physics, and it shows precisely how to define spin in
3264:
The left and right components correspond to the helicity
10808:"Experimental Test of Parity Conservation in Beta Decay"
9585:
When this is done, the resulting structure is called a
8060:
are both Lorentz covariant. The product is explicitly
6879:
Applying the defining relationship, one concludes that
1193:
that behave as Weyl fermions, leading to the notion of
1177:
are Weyl fermions. Previous to the confirmation of the
16:
Relativistic wave equation describing massless fermions
6499:
5480:
3436:
2418:
2246:
1975:
1674:
1609:
10805:
10468:
10428:
10364:
10305:
10269:
10234:
10190:
10071:
10010:
9966:
9934:
9914:
9869:
9739:
9705:
9664:
9644:
9565:
9521:
9468:
9441:
9380:
9307:
9205:
9109:
9076:
9039:
8999:
8963:
8933:
8901:
8878:
8843:
8823:
8781:
8703:
8602:
8514:
8468:
8432:
8387:
8351:
8069:
8025:
7984:
7900:
7751:
7579:
7556:
7422:
7385:
7332:
7217:
7194:
7088:
7050:
6888:
6759:
6650:
6616:
6552:
6471:
6424:
6373:
6324:
6256:
6150:
6118:
6076:
5704:
5597:
5545:
5472:
5418:
5312:
4847:
4761:
4693:
4642:
4536:
4396:
4167:
4061:
4016:
3959:
3853:
3844:, provided that the right-handed field transforms as
3819:
3589:
3534:
3492:
3340:
3315:
3293:
3270:
3134:
3024:
2995:
2970:
2707:
2586:
2475:
2406:
2213:
2129:
2074:
1947:
1886:
1855:
1814:
1784:
1757:
1590:
1422:
1395:
1379:{\displaystyle \sigma ^{\mu }\partial _{\mu }\psi =0}
1346:
10892:
10055:{\displaystyle \mathrm {Spin} ^{\mathbb {C} }(p,q).}
7413:
Define a pair of operators, the Majorana operators,
3999:
is related to the Lorentz transform by means of the
10933:. Vol. 2010, no. 2. pp. 1776â1809.
10543:ITEP Lectures on Particle Physics and Field Theory
10485:
10454:
10406:
10350:
10291:
10251:
10220:
10173:
10054:
9984:
9952:
9920:
9896:
9855:
9725:
9691:
9650:
9577:
9547:
9500:
9450:
9416:
9362:
9286:
9185:
9092:
9058:
9025:
8981:
8949:
8916:
8887:
8861:
8829:
8787:
8709:
8682:
8585:
8486:
8454:
8414:
8373:
8334:
8052:
8011:
7975:as a pair of complex 2-spinor Majorana equations.
7964:
7880:
7731:
7562:
7539:
7397:
7368:{\displaystyle \omega \psi ^{*}=i\sigma ^{2}\psi }
7367:
7315:
7200:
7173:
7072:
7033:
6868:
6739:
6629:
6602:
6535:
6451:
6407:
6352:
6296:
6239:
6133:
6104:
6059:
5684:
5580:
5527:
5455:
5398:
5292:
4827:
4741:
4677:{\displaystyle R\in \mathrm {SL} (2,\mathbb {C} )}
4676:
4625:
4516:
4387:provided that the left-handed field transforms as
4376:
4147:
4044:
3991:
3942:
3832:
3802:
3572:
3520:
3464:
3419:
3323:
3301:
3276:
3242:
3114:
3003:
2978:
2945:
2686:
2566:
2455:
2385:
2185:
2115:
2049:
1927:
1861:
1841:
1796:
1770:
1733:
1570:
1401:
1378:
10782:
10780:
10778:
6750:and so the complex conjugate field transforms as
6297:{\displaystyle L=\left(S^{\dagger }\right)^{-1}.}
11256:
11171:"Weyl semimetals, Fermi arcs and chiral anomaly"
10351:{\displaystyle \mathrm {Spin} (p,q)\times S^{1}}
8803:. The spinors transform under the action of the
5528:{\displaystyle \eta ={\mbox{diag}}(+1,-1,-1,-1)}
3573:{\displaystyle \Lambda \in \mathrm {SO} (1,3)~.}
10898:
10645:Proceedings of the National Academy of Sciences
6452:{\displaystyle S^{\mathsf {T}}\omega S=\omega }
6408:{\displaystyle \mathrm {Sp} (2,\mathbb {C} )~.}
5581:{\displaystyle \Lambda \in \mathrm {SO} (1,3).}
1301:was demonstrated experimentally in crystalline
10775:
10712:
10710:
10708:
10706:
10704:
10702:
10455:{\displaystyle \mathrm {spin} ^{\mathbb {C} }}
10228:is the circle, and can be identified with the
9100:, one may construct a pair of Weyl spinors as
6546:The defining relationship can be rewritten as
3521:{\displaystyle x\mapsto x^{\prime }=\Lambda x}
9904:. A left-handed (respectively, right-handed)
8345:Verifying this requires keeping in mind that
6353:{\displaystyle \mathrm {SL} (2,\mathbb {C} )}
4045:{\displaystyle \mathrm {SL} (2,\mathbb {C} )}
1111:
11235:http://aesop.phys.utk.edu/qft/2004-5/2-2.pdf
10924:
10922:
10299:is just fancy notation denoting the product
9053:
9040:
11240:http://www.nbi.dk/~kleppe/random/ll/l2.html
10842:
10699:
10534:
6306:
3580:More precisely, the equations transform as
11168:
11147:
11019:
11000:
10964:Riemannian Geometry and Geometric Analysis
10876:The Cambridge Handbook of Physics Formulas
10786:
10566:
10564:
10562:
10292:{\displaystyle \times _{\mathbb {Z} _{2}}}
10221:{\displaystyle S^{1}\cong \mathrm {U} (1)}
9627:
8757:. This is informally sketched as follows.
3465:{\textstyle \lambda =\pm {\frac {1}{2}}~.}
2698:By direct manipulation, one obtains that
1118:
1104:
29:
11186:
11122:
10938:
10919:
10833:
10734:
10682:
10664:
10588:
10446:
10277:
10149:
10089:
10028:
9796:
9756:
9707:
9666:
6392:
6343:
4667:
4035:
3982:
3083:
3079:
1325:) also observed Weyl-like excitations in
1251:proposed that Weyl fermions may exist as
11087:"Weyl fermions are spotted at long last"
11084:
10867:
10629:
9363:{\displaystyle w_{j}w_{m}=-w_{m}w_{j}~.}
11148:Vishwanath, Ashvin (8 September 2015).
11063:
11042:
11020:Martin, Brian R.; Shaw, Graham (2013).
10979:
10902:An Introduction to Quantum Field Theory
10848:
10716:
10559:
10540:
8497:
3992:{\displaystyle S\in SL(2,\mathbb {C} )}
2195:
1389:Expanding this equation, and inserting
1317:) teams. Independently, the same year,
11257:
11104:
10966:(3rd ed.). Springer Universitext.
10899:Peskin, M.E.; Schroeder, D.V. (1995).
10719:"The history of neutrino oscillations"
8053:{\displaystyle D_{\rm {R}}D_{\rm {L}}}
8012:{\displaystyle D_{\rm {L}}D_{\rm {R}}}
7742:whereas the Weyl spinors transform as
6431:
6070:One thus regains the original form if
5827:
5658:
5428:
1181:, it was considered possible that the
11250:http://www.weylmann.com/weyldirac.pdf
10853:. Addison Wesley, Prentice Hall Inc.
10799:
5303:In order to make use of the Weyl map
3474:
1323:Massachusetts Institute of Technology
10961:
10928:
10873:
10635:
10573:"Dirac, Majorana, and Weyl fermions"
9928:-dimensional space is an element of
9692:{\displaystyle \mathbb {C} l^{0}(n)}
9501:{\displaystyle \mathrm {Spin} (p,q)}
8502:The equations are obtained from the
3284:of the particles, the projection of
10570:
9634:classification of Clifford algebras
7410:and Majorana equation for details.
13:
10973:
10470:
10440:
10437:
10434:
10431:
10316:
10313:
10310:
10307:
10236:
10205:
10123:
10120:
10117:
10114:
10083:
10080:
10077:
10074:
10022:
10019:
10016:
10013:
9968:
9936:
9839:
9821:
9787:
9784:
9781:
9747:
9744:
9741:
9548:{\displaystyle \mathrm {SO} (p,q)}
9526:
9523:
9479:
9476:
9473:
9470:
9026:{\displaystyle \mathrm {Cl} (p,q)}
9004:
9001:
8668:
8653:
8622:
8605:
8571:
8556:
8534:
8517:
8247:
8232:
8212:
8171:
8116:
8088:
8076:
8044:
8032:
8003:
7991:
7950:
7938:
7919:
7907:
7872:
7855:
7849:
7834:
7821:
7779:
7773:
7758:
7710:
7668:
7662:
7647:
7624:
7607:
7601:
7586:
7513:
7490:
7459:
7429:
7287:
7254:
7239:
7151:
7118:
7103:
7011:
6959:
6942:
6936:
6901:
6846:
6818:
6801:
6795:
6766:
6722:
6701:
6687:
6681:
6657:
6378:
6375:
6329:
6326:
6222:
6201:
6187:
6181:
6157:
6038:
6007:
5943:
5925:
5908:
5870:
5852:
5815:
5775:
5757:
5727:
5646:
5606:
5556:
5553:
5546:
5441:
5423:
5326:
5271:
5253:
5223:
5180:
5155:
5151:
5120:
5077:
5052:
5048:
5035:
5027:
5012:
4980:
4966:
4960:
4943:
4935:
4931:
4903:
4889:
4883:
4872:
4863:
4817:
4783:
4713:
4699:
4653:
4650:
4608:
4587:
4573:
4567:
4543:
4493:
4447:
4433:
4427:
4403:
4359:
4337:
4333:
4299:
4285:
4279:
4262:
4254:
4250:
4213:
4191:
4187:
4075:
4021:
4018:
3925:
3904:
3890:
3884:
3860:
3785:
3763:
3759:
3707:
3693:
3687:
3670:
3662:
3658:
3628:
3606:
3602:
3545:
3542:
3535:
3512:
3504:
3220:
3205:
3165:
3150:
2165:
2150:
2101:
2086:
1907:
1553:
1545:
1520:
1512:
1487:
1479:
1454:
1446:
1358:
14:
11276:
11228:
11085:Johnston, Hamish (23 July 2015).
10787:Vishwanath, Ashvin (2015-09-08).
8737:as geometric objects living on a
8455:{\displaystyle \eta \zeta ^{*}=1}
3043:
2378:
11105:Ciudad, David (20 August 2015).
10983:Quantum Field Theory Demystified
10414:identified (a double covering).
9995:
9726:{\displaystyle \mathbb {C} l(n)}
9699:of the complex Clifford algebra
7073:{\displaystyle \eta =e^{i\phi }}
3402:
3387:
3360:
3350:
3342:
3317:
3295:
3090:
3052:
3031:
2997:
2972:
2961:. As a result, the magnitude of
2357:
2349:
2310:
2302:
2223:
11001:Martin, B.R.; Shaw, G. (2008).
10955:
10849:Pearson, E. Abers, ed. (2004).
10753:10.1088/0031-8949/2005/T121/001
10510:
10486:{\displaystyle \mathrm {U} (1)}
10422:scalar under the action of the
10252:{\displaystyle \mathrm {U} (1)}
9985:{\displaystyle \Delta _{n}^{-}}
9953:{\displaystyle \Delta _{n}^{+}}
8720:
8693:By treating the spinor and its
7931:
7827:
7640:
7483:
1305:(TaAs) by the collaboration of
10878:. Cambridge University Press.
10641:"Gravitation and the electron"
10496:The third special case is the
10480:
10474:
10401:
10383:
10377:
10365:
10332:
10320:
10246:
10240:
10215:
10209:
10139:
10127:
10107:
10095:
10062:This group has the structure
10046:
10034:
9814:
9791:
9774:
9751:
9720:
9714:
9686:
9680:
9542:
9530:
9495:
9483:
9411:
9399:
9393:
9381:
9020:
9008:
8976:
8964:
8856:
8844:
8640:
8487:{\displaystyle \eta =\zeta ~.}
8374:{\displaystyle \omega ^{2}=-1}
8250:
8235:
7840:
7764:
7653:
7592:
7304:
7298:
7266:
7260:
7168:
7162:
7130:
7124:
7028:
7022:
6921:
6918:
6912:
6863:
6857:
6786:
6783:
6777:
6734:
6728:
6672:
6669:
6663:
6396:
6382:
6347:
6333:
6234:
6228:
6172:
6169:
6163:
6050:
6044:
5955:
5949:
5882:
5876:
5787:
5781:
5572:
5560:
5522:
5486:
5283:
5277:
5192:
5186:
5089:
5083:
4671:
4657:
4620:
4614:
4558:
4555:
4549:
4505:
4499:
4418:
4415:
4409:
4371:
4365:
4228:
4225:
4219:
4039:
4025:
3986:
3972:
3937:
3931:
3875:
3872:
3866:
3797:
3791:
3643:
3640:
3634:
3561:
3549:
3496:
3392:
3382:
3192:
3095:
3085:
3080:
3057:
3047:
3036:
3026:
2923:
2908:
2825:
2720:
2664:
2649:
2594:
2544:
2529:
2370:
2345:
2323:
2298:
2137:
1955:
1894:
1222:relativistic quantum mechanics
1220:particles in the framework of
1216:, and was first used to model
1154:. The equation is named after
1:
10528:
10407:{\displaystyle (s,u)=(-s,-u)}
8795:itself does change. Ignoring
1748:whose components are the 2Ă2
1289:, it was not until 1998 that
1189:, some materials can display
158:Spontaneous symmetry breaking
118:Symmetry in quantum mechanics
10545:. Vol. 1. p. 292.
8159:
7447:
7227:
4320:
4237:
4174:
3833:{\displaystyle S^{\dagger }}
3324:{\displaystyle \mathbf {p} }
3302:{\displaystyle \mathbf {J} }
3004:{\displaystyle \mathbf {k} }
2979:{\displaystyle \mathbf {p} }
7:
11150:"Where the Weyl things are"
10789:"Where the Weyl things are"
10577:American Journal of Physics
9417:{\displaystyle (p,q)=(1,3)}
3260:Helicity (particle physics)
3253:
1842:{\displaystyle \mu =1,2,3,}
1332:
1315:Chinese Academy of Sciences
10:
11281:
10493:action of the spin group.
9631:
8816:pseudo-Riemannian manifold
8755:pseudo-Riemannian manifold
6315:. This arises because the
6105:{\displaystyle S^{-1}R=1,}
3257:
1203:
1144:relativistic wave equation
153:Explicit symmetry breaking
11059:– via Google Books.
11046:Supersymmetry Demystified
11036:– via Google Books.
10915:– via Google Books.
10541:Shifman, Mikhail (1999).
9897:{\displaystyle N=2^{n/2}}
9372:Pauli exclusion principle
9059:{\displaystyle \{e_{i}\}}
7375:can be recognized as the
5588:One takes the transpose:
3309:onto the linear momentum
3286:angular momentum operator
1266:could be violated by the
1212:was published in 1928 by
309:BargmannâWigner equations
10835:10.1103/PhysRev.105.1413
10503:
9514:special orthogonal group
9301:, that is, one has that
8710:{\displaystyle \dagger }
8415:{\displaystyle Ki=-iK~.}
6307:Relationship to Majorana
3277:{\displaystyle \lambda }
2986:relates directly to the
1281:While Italian physicist
1187:condensed matter physics
1146:for describing massless
11064:Penrose, Roger (2007).
10571:Pal, Palash B. (2011).
9628:Mathematical definition
8917:{\displaystyle x\in M,}
8869:, one may consider its
8422:The RHS reduces to the
7398:{\displaystyle \psi ~.}
334:Electroweak interaction
329:Quantum electrodynamics
304:WheelerâDeWitt equation
191:Background field method
10874:Woan, G., ed. (2010).
10717:Bilenky, S.M. (2005).
10487:
10456:
10408:
10352:
10293:
10253:
10222:
10175:
10056:
9986:
9954:
9922:
9898:
9857:
9727:
9693:
9658:, the even subalgebra
9652:
9594:local coordinate frame
9579:
9549:
9502:
9452:
9418:
9364:
9288:
9187:
9094:
9093:{\displaystyle T_{x}M}
9060:
9027:
8983:
8951:
8950:{\displaystyle T_{x}M}
8918:
8889:
8863:
8831:
8789:
8749:, or, indeed, for any
8711:
8684:
8587:
8488:
8456:
8416:
8375:
8336:
8054:
8013:
7966:
7882:
7733:
7564:
7541:
7399:
7369:
7317:
7202:
7201:{\displaystyle \zeta }
7175:
7074:
7035:
6870:
6741:
6631:
6604:
6537:
6453:
6409:
6354:
6298:
6241:
6135:
6106:
6061:
5686:
5582:
5529:
5457:
5400:
5294:
4829:
4743:
4678:
4627:
4518:
4378:
4149:
4046:
3993:
3944:
3834:
3804:
3574:
3522:
3485:Lorentz transformation
3466:
3421:
3325:
3303:
3278:
3244:
3116:
3005:
2980:
2947:
2688:
2568:
2457:
2387:
2187:
2117:
2051:
1929:
1863:
1843:
1798:
1797:{\displaystyle \mu =0}
1772:
1735:
1572:
1403:
1380:
339:Quantum chromodynamics
216:Effective field theory
10962:Jost, Jurgen (2002).
10666:10.1073/pnas.15.4.323
10488:
10457:
10409:
10358:with opposite points
10353:
10294:
10254:
10223:
10176:
10057:
9987:
9955:
9923:
9899:
9858:
9728:
9694:
9653:
9580:
9550:
9503:
9453:
9419:
9365:
9289:
9188:
9095:
9061:
9028:
8984:
8982:{\displaystyle (p,q)}
8952:
8919:
8895:. At any given point
8890:
8864:
8862:{\displaystyle (p,q)}
8832:
8790:
8788:{\displaystyle \psi }
8760:The Weyl equation is
8712:
8685:
8588:
8489:
8457:
8424:KleinâGordon operator
8417:
8376:
8337:
8055:
8014:
7967:
7883:
7734:
7565:
7542:
7400:
7370:
7318:
7203:
7176:
7075:
7036:
6871:
6742:
6632:
6630:{\displaystyle S^{*}}
6605:
6538:
6454:
6410:
6355:
6299:
6242:
6136:
6107:
6062:
5687:
5583:
5530:
5458:
5401:
5295:
4830:
4744:
4679:
4628:
4519:
4379:
4150:
4047:
3994:
3945:
3835:
3805:
3575:
3523:
3467:
3422:
3326:
3304:
3279:
3245:
3117:
3006:
2981:
2948:
2689:
2569:
2458:
2388:
2188:
2118:
2052:
1930:
1864:
1862:{\displaystyle \psi }
1844:
1799:
1773:
1771:{\displaystyle I_{2}}
1736:
1573:
1404:
1381:
1287:neutrino oscillations
1255:in condensed matter.
1179:neutrino oscillations
294:KleinâGordon equation
236:LSZ reduction formula
11049:. USA: McGraw-Hill.
11043:LaBelle, P. (2010).
10986:. USA: McGraw-Hill.
10980:McMahon, D. (2008).
10466:
10426:
10362:
10303:
10267:
10232:
10188:
10069:
10008:
9964:
9932:
9912:
9867:
9737:
9703:
9662:
9642:
9614:covariant derivative
9604:, the analog of the
9563:
9519:
9466:
9439:
9426:Minkowski space-time
9378:
9305:
9203:
9107:
9074:
9037:
8997:
8961:
8931:
8899:
8876:
8841:
8821:
8779:
8701:
8600:
8512:
8504:Lagrangian densities
8498:Lagrangian densities
8466:
8430:
8385:
8349:
8067:
8023:
7982:
7898:
7749:
7577:
7554:
7420:
7383:
7330:
7215:
7192:
7086:
7048:
6886:
6757:
6648:
6614:
6550:
6469:
6422:
6371:
6322:
6317:special linear group
6254:
6148:
6134:{\displaystyle R=S.}
6116:
6074:
5702:
5595:
5543:
5470:
5416:
5310:
4845:
4759:
4691:
4640:
4534:
4394:
4165:
4059:
4014:
4009:special linear group
3957:
3851:
3817:
3587:
3532:
3490:
3434:
3338:
3313:
3291:
3268:
3132:
3022:
3013:de Broglie relations
2993:
2968:
2705:
2584:
2473:
2404:
2211:
2196:Plane wave solutions
2127:
2072:
1945:
1884:
1853:
1812:
1782:
1755:
1588:
1420:
1393:
1344:
1311:Princeton University
1171:elementary particles
1136:quantum field theory
377:Theory of everything
231:Lattice field theory
201:Correlation function
23:Quantum field theory
11197:2016NatMa..15.1140J
11107:"Massless yet real"
11067:The Road to Reality
10826:1957PhRv..105.1413W
10745:2005PhST..121...17B
10657:1929PNAS...15..323W
10599:2011AmJPh..79..485P
9981:
9949:
9906:complex Weyl spinor
9852:
9834:
9578:{\displaystyle M~.}
9220:
8814:Given an arbitrary
8751:Riemannian manifold
8632:
8544:
8225:
7859:
7783:
7672:
7611:
7297:
7161:
7021:
6949:
6911:
6856:
6808:
6776:
6691:
6191:
4970:
4893:
4876:
4577:
4437:
4289:
3894:
3842:Hermitian transpose
3697:
3479:Both equations are
1297:In 2015, the first
356:Incomplete theories
10940:10.3390/sym2041776
10905:. Addison-Wesley.
10483:
10452:
10404:
10348:
10289:
10249:
10218:
10171:
10052:
9982:
9967:
9950:
9935:
9918:
9894:
9853:
9838:
9820:
9723:
9689:
9648:
9597:boosts/rotations.
9575:
9545:
9498:
9451:{\displaystyle M,}
9448:
9414:
9360:
9284:
9206:
9183:
9090:
9068:vector space basis
9056:
9023:
8979:
8947:
8914:
8888:{\displaystyle TM}
8885:
8859:
8827:
8785:
8747:General Relativity
8707:
8680:
8616:
8583:
8528:
8484:
8452:
8412:
8371:
8332:
8211:
8050:
8009:
7962:
7878:
7843:
7767:
7729:
7656:
7595:
7560:
7537:
7408:charge conjugation
7395:
7365:
7313:
7281:
7198:
7171:
7145:
7070:
7031:
7005:
6930:
6895:
6866:
6840:
6789:
6760:
6737:
6675:
6627:
6600:
6533:
6527:
6449:
6405:
6350:
6294:
6237:
6175:
6131:
6102:
6057:
6055:
5682:
5578:
5535:is the flat-space
5525:
5484:
5453:
5396:
5290:
5288:
4954:
4877:
4862:
4825:
4752:or, equivalently,
4739:
4674:
4623:
4561:
4514:
4421:
4374:
4273:
4145:
4042:
3989:
3940:
3878:
3830:
3800:
3681:
3570:
3518:
3475:Lorentz invariance
3462:
3417:
3321:
3299:
3274:
3240:
3238:
3112:
3001:
2976:
2943:
2684:
2564:
2453:
2447:
2383:
2275:
2183:
2113:
2047:
2035:
1925:
1873:â one of the Weyl
1859:
1839:
1794:
1768:
1731:
1725:
1660:
1568:
1399:
1376:
1134:, particularly in
241:Partition function
168:Topological charge
88:General relativity
83:Special relativity
11265:Quantum mechanics
11181:(11): 1140â1144.
11077:978-0-679-77631-4
11070:. Vintage Books.
11056:978-0-07-163641-4
11012:978-0-470-03294-7
10993:978-0-07-154382-8
10885:978-0-521-57507-2
10860:978-0-13-146100-0
10851:Quantum Mechanics
10607:10.1119/1.3549729
9921:{\displaystyle n}
9733:is isomorphic to
9651:{\displaystyle n}
9606:metric connection
9571:
9458:one constructs a
9356:
9234:
9233:
9133:
9132:
8830:{\displaystyle M}
8676:
8643:
8579:
8480:
8408:
8253:
8238:
8162:
7563:{\displaystyle K}
7450:
7391:
7230:
7186:Majorana equation
6639:complex conjugate
6401:
6313:Majorana equation
5483:
5169:
5066:
5044:
4952:
4636:for some unknown
4510:
4351:
4323:
4271:
4240:
4205:
4177:
3777:
3679:
3620:
3566:
3481:Lorentz invariant
3458:
3454:
3195:
3110:
3077:
2926:
2911:
2828:
2723:
2667:
2652:
2597:
2547:
2532:
2179:
2140:
2043:
1958:
1897:
1560:
1527:
1494:
1461:
1441:
1402:{\displaystyle c}
1327:photonic crystals
1313:) and H. Ding's (
1303:tantalum arsenide
1240:, to explain the
1164:Majorana fermions
1150:particles called
1128:
1127:
221:Expectation value
196:BRST quantization
143:Poincaré symmetry
98:YangâMills theory
78:Quantum mechanics
11272:
11224:
11205:10.1038/nmat4787
11190:
11175:Nature Materials
11165:
11163:
11161:
11144:
11126:
11124:10.1038/nmat4411
11111:Nature Materials
11101:
11099:
11097:
11081:
11060:
11037:
11026:(3rd ed.).
11023:Particle Physics
11016:
11003:Particle Physics
10997:
10968:
10967:
10959:
10953:
10952:
10942:
10926:
10917:
10916:
10896:
10890:
10889:
10871:
10865:
10864:
10846:
10840:
10839:
10837:
10820:(4): 1413â1415.
10803:
10797:
10796:
10784:
10773:
10772:
10738:
10714:
10697:
10696:
10686:
10668:
10633:
10627:
10626:
10592:
10568:
10557:
10556:
10538:
10522:
10514:
10492:
10490:
10489:
10484:
10473:
10461:
10459:
10458:
10453:
10451:
10450:
10449:
10443:
10413:
10411:
10410:
10405:
10357:
10355:
10354:
10349:
10347:
10346:
10319:
10298:
10296:
10295:
10290:
10288:
10287:
10286:
10285:
10280:
10261:electromagnetism
10258:
10256:
10255:
10250:
10239:
10227:
10225:
10224:
10219:
10208:
10200:
10199:
10180:
10178:
10177:
10172:
10170:
10169:
10160:
10159:
10158:
10157:
10152:
10126:
10094:
10093:
10092:
10086:
10061:
10059:
10058:
10053:
10033:
10032:
10031:
10025:
9991:
9989:
9988:
9983:
9980:
9975:
9959:
9957:
9956:
9951:
9948:
9943:
9927:
9925:
9924:
9919:
9903:
9901:
9900:
9895:
9893:
9892:
9888:
9862:
9860:
9859:
9854:
9851:
9846:
9833:
9828:
9813:
9812:
9808:
9799:
9790:
9773:
9772:
9768:
9759:
9750:
9732:
9730:
9729:
9724:
9710:
9698:
9696:
9695:
9690:
9679:
9678:
9669:
9657:
9655:
9654:
9649:
9584:
9582:
9581:
9576:
9569:
9554:
9552:
9551:
9546:
9529:
9507:
9505:
9504:
9499:
9482:
9457:
9455:
9454:
9449:
9423:
9421:
9420:
9415:
9369:
9367:
9366:
9361:
9354:
9353:
9352:
9343:
9342:
9327:
9326:
9317:
9316:
9293:
9291:
9290:
9285:
9283:
9279:
9278:
9277:
9253:
9252:
9235:
9229:
9225:
9219:
9214:
9192:
9190:
9189:
9184:
9182:
9178:
9177:
9176:
9152:
9151:
9134:
9128:
9124:
9119:
9118:
9099:
9097:
9096:
9091:
9086:
9085:
9065:
9063:
9062:
9057:
9052:
9051:
9032:
9030:
9029:
9024:
9007:
8988:
8986:
8985:
8980:
8956:
8954:
8953:
8948:
8943:
8942:
8923:
8921:
8920:
8915:
8894:
8892:
8891:
8886:
8868:
8866:
8865:
8860:
8836:
8834:
8833:
8828:
8801:Clifford algebra
8794:
8792:
8791:
8786:
8716:
8714:
8713:
8708:
8689:
8687:
8686:
8681:
8674:
8673:
8672:
8671:
8661:
8660:
8651:
8650:
8645:
8644:
8636:
8631:
8626:
8625:
8609:
8608:
8592:
8590:
8589:
8584:
8577:
8576:
8575:
8574:
8564:
8563:
8554:
8553:
8543:
8538:
8537:
8521:
8520:
8493:
8491:
8490:
8485:
8478:
8461:
8459:
8458:
8453:
8445:
8444:
8421:
8419:
8418:
8413:
8406:
8380:
8378:
8377:
8372:
8361:
8360:
8341:
8339:
8338:
8333:
8331:
8327:
8326:
8325:
8316:
8315:
8286:
8282:
8281:
8280:
8271:
8270:
8255:
8254:
8246:
8240:
8239:
8231:
8224:
8219:
8199:
8195:
8179:
8178:
8169:
8168:
8163:
8155:
8144:
8140:
8124:
8123:
8114:
8113:
8093:
8092:
8091:
8081:
8080:
8079:
8059:
8057:
8056:
8051:
8049:
8048:
8047:
8037:
8036:
8035:
8018:
8016:
8015:
8010:
8008:
8007:
8006:
7996:
7995:
7994:
7971:
7969:
7968:
7963:
7955:
7954:
7953:
7943:
7942:
7941:
7924:
7923:
7922:
7912:
7911:
7910:
7887:
7885:
7884:
7879:
7877:
7876:
7875:
7858:
7853:
7852:
7839:
7838:
7837:
7826:
7825:
7824:
7814:
7813:
7805:
7801:
7800:
7782:
7777:
7776:
7763:
7762:
7761:
7738:
7736:
7735:
7730:
7728:
7727:
7715:
7714:
7713:
7703:
7702:
7694:
7690:
7689:
7671:
7666:
7665:
7652:
7651:
7650:
7639:
7638:
7629:
7628:
7627:
7610:
7605:
7604:
7591:
7590:
7589:
7569:
7567:
7566:
7561:
7546:
7544:
7543:
7538:
7521:
7520:
7511:
7510:
7495:
7494:
7493:
7467:
7466:
7457:
7456:
7451:
7443:
7434:
7433:
7432:
7404:
7402:
7401:
7396:
7389:
7377:charge conjugate
7374:
7372:
7371:
7366:
7361:
7360:
7345:
7344:
7322:
7320:
7319:
7314:
7296:
7291:
7290:
7259:
7258:
7257:
7247:
7246:
7237:
7236:
7231:
7223:
7207:
7205:
7204:
7199:
7180:
7178:
7177:
7172:
7160:
7155:
7154:
7123:
7122:
7121:
7111:
7110:
7101:
7100:
7079:
7077:
7076:
7071:
7069:
7068:
7040:
7038:
7037:
7032:
7020:
7015:
7014:
6998:
6997:
6989:
6985:
6984:
6967:
6963:
6962:
6948:
6940:
6939:
6910:
6905:
6904:
6875:
6873:
6872:
6867:
6855:
6850:
6849:
6839:
6838:
6826:
6822:
6821:
6807:
6799:
6798:
6775:
6770:
6769:
6746:
6744:
6743:
6738:
6727:
6726:
6725:
6709:
6705:
6704:
6690:
6685:
6684:
6662:
6661:
6660:
6636:
6634:
6633:
6628:
6626:
6625:
6609:
6607:
6606:
6601:
6596:
6595:
6587:
6583:
6582:
6565:
6564:
6542:
6540:
6539:
6534:
6532:
6531:
6490:
6489:
6458:
6456:
6455:
6450:
6436:
6435:
6434:
6414:
6412:
6411:
6406:
6399:
6395:
6381:
6366:symplectic group
6359:
6357:
6356:
6351:
6346:
6332:
6303:
6301:
6300:
6295:
6290:
6289:
6281:
6277:
6276:
6246:
6244:
6243:
6238:
6227:
6226:
6225:
6209:
6205:
6204:
6190:
6185:
6184:
6162:
6161:
6160:
6140:
6138:
6137:
6132:
6111:
6109:
6108:
6103:
6089:
6088:
6066:
6064:
6063:
6058:
6056:
6043:
6042:
6041:
6028:
6027:
6015:
6014:
6005:
6004:
5995:
5994:
5989:
5985:
5984:
5961:
5948:
5947:
5946:
5933:
5932:
5923:
5922:
5917:
5916:
5915:
5904:
5903:
5888:
5875:
5874:
5873:
5860:
5859:
5850:
5849:
5844:
5843:
5842:
5837:
5833:
5832:
5831:
5830:
5806:
5805:
5780:
5779:
5778:
5765:
5764:
5755:
5754:
5749:
5748:
5747:
5742:
5738:
5737:
5718:
5717:
5691:
5689:
5688:
5683:
5681:
5680:
5675:
5674:
5673:
5668:
5664:
5663:
5662:
5661:
5634:
5633:
5628:
5627:
5626:
5621:
5617:
5616:
5587:
5585:
5584:
5579:
5559:
5537:Minkowski metric
5534:
5532:
5531:
5526:
5485:
5481:
5462:
5460:
5459:
5454:
5452:
5451:
5433:
5432:
5431:
5405:
5403:
5402:
5397:
5395:
5394:
5382:
5381:
5372:
5371:
5366:
5362:
5361:
5341:
5340:
5335:
5334:
5333:
5322:
5321:
5299:
5297:
5296:
5291:
5289:
5276:
5275:
5274:
5261:
5260:
5251:
5250:
5245:
5244:
5243:
5238:
5234:
5233:
5214:
5213:
5198:
5185:
5184:
5183:
5170:
5168:
5167:
5166:
5150:
5148:
5147:
5142:
5141:
5140:
5135:
5131:
5130:
5111:
5110:
5095:
5082:
5081:
5080:
5067:
5065:
5064:
5063:
5047:
5045:
5043:
5042:
5041:
5025:
5024:
5023:
5010:
5008:
5007:
4992:
4988:
4984:
4983:
4969:
4964:
4963:
4953:
4951:
4950:
4949:
4930:
4928:
4927:
4911:
4907:
4906:
4892:
4887:
4886:
4875:
4870:
4861:
4860:
4834:
4832:
4831:
4826:
4824:
4823:
4811:
4810:
4805:
4804:
4803:
4798:
4794:
4793:
4771:
4770:
4748:
4746:
4745:
4740:
4738:
4737:
4728:
4727:
4722:
4721:
4720:
4706:
4705:
4683:
4681:
4680:
4675:
4670:
4656:
4632:
4630:
4629:
4624:
4613:
4612:
4611:
4595:
4591:
4590:
4576:
4571:
4570:
4548:
4547:
4546:
4523:
4521:
4520:
4515:
4508:
4498:
4497:
4496:
4486:
4485:
4477:
4473:
4472:
4455:
4451:
4450:
4436:
4431:
4430:
4408:
4407:
4406:
4383:
4381:
4380:
4375:
4364:
4363:
4362:
4352:
4350:
4349:
4348:
4332:
4330:
4329:
4324:
4316:
4307:
4303:
4302:
4288:
4283:
4282:
4272:
4270:
4269:
4268:
4249:
4247:
4246:
4241:
4233:
4218:
4217:
4216:
4206:
4204:
4203:
4202:
4186:
4184:
4183:
4178:
4170:
4154:
4152:
4151:
4146:
4144:
4143:
4131:
4130:
4121:
4120:
4115:
4111:
4110:
4090:
4089:
4084:
4083:
4082:
4071:
4070:
4051:
4049:
4048:
4043:
4038:
4024:
3998:
3996:
3995:
3990:
3985:
3949:
3947:
3946:
3941:
3930:
3929:
3928:
3912:
3908:
3907:
3893:
3888:
3887:
3865:
3864:
3863:
3839:
3837:
3836:
3831:
3829:
3828:
3809:
3807:
3806:
3801:
3790:
3789:
3788:
3778:
3776:
3775:
3774:
3758:
3756:
3755:
3746:
3745:
3740:
3736:
3735:
3715:
3711:
3710:
3696:
3691:
3690:
3680:
3678:
3677:
3676:
3657:
3655:
3654:
3633:
3632:
3631:
3621:
3619:
3618:
3617:
3601:
3599:
3598:
3579:
3577:
3576:
3571:
3564:
3548:
3527:
3525:
3524:
3519:
3508:
3507:
3471:
3469:
3468:
3463:
3456:
3455:
3447:
3426:
3424:
3423:
3418:
3416:
3412:
3405:
3395:
3390:
3385:
3374:
3370:
3363:
3353:
3345:
3330:
3328:
3327:
3322:
3320:
3308:
3306:
3305:
3300:
3298:
3283:
3281:
3280:
3275:
3249:
3247:
3246:
3241:
3239:
3225:
3224:
3223:
3213:
3212:
3203:
3202:
3197:
3196:
3188:
3170:
3169:
3168:
3158:
3157:
3148:
3147:
3121:
3119:
3118:
3113:
3111:
3103:
3098:
3093:
3088:
3078:
3073:
3065:
3060:
3055:
3050:
3039:
3034:
3029:
3010:
3008:
3007:
3002:
3000:
2985:
2983:
2982:
2977:
2975:
2952:
2950:
2949:
2944:
2933:
2929:
2928:
2927:
2919:
2913:
2912:
2904:
2898:
2897:
2877:
2876:
2867:
2866:
2851:
2847:
2846:
2845:
2836:
2835:
2830:
2829:
2821:
2812:
2808:
2807:
2806:
2797:
2796:
2776:
2772:
2771:
2770:
2761:
2760:
2746:
2742:
2741:
2740:
2731:
2730:
2725:
2724:
2716:
2693:
2691:
2690:
2685:
2674:
2670:
2669:
2668:
2660:
2654:
2653:
2645:
2636:
2635:
2615:
2614:
2605:
2604:
2599:
2598:
2590:
2573:
2571:
2570:
2565:
2554:
2550:
2549:
2548:
2540:
2534:
2533:
2525:
2516:
2515:
2495:
2494:
2485:
2484:
2462:
2460:
2459:
2454:
2452:
2451:
2444:
2443:
2430:
2429:
2392:
2390:
2389:
2384:
2382:
2381:
2377:
2360:
2352:
2327:
2326:
2313:
2305:
2280:
2279:
2272:
2271:
2258:
2257:
2237:
2233:
2226:
2192:
2190:
2189:
2184:
2177:
2170:
2169:
2168:
2158:
2157:
2148:
2147:
2142:
2141:
2133:
2122:
2120:
2119:
2114:
2106:
2105:
2104:
2094:
2093:
2084:
2083:
2056:
2054:
2053:
2048:
2041:
2040:
2039:
2032:
2031:
2017:
2016:
2002:
2001:
1987:
1986:
1966:
1965:
1960:
1959:
1951:
1934:
1932:
1931:
1926:
1915:
1914:
1905:
1904:
1899:
1898:
1890:
1868:
1866:
1865:
1860:
1848:
1846:
1845:
1840:
1803:
1801:
1800:
1795:
1777:
1775:
1774:
1769:
1767:
1766:
1740:
1738:
1737:
1732:
1730:
1729:
1722:
1721:
1710:
1709:
1698:
1697:
1686:
1685:
1665:
1664:
1657:
1656:
1645:
1644:
1633:
1632:
1621:
1620:
1600:
1599:
1577:
1575:
1574:
1569:
1561:
1559:
1551:
1543:
1541:
1540:
1528:
1526:
1518:
1510:
1508:
1507:
1495:
1493:
1485:
1477:
1475:
1474:
1462:
1460:
1452:
1444:
1442:
1434:
1432:
1431:
1408:
1406:
1405:
1400:
1385:
1383:
1382:
1377:
1366:
1365:
1356:
1355:
1291:Super-Kamiokande
1283:Bruno Pontecorvo
1268:weak interaction
1120:
1113:
1106:
211:Effective action
138:Lorentz symmetry
63:Electromagnetism
33:
19:
18:
11280:
11279:
11275:
11274:
11273:
11271:
11270:
11269:
11255:
11254:
11231:
11159:
11157:
11095:
11093:
11078:
11057:
11034:
11013:
10994:
10976:
10974:Further reading
10971:
10960:
10956:
10927:
10920:
10913:
10897:
10893:
10886:
10872:
10868:
10861:
10847:
10843:
10813:Physical Review
10804:
10800:
10785:
10776:
10723:Physica Scripta
10715:
10700:
10634:
10630:
10569:
10560:
10553:
10539:
10535:
10531:
10526:
10525:
10515:
10511:
10506:
10469:
10467:
10464:
10463:
10445:
10444:
10430:
10429:
10427:
10424:
10423:
10419:Majorana spinor
10363:
10360:
10359:
10342:
10338:
10306:
10304:
10301:
10300:
10281:
10276:
10275:
10274:
10270:
10268:
10265:
10264:
10235:
10233:
10230:
10229:
10204:
10195:
10191:
10189:
10186:
10185:
10165:
10161:
10153:
10148:
10147:
10146:
10142:
10113:
10088:
10087:
10073:
10072:
10070:
10067:
10066:
10027:
10026:
10012:
10011:
10009:
10006:
10005:
9998:
9976:
9971:
9965:
9962:
9961:
9960:(respectively,
9944:
9939:
9933:
9930:
9929:
9913:
9910:
9909:
9884:
9880:
9876:
9868:
9865:
9864:
9847:
9842:
9829:
9824:
9804:
9800:
9795:
9794:
9780:
9764:
9760:
9755:
9754:
9740:
9738:
9735:
9734:
9706:
9704:
9701:
9700:
9674:
9670:
9665:
9663:
9660:
9659:
9643:
9640:
9639:
9636:
9630:
9618:Clifford bundle
9610:spin connection
9564:
9561:
9560:
9522:
9520:
9517:
9516:
9469:
9467:
9464:
9463:
9440:
9437:
9436:
9379:
9376:
9375:
9348:
9344:
9338:
9334:
9322:
9318:
9312:
9308:
9306:
9303:
9302:
9264:
9260:
9245:
9241:
9240:
9236:
9224:
9215:
9210:
9204:
9201:
9200:
9163:
9159:
9144:
9140:
9139:
9135:
9123:
9114:
9110:
9108:
9105:
9104:
9081:
9077:
9075:
9072:
9071:
9047:
9043:
9038:
9035:
9034:
9000:
8998:
8995:
8994:
8962:
8959:
8958:
8938:
8934:
8932:
8929:
8928:
8900:
8897:
8896:
8877:
8874:
8873:
8842:
8839:
8838:
8822:
8819:
8818:
8780:
8777:
8776:
8731:Clifford module
8723:
8702:
8699:
8698:
8667:
8666:
8662:
8656:
8652:
8646:
8635:
8634:
8633:
8627:
8621:
8620:
8604:
8603:
8601:
8598:
8597:
8570:
8569:
8565:
8559:
8555:
8549:
8545:
8539:
8533:
8532:
8516:
8515:
8513:
8510:
8509:
8500:
8467:
8464:
8463:
8440:
8436:
8431:
8428:
8427:
8386:
8383:
8382:
8356:
8352:
8350:
8347:
8346:
8321:
8317:
8311:
8307:
8297:
8293:
8276:
8272:
8266:
8262:
8245:
8244:
8230:
8229:
8220:
8215:
8210:
8206:
8174:
8170:
8164:
8154:
8153:
8149:
8145:
8119:
8115:
8109:
8105:
8101:
8097:
8087:
8086:
8082:
8075:
8074:
8070:
8068:
8065:
8064:
8043:
8042:
8038:
8031:
8030:
8026:
8024:
8021:
8020:
8002:
8001:
7997:
7990:
7989:
7985:
7983:
7980:
7979:
7949:
7948:
7944:
7937:
7936:
7932:
7918:
7917:
7913:
7906:
7905:
7901:
7899:
7896:
7895:
7871:
7870:
7866:
7854:
7848:
7847:
7833:
7832:
7828:
7820:
7819:
7815:
7806:
7796:
7792:
7788:
7787:
7778:
7772:
7771:
7757:
7756:
7752:
7750:
7747:
7746:
7720:
7716:
7709:
7708:
7704:
7695:
7685:
7681:
7677:
7676:
7667:
7661:
7660:
7646:
7645:
7641:
7634:
7630:
7623:
7622:
7618:
7606:
7600:
7599:
7585:
7584:
7580:
7578:
7575:
7574:
7555:
7552:
7551:
7516:
7512:
7506:
7502:
7489:
7488:
7484:
7462:
7458:
7452:
7442:
7441:
7428:
7427:
7423:
7421:
7418:
7417:
7384:
7381:
7380:
7356:
7352:
7340:
7336:
7331:
7328:
7327:
7292:
7286:
7285:
7253:
7252:
7248:
7242:
7238:
7232:
7222:
7221:
7216:
7213:
7212:
7193:
7190:
7189:
7156:
7150:
7149:
7117:
7116:
7112:
7106:
7102:
7096:
7092:
7087:
7084:
7083:
7061:
7057:
7049:
7046:
7045:
7016:
7010:
7009:
6990:
6980:
6976:
6972:
6971:
6958:
6954:
6950:
6941:
6935:
6934:
6906:
6900:
6899:
6887:
6884:
6883:
6851:
6845:
6844:
6834:
6830:
6817:
6813:
6809:
6800:
6794:
6793:
6771:
6765:
6764:
6758:
6755:
6754:
6721:
6720:
6716:
6700:
6696:
6692:
6686:
6680:
6679:
6656:
6655:
6651:
6649:
6646:
6645:
6621:
6617:
6615:
6612:
6611:
6588:
6578:
6574:
6570:
6569:
6560:
6556:
6551:
6548:
6547:
6526:
6525:
6520:
6511:
6510:
6505:
6495:
6494:
6485:
6481:
6470:
6467:
6466:
6430:
6429:
6425:
6423:
6420:
6419:
6391:
6374:
6372:
6369:
6368:
6342:
6325:
6323:
6320:
6319:
6309:
6282:
6272:
6268:
6264:
6263:
6255:
6252:
6251:
6221:
6220:
6216:
6200:
6196:
6192:
6186:
6180:
6179:
6156:
6155:
6151:
6149:
6146:
6145:
6117:
6114:
6113:
6081:
6077:
6075:
6072:
6071:
6054:
6053:
6037:
6036:
6032:
6020:
6016:
6010:
6006:
6000:
5996:
5990:
5977:
5973:
5969:
5968:
5959:
5958:
5942:
5941:
5937:
5928:
5924:
5918:
5911:
5907:
5906:
5905:
5899:
5895:
5886:
5885:
5869:
5868:
5864:
5855:
5851:
5845:
5838:
5826:
5825:
5818:
5814:
5810:
5809:
5808:
5807:
5801:
5797:
5790:
5774:
5773:
5769:
5760:
5756:
5750:
5743:
5730:
5726:
5722:
5721:
5720:
5719:
5713:
5709:
5705:
5703:
5700:
5699:
5676:
5669:
5657:
5656:
5649:
5645:
5641:
5640:
5639:
5638:
5629:
5622:
5609:
5605:
5601:
5600:
5599:
5598:
5596:
5593:
5592:
5552:
5544:
5541:
5540:
5479:
5471:
5468:
5467:
5444:
5440:
5427:
5426:
5422:
5417:
5414:
5413:
5387:
5383:
5377:
5373:
5367:
5354:
5350:
5346:
5345:
5336:
5329:
5325:
5324:
5323:
5317:
5313:
5311:
5308:
5307:
5287:
5286:
5270:
5269:
5265:
5256:
5252:
5246:
5239:
5226:
5222:
5218:
5217:
5216:
5215:
5209:
5205:
5196:
5195:
5179:
5178:
5174:
5162:
5158:
5154:
5149:
5143:
5136:
5123:
5119:
5115:
5114:
5113:
5112:
5106:
5102:
5093:
5092:
5076:
5075:
5071:
5059:
5055:
5051:
5046:
5034:
5030:
5026:
5019:
5015:
5011:
5009:
5003:
4999:
4990:
4989:
4979:
4975:
4971:
4965:
4959:
4958:
4942:
4938:
4934:
4929:
4923:
4919:
4912:
4902:
4898:
4894:
4888:
4882:
4881:
4871:
4866:
4856:
4852:
4848:
4846:
4843:
4842:
4816:
4812:
4806:
4799:
4786:
4782:
4778:
4777:
4776:
4775:
4766:
4762:
4760:
4757:
4756:
4733:
4729:
4723:
4716:
4712:
4711:
4710:
4698:
4694:
4692:
4689:
4688:
4666:
4649:
4641:
4638:
4637:
4607:
4606:
4602:
4586:
4582:
4578:
4572:
4566:
4565:
4542:
4541:
4537:
4535:
4532:
4531:
4492:
4491:
4487:
4478:
4468:
4464:
4460:
4459:
4446:
4442:
4438:
4432:
4426:
4425:
4402:
4401:
4397:
4395:
4392:
4391:
4358:
4357:
4353:
4344:
4340:
4336:
4331:
4325:
4315:
4314:
4298:
4294:
4290:
4284:
4278:
4277:
4261:
4257:
4253:
4248:
4242:
4232:
4231:
4212:
4211:
4207:
4198:
4194:
4190:
4185:
4179:
4169:
4168:
4166:
4163:
4162:
4136:
4132:
4126:
4122:
4116:
4103:
4099:
4095:
4094:
4085:
4078:
4074:
4073:
4072:
4066:
4062:
4060:
4057:
4056:
4034:
4017:
4015:
4012:
4011:
4001:double covering
3981:
3958:
3955:
3954:
3924:
3923:
3919:
3903:
3899:
3895:
3889:
3883:
3882:
3859:
3858:
3854:
3852:
3849:
3848:
3824:
3820:
3818:
3815:
3814:
3784:
3783:
3779:
3770:
3766:
3762:
3757:
3751:
3747:
3741:
3728:
3724:
3720:
3719:
3706:
3702:
3698:
3692:
3686:
3685:
3669:
3665:
3661:
3656:
3650:
3646:
3627:
3626:
3622:
3613:
3609:
3605:
3600:
3594:
3590:
3588:
3585:
3584:
3541:
3533:
3530:
3529:
3503:
3499:
3491:
3488:
3487:
3477:
3446:
3435:
3432:
3431:
3401:
3400:
3396:
3391:
3386:
3381:
3359:
3358:
3354:
3349:
3341:
3339:
3336:
3335:
3316:
3314:
3311:
3310:
3294:
3292:
3289:
3288:
3269:
3266:
3265:
3262:
3256:
3237:
3236:
3226:
3219:
3218:
3214:
3208:
3204:
3198:
3187:
3186:
3185:
3182:
3181:
3171:
3164:
3163:
3159:
3153:
3149:
3143:
3139:
3135:
3133:
3130:
3129:
3102:
3094:
3089:
3084:
3066:
3064:
3056:
3051:
3046:
3035:
3030:
3025:
3023:
3020:
3019:
2996:
2994:
2991:
2990:
2971:
2969:
2966:
2965:
2918:
2917:
2903:
2902:
2893:
2889:
2888:
2884:
2872:
2868:
2862:
2858:
2841:
2837:
2831:
2820:
2819:
2818:
2817:
2813:
2802:
2798:
2792:
2788:
2787:
2783:
2766:
2762:
2756:
2752:
2751:
2747:
2736:
2732:
2726:
2715:
2714:
2713:
2712:
2708:
2706:
2703:
2702:
2659:
2658:
2644:
2643:
2631:
2627:
2626:
2622:
2610:
2606:
2600:
2589:
2588:
2587:
2585:
2582:
2581:
2539:
2538:
2524:
2523:
2511:
2507:
2506:
2502:
2490:
2486:
2480:
2476:
2474:
2471:
2470:
2446:
2445:
2439:
2435:
2432:
2431:
2425:
2421:
2414:
2413:
2405:
2402:
2401:
2373:
2356:
2348:
2338:
2334:
2309:
2301:
2291:
2287:
2274:
2273:
2267:
2263:
2260:
2259:
2253:
2249:
2242:
2241:
2222:
2221:
2217:
2212:
2209:
2208:
2198:
2164:
2163:
2159:
2153:
2149:
2143:
2132:
2131:
2130:
2128:
2125:
2124:
2100:
2099:
2095:
2089:
2085:
2079:
2075:
2073:
2070:
2069:
2034:
2033:
2027:
2023:
2018:
2012:
2008:
2003:
1997:
1993:
1988:
1982:
1978:
1971:
1970:
1961:
1950:
1949:
1948:
1946:
1943:
1942:
1910:
1906:
1900:
1889:
1888:
1887:
1885:
1882:
1881:
1854:
1851:
1850:
1813:
1810:
1809:
1783:
1780:
1779:
1762:
1758:
1756:
1753:
1752:
1750:identity matrix
1724:
1723:
1717:
1713:
1711:
1705:
1701:
1699:
1693:
1689:
1687:
1681:
1677:
1670:
1669:
1659:
1658:
1652:
1648:
1646:
1640:
1636:
1634:
1628:
1624:
1622:
1616:
1612:
1605:
1604:
1595:
1591:
1589:
1586:
1585:
1552:
1544:
1542:
1536:
1532:
1519:
1511:
1509:
1503:
1499:
1486:
1478:
1476:
1470:
1466:
1453:
1445:
1443:
1433:
1427:
1423:
1421:
1418:
1417:
1394:
1391:
1390:
1361:
1357:
1351:
1347:
1345:
1342:
1341:
1335:
1249:Conyers Herring
1206:
1195:Weyl semimetals
1124:
1095:
1094:
1093:
1091:
395:
387:
386:
382:Quantum gravity
357:
349:
348:
344:Higgs mechanism
324:
314:
313:
299:Proca equations
284:
276:
275:
261:Renormalization
226:Feynman diagram
181:
173:
172:
113:
103:
102:
53:
38:
36:Feynman diagram
17:
12:
11:
5:
11278:
11268:
11267:
11253:
11252:
11247:
11242:
11237:
11230:
11229:External links
11227:
11226:
11225:
11166:
11145:
11102:
11082:
11076:
11061:
11055:
11040:
11039:
11038:
11032:
11011:
10998:
10992:
10975:
10972:
10970:
10969:
10954:
10918:
10911:
10891:
10884:
10866:
10859:
10841:
10798:
10795:. Vol. 8.
10774:
10736:hep-ph/0410090
10698:
10651:(4): 323â334.
10639:(1929-04-15).
10628:
10583:(5): 485â498.
10558:
10551:
10532:
10530:
10527:
10524:
10523:
10508:
10507:
10505:
10502:
10482:
10479:
10476:
10472:
10448:
10442:
10439:
10436:
10433:
10403:
10400:
10397:
10394:
10391:
10388:
10385:
10382:
10379:
10376:
10373:
10370:
10367:
10345:
10341:
10337:
10334:
10331:
10328:
10325:
10322:
10318:
10315:
10312:
10309:
10284:
10279:
10273:
10263:. The product
10248:
10245:
10242:
10238:
10217:
10214:
10211:
10207:
10203:
10198:
10194:
10182:
10181:
10168:
10164:
10156:
10151:
10145:
10141:
10138:
10135:
10132:
10129:
10125:
10122:
10119:
10116:
10112:
10109:
10106:
10103:
10100:
10097:
10091:
10085:
10082:
10079:
10076:
10051:
10048:
10045:
10042:
10039:
10036:
10030:
10024:
10021:
10018:
10015:
9997:
9994:
9979:
9974:
9970:
9947:
9942:
9938:
9917:
9891:
9887:
9883:
9879:
9875:
9872:
9850:
9845:
9841:
9837:
9832:
9827:
9823:
9819:
9816:
9811:
9807:
9803:
9798:
9793:
9789:
9786:
9783:
9779:
9776:
9771:
9767:
9763:
9758:
9753:
9749:
9746:
9743:
9722:
9719:
9716:
9713:
9709:
9688:
9685:
9682:
9677:
9673:
9668:
9647:
9629:
9626:
9587:spin structure
9574:
9568:
9544:
9541:
9538:
9535:
9532:
9528:
9525:
9497:
9494:
9491:
9488:
9485:
9481:
9478:
9475:
9472:
9447:
9444:
9413:
9410:
9407:
9404:
9401:
9398:
9395:
9392:
9389:
9386:
9383:
9359:
9351:
9347:
9341:
9337:
9333:
9330:
9325:
9321:
9315:
9311:
9299:anti-commuting
9295:
9294:
9282:
9276:
9273:
9270:
9267:
9263:
9259:
9256:
9251:
9248:
9244:
9239:
9232:
9228:
9223:
9218:
9213:
9209:
9194:
9193:
9181:
9175:
9172:
9169:
9166:
9162:
9158:
9155:
9150:
9147:
9143:
9138:
9131:
9127:
9122:
9117:
9113:
9089:
9084:
9080:
9055:
9050:
9046:
9042:
9022:
9019:
9016:
9013:
9010:
9006:
9003:
8978:
8975:
8972:
8969:
8966:
8946:
8941:
8937:
8913:
8910:
8907:
8904:
8884:
8881:
8871:tangent bundle
8858:
8855:
8852:
8849:
8846:
8826:
8809:rotation group
8784:
8722:
8719:
8706:
8691:
8690:
8679:
8670:
8665:
8659:
8655:
8649:
8642:
8639:
8630:
8624:
8619:
8615:
8612:
8607:
8594:
8593:
8582:
8573:
8568:
8562:
8558:
8552:
8548:
8542:
8536:
8531:
8527:
8524:
8519:
8499:
8496:
8483:
8477:
8474:
8471:
8451:
8448:
8443:
8439:
8435:
8426:provided that
8411:
8405:
8402:
8399:
8396:
8393:
8390:
8370:
8367:
8364:
8359:
8355:
8343:
8342:
8330:
8324:
8320:
8314:
8310:
8306:
8303:
8300:
8296:
8292:
8289:
8285:
8279:
8275:
8269:
8265:
8261:
8258:
8252:
8249:
8243:
8237:
8234:
8228:
8223:
8218:
8214:
8209:
8205:
8202:
8198:
8194:
8191:
8188:
8185:
8182:
8177:
8173:
8167:
8161:
8158:
8152:
8148:
8143:
8139:
8136:
8133:
8130:
8127:
8122:
8118:
8112:
8108:
8104:
8100:
8096:
8090:
8085:
8078:
8073:
8046:
8041:
8034:
8029:
8005:
8000:
7993:
7988:
7973:
7972:
7961:
7958:
7952:
7947:
7940:
7935:
7930:
7927:
7921:
7916:
7909:
7904:
7889:
7888:
7874:
7869:
7865:
7862:
7857:
7851:
7846:
7842:
7836:
7831:
7823:
7818:
7812:
7809:
7804:
7799:
7795:
7791:
7786:
7781:
7775:
7770:
7766:
7760:
7755:
7740:
7739:
7726:
7723:
7719:
7712:
7707:
7701:
7698:
7693:
7688:
7684:
7680:
7675:
7670:
7664:
7659:
7655:
7649:
7644:
7637:
7633:
7626:
7621:
7617:
7614:
7609:
7603:
7598:
7594:
7588:
7583:
7559:
7548:
7547:
7536:
7533:
7530:
7527:
7524:
7519:
7515:
7509:
7505:
7501:
7498:
7492:
7487:
7482:
7479:
7476:
7473:
7470:
7465:
7461:
7455:
7449:
7446:
7440:
7437:
7431:
7426:
7394:
7388:
7364:
7359:
7355:
7351:
7348:
7343:
7339:
7335:
7324:
7323:
7312:
7309:
7306:
7303:
7300:
7295:
7289:
7284:
7280:
7277:
7274:
7271:
7268:
7265:
7262:
7256:
7251:
7245:
7241:
7235:
7229:
7226:
7220:
7197:
7182:
7181:
7170:
7167:
7164:
7159:
7153:
7148:
7144:
7141:
7138:
7135:
7132:
7129:
7126:
7120:
7115:
7109:
7105:
7099:
7095:
7091:
7067:
7064:
7060:
7056:
7053:
7042:
7041:
7030:
7027:
7024:
7019:
7013:
7008:
7004:
7001:
6996:
6993:
6988:
6983:
6979:
6975:
6970:
6966:
6961:
6957:
6953:
6947:
6944:
6938:
6933:
6929:
6926:
6923:
6920:
6917:
6914:
6909:
6903:
6898:
6894:
6891:
6877:
6876:
6865:
6862:
6859:
6854:
6848:
6843:
6837:
6833:
6829:
6825:
6820:
6816:
6812:
6806:
6803:
6797:
6792:
6788:
6785:
6782:
6779:
6774:
6768:
6763:
6748:
6747:
6736:
6733:
6730:
6724:
6719:
6715:
6712:
6708:
6703:
6699:
6695:
6689:
6683:
6678:
6674:
6671:
6668:
6665:
6659:
6654:
6624:
6620:
6599:
6594:
6591:
6586:
6581:
6577:
6573:
6568:
6563:
6559:
6555:
6544:
6543:
6530:
6524:
6521:
6519:
6516:
6513:
6512:
6509:
6506:
6504:
6501:
6500:
6498:
6493:
6488:
6484:
6480:
6477:
6474:
6460:
6459:
6448:
6445:
6442:
6439:
6433:
6428:
6404:
6398:
6394:
6390:
6387:
6384:
6380:
6377:
6349:
6345:
6341:
6338:
6335:
6331:
6328:
6308:
6305:
6293:
6288:
6285:
6280:
6275:
6271:
6267:
6262:
6259:
6248:
6247:
6236:
6233:
6230:
6224:
6219:
6215:
6212:
6208:
6203:
6199:
6195:
6189:
6183:
6178:
6174:
6171:
6168:
6165:
6159:
6154:
6130:
6127:
6124:
6121:
6101:
6098:
6095:
6092:
6087:
6084:
6080:
6068:
6067:
6052:
6049:
6046:
6040:
6035:
6031:
6026:
6023:
6019:
6013:
6009:
6003:
5999:
5993:
5988:
5983:
5980:
5976:
5972:
5967:
5964:
5962:
5960:
5957:
5954:
5951:
5945:
5940:
5936:
5931:
5927:
5921:
5914:
5910:
5902:
5898:
5894:
5891:
5889:
5887:
5884:
5881:
5878:
5872:
5867:
5863:
5858:
5854:
5848:
5841:
5836:
5829:
5824:
5821:
5817:
5813:
5804:
5800:
5796:
5793:
5791:
5789:
5786:
5783:
5777:
5772:
5768:
5763:
5759:
5753:
5746:
5741:
5736:
5733:
5729:
5725:
5716:
5712:
5708:
5707:
5693:
5692:
5679:
5672:
5667:
5660:
5655:
5652:
5648:
5644:
5637:
5632:
5625:
5620:
5615:
5612:
5608:
5604:
5577:
5574:
5571:
5568:
5565:
5562:
5558:
5555:
5551:
5548:
5524:
5521:
5518:
5515:
5512:
5509:
5506:
5503:
5500:
5497:
5494:
5491:
5488:
5478:
5475:
5464:
5463:
5450:
5447:
5443:
5439:
5436:
5430:
5425:
5421:
5407:
5406:
5393:
5390:
5386:
5380:
5376:
5370:
5365:
5360:
5357:
5353:
5349:
5344:
5339:
5332:
5328:
5320:
5316:
5301:
5300:
5285:
5282:
5279:
5273:
5268:
5264:
5259:
5255:
5249:
5242:
5237:
5232:
5229:
5225:
5221:
5212:
5208:
5204:
5201:
5199:
5197:
5194:
5191:
5188:
5182:
5177:
5173:
5165:
5161:
5157:
5153:
5146:
5139:
5134:
5129:
5126:
5122:
5118:
5109:
5105:
5101:
5098:
5096:
5094:
5091:
5088:
5085:
5079:
5074:
5070:
5062:
5058:
5054:
5050:
5040:
5037:
5033:
5029:
5022:
5018:
5014:
5006:
5002:
4998:
4995:
4993:
4991:
4987:
4982:
4978:
4974:
4968:
4962:
4957:
4948:
4945:
4941:
4937:
4933:
4926:
4922:
4918:
4915:
4913:
4910:
4905:
4901:
4897:
4891:
4885:
4880:
4874:
4869:
4865:
4859:
4855:
4851:
4850:
4838:This leads to
4836:
4835:
4822:
4819:
4815:
4809:
4802:
4797:
4792:
4789:
4785:
4781:
4774:
4769:
4765:
4750:
4749:
4736:
4732:
4726:
4719:
4715:
4709:
4704:
4701:
4697:
4673:
4669:
4665:
4662:
4659:
4655:
4652:
4648:
4645:
4634:
4633:
4622:
4619:
4616:
4610:
4605:
4601:
4598:
4594:
4589:
4585:
4581:
4575:
4569:
4564:
4560:
4557:
4554:
4551:
4545:
4540:
4525:
4524:
4513:
4507:
4504:
4501:
4495:
4490:
4484:
4481:
4476:
4471:
4467:
4463:
4458:
4454:
4449:
4445:
4441:
4435:
4429:
4424:
4420:
4417:
4414:
4411:
4405:
4400:
4385:
4384:
4373:
4370:
4367:
4361:
4356:
4347:
4343:
4339:
4335:
4328:
4322:
4319:
4313:
4310:
4306:
4301:
4297:
4293:
4287:
4281:
4276:
4267:
4264:
4260:
4256:
4252:
4245:
4239:
4236:
4230:
4227:
4224:
4221:
4215:
4210:
4201:
4197:
4193:
4189:
4182:
4176:
4173:
4156:
4155:
4142:
4139:
4135:
4129:
4125:
4119:
4114:
4109:
4106:
4102:
4098:
4093:
4088:
4081:
4077:
4069:
4065:
4041:
4037:
4033:
4030:
4027:
4023:
4020:
3988:
3984:
3980:
3977:
3974:
3971:
3968:
3965:
3962:
3951:
3950:
3939:
3936:
3933:
3927:
3922:
3918:
3915:
3911:
3906:
3902:
3898:
3892:
3886:
3881:
3877:
3874:
3871:
3868:
3862:
3857:
3827:
3823:
3811:
3810:
3799:
3796:
3793:
3787:
3782:
3773:
3769:
3765:
3761:
3754:
3750:
3744:
3739:
3734:
3731:
3727:
3723:
3718:
3714:
3709:
3705:
3701:
3695:
3689:
3684:
3675:
3672:
3668:
3664:
3660:
3653:
3649:
3645:
3642:
3639:
3636:
3630:
3625:
3616:
3612:
3608:
3604:
3597:
3593:
3569:
3563:
3560:
3557:
3554:
3551:
3547:
3544:
3540:
3537:
3517:
3514:
3511:
3506:
3502:
3498:
3495:
3476:
3473:
3461:
3453:
3450:
3445:
3442:
3439:
3428:
3427:
3415:
3411:
3408:
3404:
3399:
3394:
3389:
3384:
3380:
3377:
3373:
3369:
3366:
3362:
3357:
3352:
3348:
3344:
3319:
3297:
3273:
3258:Main article:
3255:
3252:
3251:
3250:
3235:
3232:
3229:
3227:
3222:
3217:
3211:
3207:
3201:
3194:
3191:
3184:
3183:
3180:
3177:
3174:
3172:
3167:
3162:
3156:
3152:
3146:
3142:
3138:
3137:
3123:
3122:
3109:
3106:
3101:
3097:
3092:
3087:
3082:
3076:
3072:
3069:
3063:
3059:
3054:
3049:
3045:
3042:
3038:
3033:
3028:
2999:
2974:
2955:
2954:
2942:
2939:
2936:
2932:
2925:
2922:
2916:
2910:
2907:
2901:
2896:
2892:
2887:
2883:
2880:
2875:
2871:
2865:
2861:
2857:
2854:
2850:
2844:
2840:
2834:
2827:
2824:
2816:
2811:
2805:
2801:
2795:
2791:
2786:
2782:
2779:
2775:
2769:
2765:
2759:
2755:
2750:
2745:
2739:
2735:
2729:
2722:
2719:
2711:
2696:
2695:
2683:
2680:
2677:
2673:
2666:
2663:
2657:
2651:
2648:
2642:
2639:
2634:
2630:
2625:
2621:
2618:
2613:
2609:
2603:
2596:
2593:
2575:
2574:
2563:
2560:
2557:
2553:
2546:
2543:
2537:
2531:
2528:
2522:
2519:
2514:
2510:
2505:
2501:
2498:
2493:
2489:
2483:
2479:
2464:
2463:
2450:
2442:
2438:
2434:
2433:
2428:
2424:
2420:
2419:
2417:
2412:
2409:
2395:
2394:
2380:
2376:
2372:
2369:
2366:
2363:
2359:
2355:
2351:
2347:
2344:
2341:
2337:
2333:
2330:
2325:
2322:
2319:
2316:
2312:
2308:
2304:
2300:
2297:
2294:
2290:
2286:
2283:
2278:
2270:
2266:
2262:
2261:
2256:
2252:
2248:
2247:
2245:
2240:
2236:
2232:
2229:
2225:
2220:
2216:
2197:
2194:
2182:
2176:
2173:
2167:
2162:
2156:
2152:
2146:
2139:
2136:
2112:
2109:
2103:
2098:
2092:
2088:
2082:
2078:
2058:
2057:
2046:
2038:
2030:
2026:
2022:
2019:
2015:
2011:
2007:
2004:
2000:
1996:
1992:
1989:
1985:
1981:
1977:
1976:
1974:
1969:
1964:
1957:
1954:
1936:
1935:
1924:
1921:
1918:
1913:
1909:
1903:
1896:
1893:
1858:
1838:
1835:
1832:
1829:
1826:
1823:
1820:
1817:
1806:Pauli matrices
1793:
1790:
1787:
1765:
1761:
1742:
1741:
1728:
1720:
1716:
1712:
1708:
1704:
1700:
1696:
1692:
1688:
1684:
1680:
1676:
1675:
1673:
1668:
1663:
1655:
1651:
1647:
1643:
1639:
1635:
1631:
1627:
1623:
1619:
1615:
1611:
1610:
1608:
1603:
1598:
1594:
1579:
1578:
1567:
1564:
1558:
1555:
1550:
1547:
1539:
1535:
1531:
1525:
1522:
1517:
1514:
1506:
1502:
1498:
1492:
1489:
1484:
1481:
1473:
1469:
1465:
1459:
1456:
1451:
1448:
1440:
1437:
1430:
1426:
1411:speed of light
1398:
1387:
1386:
1375:
1372:
1369:
1364:
1360:
1354:
1350:
1334:
1331:
1299:Weyl semimetal
1276:Standard Model
1253:quasiparticles
1226:Wolfgang Pauli
1210:Dirac equation
1205:
1202:
1191:quasiparticles
1175:Standard Model
1126:
1125:
1123:
1122:
1115:
1108:
1100:
1097:
1096:
1089:
1088:
1083:
1078:
1073:
1068:
1063:
1058:
1053:
1048:
1043:
1038:
1033:
1028:
1023:
1018:
1013:
1008:
1003:
998:
993:
988:
983:
978:
973:
968:
963:
958:
953:
948:
943:
938:
933:
928:
923:
918:
913:
908:
903:
898:
893:
888:
883:
878:
873:
868:
863:
858:
853:
848:
843:
838:
833:
828:
823:
818:
813:
808:
803:
798:
793:
788:
783:
778:
773:
768:
763:
758:
753:
748:
743:
738:
733:
728:
723:
718:
713:
708:
703:
698:
693:
688:
683:
678:
673:
668:
663:
658:
653:
648:
643:
638:
633:
628:
623:
618:
613:
608:
603:
598:
593:
588:
583:
578:
573:
568:
563:
558:
553:
548:
543:
538:
533:
528:
523:
518:
513:
508:
503:
498:
493:
488:
483:
478:
473:
468:
463:
458:
453:
448:
443:
438:
433:
428:
423:
418:
413:
408:
403:
397:
396:
393:
392:
389:
388:
385:
384:
379:
374:
369:
364:
358:
355:
354:
351:
350:
347:
346:
341:
336:
331:
325:
322:Standard Model
320:
319:
316:
315:
312:
311:
306:
301:
296:
291:
289:Dirac equation
285:
282:
281:
278:
277:
274:
273:
271:Wick's theorem
268:
263:
258:
256:Regularization
253:
248:
243:
238:
233:
228:
223:
218:
213:
208:
203:
198:
193:
188:
182:
179:
178:
175:
174:
171:
170:
165:
163:Noether charge
160:
155:
150:
148:Gauge symmetry
145:
140:
135:
130:
125:
120:
114:
109:
108:
105:
104:
101:
100:
95:
90:
85:
80:
75:
70:
65:
60:
54:
51:
50:
47:
46:
40:
39:
34:
26:
25:
15:
9:
6:
4:
3:
2:
11277:
11266:
11263:
11262:
11260:
11251:
11248:
11246:
11243:
11241:
11238:
11236:
11233:
11232:
11222:
11218:
11214:
11210:
11206:
11202:
11198:
11194:
11189:
11184:
11180:
11176:
11172:
11167:
11156:. Vol. 8
11155:
11151:
11146:
11142:
11138:
11134:
11130:
11125:
11120:
11116:
11112:
11108:
11103:
11092:
11091:Physics World
11088:
11083:
11079:
11073:
11069:
11068:
11062:
11058:
11052:
11048:
11047:
11041:
11035:
11033:9781118681664
11029:
11025:
11024:
11018:
11017:
11014:
11008:
11004:
10999:
10995:
10989:
10985:
10984:
10978:
10977:
10965:
10958:
10950:
10946:
10941:
10936:
10932:
10925:
10923:
10914:
10912:0-201-50397-2
10908:
10904:
10903:
10895:
10887:
10881:
10877:
10870:
10862:
10856:
10852:
10845:
10836:
10831:
10827:
10823:
10819:
10815:
10814:
10809:
10802:
10794:
10790:
10783:
10781:
10779:
10770:
10766:
10762:
10758:
10754:
10750:
10746:
10742:
10737:
10732:
10728:
10724:
10720:
10713:
10711:
10709:
10707:
10705:
10703:
10694:
10690:
10685:
10680:
10676:
10672:
10667:
10662:
10658:
10654:
10650:
10646:
10642:
10638:
10637:Weyl, Hermann
10632:
10624:
10620:
10616:
10612:
10608:
10604:
10600:
10596:
10591:
10586:
10582:
10578:
10574:
10567:
10565:
10563:
10554:
10552:9789810239480
10548:
10544:
10537:
10533:
10520:
10519:Lorentz group
10513:
10509:
10501:
10499:
10494:
10477:
10420:
10415:
10398:
10395:
10392:
10389:
10386:
10380:
10374:
10371:
10368:
10343:
10339:
10335:
10329:
10326:
10323:
10282:
10271:
10262:
10243:
10212:
10201:
10196:
10192:
10166:
10162:
10154:
10143:
10136:
10133:
10130:
10110:
10104:
10101:
10098:
10065:
10064:
10063:
10049:
10043:
10040:
10037:
10003:
9996:Special cases
9993:
9977:
9972:
9945:
9940:
9915:
9907:
9889:
9885:
9881:
9877:
9873:
9870:
9848:
9843:
9835:
9830:
9825:
9817:
9809:
9805:
9801:
9777:
9769:
9765:
9761:
9717:
9711:
9683:
9675:
9671:
9645:
9635:
9625:
9623:
9622:spin geometry
9619:
9615:
9611:
9607:
9603:
9602:spin manifold
9598:
9595:
9590:
9588:
9572:
9566:
9558:
9539:
9536:
9533:
9515:
9511:
9492:
9489:
9486:
9461:
9445:
9442:
9433:
9431:
9427:
9408:
9405:
9402:
9396:
9390:
9387:
9384:
9373:
9357:
9349:
9345:
9339:
9335:
9331:
9328:
9323:
9319:
9313:
9309:
9300:
9280:
9274:
9271:
9268:
9265:
9261:
9257:
9254:
9249:
9246:
9242:
9237:
9230:
9226:
9221:
9216:
9211:
9207:
9199:
9198:
9197:
9179:
9173:
9170:
9167:
9164:
9160:
9156:
9153:
9148:
9145:
9141:
9136:
9129:
9125:
9120:
9115:
9111:
9103:
9102:
9101:
9087:
9082:
9078:
9069:
9048:
9044:
9017:
9014:
9011:
8992:
8973:
8970:
8967:
8944:
8939:
8935:
8927:
8926:tangent space
8911:
8908:
8905:
8902:
8882:
8879:
8872:
8853:
8850:
8847:
8837:of dimension
8824:
8817:
8812:
8810:
8806:
8802:
8798:
8782:
8775:
8771:
8767:
8763:
8758:
8756:
8752:
8748:
8744:
8740:
8736:
8732:
8728:
8718:
8704:
8696:
8677:
8663:
8657:
8647:
8637:
8628:
8617:
8613:
8610:
8596:
8595:
8580:
8566:
8560:
8550:
8546:
8540:
8529:
8525:
8522:
8508:
8507:
8506:
8505:
8495:
8481:
8475:
8472:
8469:
8449:
8446:
8441:
8437:
8433:
8425:
8409:
8403:
8400:
8397:
8394:
8391:
8388:
8368:
8365:
8362:
8357:
8353:
8328:
8322:
8318:
8312:
8308:
8304:
8301:
8298:
8294:
8290:
8287:
8283:
8277:
8273:
8267:
8263:
8259:
8256:
8241:
8226:
8221:
8216:
8207:
8203:
8200:
8196:
8192:
8189:
8186:
8183:
8180:
8175:
8165:
8156:
8150:
8146:
8141:
8137:
8134:
8131:
8128:
8125:
8120:
8110:
8106:
8102:
8098:
8094:
8083:
8071:
8063:
8062:
8061:
8039:
8027:
7998:
7986:
7978:The products
7976:
7959:
7956:
7945:
7933:
7928:
7925:
7914:
7902:
7894:
7893:
7892:
7867:
7863:
7860:
7844:
7829:
7816:
7810:
7807:
7802:
7797:
7793:
7789:
7784:
7768:
7753:
7745:
7744:
7743:
7724:
7721:
7717:
7705:
7699:
7696:
7691:
7686:
7682:
7678:
7673:
7657:
7642:
7635:
7631:
7619:
7615:
7612:
7596:
7581:
7573:
7572:
7571:
7557:
7534:
7531:
7528:
7525:
7522:
7517:
7507:
7503:
7499:
7496:
7485:
7480:
7477:
7474:
7471:
7468:
7463:
7453:
7444:
7438:
7435:
7424:
7416:
7415:
7414:
7411:
7409:
7392:
7386:
7378:
7362:
7357:
7353:
7349:
7346:
7341:
7337:
7333:
7310:
7307:
7301:
7293:
7282:
7278:
7275:
7272:
7269:
7263:
7249:
7243:
7233:
7224:
7218:
7211:
7210:
7209:
7195:
7187:
7165:
7157:
7146:
7142:
7139:
7136:
7133:
7127:
7113:
7107:
7097:
7093:
7089:
7082:
7081:
7080:
7065:
7062:
7058:
7054:
7051:
7025:
7017:
7006:
7002:
6999:
6994:
6991:
6986:
6981:
6977:
6973:
6968:
6964:
6955:
6951:
6945:
6931:
6927:
6924:
6915:
6907:
6896:
6892:
6889:
6882:
6881:
6880:
6860:
6852:
6841:
6835:
6831:
6827:
6823:
6814:
6810:
6804:
6790:
6780:
6772:
6761:
6753:
6752:
6751:
6731:
6717:
6713:
6710:
6706:
6697:
6693:
6676:
6666:
6652:
6644:
6643:
6642:
6640:
6622:
6618:
6597:
6592:
6589:
6584:
6579:
6575:
6571:
6566:
6561:
6557:
6553:
6528:
6522:
6517:
6514:
6507:
6502:
6496:
6491:
6486:
6482:
6478:
6475:
6472:
6465:
6464:
6463:
6446:
6443:
6440:
6437:
6426:
6418:
6417:
6416:
6402:
6388:
6385:
6367:
6363:
6339:
6336:
6318:
6314:
6304:
6291:
6286:
6283:
6278:
6273:
6269:
6265:
6260:
6257:
6231:
6217:
6213:
6210:
6206:
6197:
6193:
6176:
6166:
6152:
6144:
6143:
6142:
6128:
6125:
6122:
6119:
6099:
6096:
6093:
6090:
6085:
6082:
6078:
6047:
6033:
6029:
6024:
6021:
6017:
6011:
6001:
5997:
5991:
5986:
5981:
5978:
5974:
5970:
5965:
5963:
5952:
5938:
5934:
5929:
5919:
5912:
5900:
5896:
5892:
5890:
5879:
5865:
5861:
5856:
5846:
5839:
5834:
5822:
5819:
5811:
5802:
5798:
5794:
5792:
5784:
5770:
5766:
5761:
5751:
5744:
5739:
5734:
5731:
5723:
5714:
5710:
5698:
5697:
5696:
5677:
5670:
5665:
5653:
5650:
5642:
5635:
5630:
5623:
5618:
5613:
5610:
5602:
5591:
5590:
5589:
5575:
5569:
5566:
5563:
5549:
5538:
5519:
5516:
5513:
5510:
5507:
5504:
5501:
5498:
5495:
5492:
5489:
5476:
5473:
5448:
5445:
5437:
5434:
5419:
5412:
5411:
5410:
5391:
5388:
5384:
5378:
5374:
5368:
5363:
5358:
5355:
5351:
5347:
5342:
5337:
5330:
5318:
5314:
5306:
5305:
5304:
5280:
5266:
5262:
5257:
5247:
5240:
5235:
5230:
5227:
5219:
5210:
5206:
5202:
5200:
5189:
5175:
5171:
5163:
5159:
5144:
5137:
5132:
5127:
5124:
5116:
5107:
5103:
5099:
5097:
5086:
5072:
5068:
5060:
5056:
5038:
5031:
5020:
5016:
5004:
5000:
4996:
4994:
4985:
4976:
4972:
4955:
4946:
4939:
4924:
4920:
4916:
4914:
4908:
4899:
4895:
4878:
4867:
4857:
4853:
4841:
4840:
4839:
4820:
4813:
4807:
4800:
4795:
4790:
4787:
4779:
4772:
4767:
4763:
4755:
4754:
4753:
4734:
4730:
4724:
4717:
4707:
4702:
4695:
4687:
4686:
4685:
4663:
4660:
4646:
4643:
4617:
4603:
4599:
4596:
4592:
4583:
4579:
4562:
4552:
4538:
4530:
4529:
4528:
4511:
4502:
4488:
4482:
4479:
4474:
4469:
4465:
4461:
4456:
4452:
4443:
4439:
4422:
4412:
4398:
4390:
4389:
4388:
4368:
4354:
4345:
4341:
4326:
4317:
4311:
4308:
4304:
4295:
4291:
4274:
4265:
4258:
4243:
4234:
4222:
4208:
4199:
4195:
4180:
4171:
4161:
4160:
4159:
4140:
4137:
4133:
4127:
4123:
4117:
4112:
4107:
4104:
4100:
4096:
4091:
4086:
4079:
4067:
4063:
4055:
4054:
4053:
4031:
4028:
4010:
4006:
4005:Lorentz group
4002:
3978:
3975:
3969:
3966:
3963:
3960:
3934:
3920:
3916:
3913:
3909:
3900:
3896:
3879:
3869:
3855:
3847:
3846:
3845:
3843:
3825:
3821:
3794:
3780:
3771:
3767:
3752:
3748:
3742:
3737:
3732:
3729:
3725:
3721:
3716:
3712:
3703:
3699:
3682:
3673:
3666:
3651:
3647:
3637:
3623:
3614:
3610:
3595:
3591:
3583:
3582:
3581:
3567:
3558:
3555:
3552:
3538:
3515:
3509:
3500:
3493:
3486:
3482:
3472:
3459:
3451:
3448:
3443:
3440:
3437:
3413:
3409:
3406:
3397:
3378:
3375:
3371:
3367:
3364:
3355:
3346:
3334:
3333:
3332:
3287:
3271:
3261:
3233:
3230:
3228:
3215:
3209:
3199:
3189:
3178:
3175:
3173:
3160:
3154:
3144:
3140:
3128:
3127:
3126:
3107:
3104:
3099:
3074:
3070:
3067:
3061:
3040:
3018:
3017:
3016:
3014:
2989:
2964:
2960:
2940:
2937:
2934:
2930:
2920:
2914:
2905:
2899:
2894:
2890:
2885:
2881:
2878:
2873:
2869:
2863:
2859:
2855:
2852:
2848:
2842:
2838:
2832:
2822:
2814:
2809:
2803:
2799:
2793:
2789:
2784:
2780:
2777:
2773:
2767:
2763:
2757:
2753:
2748:
2743:
2737:
2733:
2727:
2717:
2709:
2701:
2700:
2699:
2681:
2678:
2675:
2671:
2661:
2655:
2646:
2640:
2637:
2632:
2628:
2623:
2619:
2616:
2611:
2607:
2601:
2591:
2580:
2579:
2578:
2561:
2558:
2555:
2551:
2541:
2535:
2526:
2520:
2517:
2512:
2508:
2503:
2499:
2496:
2491:
2487:
2481:
2477:
2469:
2468:
2467:
2448:
2440:
2436:
2426:
2422:
2415:
2410:
2407:
2400:
2399:
2398:
2374:
2367:
2364:
2361:
2353:
2342:
2339:
2335:
2331:
2328:
2320:
2317:
2314:
2306:
2295:
2292:
2288:
2284:
2281:
2276:
2268:
2264:
2254:
2250:
2243:
2238:
2234:
2230:
2227:
2218:
2214:
2207:
2206:
2205:
2203:
2193:
2180:
2174:
2171:
2160:
2154:
2144:
2134:
2110:
2107:
2096:
2090:
2080:
2076:
2067:
2063:
2044:
2036:
2028:
2024:
2020:
2013:
2009:
2005:
1998:
1994:
1990:
1983:
1979:
1972:
1967:
1962:
1952:
1941:
1940:
1939:
1922:
1919:
1916:
1911:
1901:
1891:
1880:
1879:
1878:
1876:
1872:
1856:
1836:
1833:
1830:
1827:
1824:
1821:
1818:
1815:
1807:
1791:
1788:
1785:
1763:
1759:
1751:
1747:
1726:
1718:
1714:
1706:
1702:
1694:
1690:
1682:
1678:
1671:
1666:
1661:
1653:
1649:
1641:
1637:
1629:
1625:
1617:
1613:
1606:
1601:
1596:
1592:
1584:
1583:
1582:
1565:
1562:
1556:
1548:
1537:
1533:
1529:
1523:
1515:
1504:
1500:
1496:
1490:
1482:
1471:
1467:
1463:
1457:
1449:
1438:
1435:
1428:
1424:
1416:
1415:
1414:
1413:, it becomes
1412:
1396:
1373:
1370:
1367:
1362:
1352:
1348:
1340:
1339:
1338:
1330:
1328:
1324:
1320:
1316:
1312:
1308:
1304:
1300:
1295:
1292:
1288:
1284:
1279:
1277:
1273:
1269:
1265:
1261:
1260:Wu experiment
1256:
1254:
1250:
1245:
1243:
1239:
1235:
1231:
1227:
1223:
1219:
1215:
1211:
1201:
1198:
1196:
1192:
1188:
1184:
1180:
1176:
1172:
1167:
1165:
1161:
1157:
1153:
1152:Weyl fermions
1149:
1145:
1141:
1140:Weyl equation
1137:
1133:
1121:
1116:
1114:
1109:
1107:
1102:
1101:
1099:
1098:
1092:
1087:
1084:
1082:
1079:
1077:
1074:
1072:
1069:
1067:
1064:
1062:
1061:Zamolodchikov
1059:
1057:
1056:Zamolodchikov
1054:
1052:
1049:
1047:
1044:
1042:
1039:
1037:
1034:
1032:
1029:
1027:
1024:
1022:
1019:
1017:
1014:
1012:
1009:
1007:
1004:
1002:
999:
997:
994:
992:
989:
987:
984:
982:
979:
977:
974:
972:
969:
967:
964:
962:
959:
957:
954:
952:
949:
947:
944:
942:
939:
937:
934:
932:
929:
927:
924:
922:
919:
917:
914:
912:
909:
907:
904:
902:
899:
897:
894:
892:
889:
887:
884:
882:
879:
877:
874:
872:
869:
867:
864:
862:
859:
857:
854:
852:
849:
847:
844:
842:
839:
837:
834:
832:
829:
827:
824:
822:
819:
817:
814:
812:
809:
807:
804:
802:
799:
797:
794:
792:
789:
787:
784:
782:
779:
777:
774:
772:
769:
767:
764:
762:
759:
757:
754:
752:
749:
747:
744:
742:
739:
737:
734:
732:
729:
727:
724:
722:
719:
717:
714:
712:
709:
707:
704:
702:
699:
697:
694:
692:
689:
687:
684:
682:
679:
677:
674:
672:
669:
667:
664:
662:
659:
657:
654:
652:
649:
647:
644:
642:
639:
637:
634:
632:
629:
627:
624:
622:
619:
617:
614:
612:
609:
607:
604:
602:
599:
597:
594:
592:
589:
587:
584:
582:
579:
577:
574:
572:
569:
567:
564:
562:
559:
557:
554:
552:
549:
547:
544:
542:
539:
537:
534:
532:
529:
527:
524:
522:
519:
517:
514:
512:
509:
507:
504:
502:
499:
497:
494:
492:
489:
487:
484:
482:
479:
477:
474:
472:
469:
467:
464:
462:
459:
457:
454:
452:
449:
447:
444:
442:
439:
437:
434:
432:
429:
427:
424:
422:
419:
417:
414:
412:
409:
407:
404:
402:
399:
398:
391:
390:
383:
380:
378:
375:
373:
370:
368:
367:Supersymmetry
365:
363:
362:String theory
360:
359:
353:
352:
345:
342:
340:
337:
335:
332:
330:
327:
326:
323:
318:
317:
310:
307:
305:
302:
300:
297:
295:
292:
290:
287:
286:
280:
279:
272:
269:
267:
264:
262:
259:
257:
254:
252:
249:
247:
244:
242:
239:
237:
234:
232:
229:
227:
224:
222:
219:
217:
214:
212:
209:
207:
204:
202:
199:
197:
194:
192:
189:
187:
184:
183:
177:
176:
169:
166:
164:
161:
159:
156:
154:
151:
149:
146:
144:
141:
139:
136:
134:
131:
129:
126:
124:
121:
119:
116:
115:
112:
107:
106:
99:
96:
94:
91:
89:
86:
84:
81:
79:
76:
74:
71:
69:
66:
64:
61:
59:
56:
55:
49:
48:
45:
42:
41:
37:
32:
28:
27:
24:
21:
20:
11178:
11174:
11158:. Retrieved
11153:
11114:
11110:
11094:. Retrieved
11090:
11065:
11045:
11022:
11002:
10982:
10963:
10957:
10930:
10901:
10894:
10875:
10869:
10850:
10844:
10817:
10811:
10801:
10792:
10726:
10722:
10648:
10644:
10631:
10580:
10576:
10542:
10536:
10512:
10495:
10416:
10183:
10002:Dirac spinor
9999:
9905:
9637:
9599:
9591:
9557:frame bundle
9510:double cover
9460:fiber bundle
9434:
9424:dimensional
9296:
9195:
8991:vector space
8989:dimensional
8813:
8759:
8726:
8724:
8721:Weyl spinors
8697:(denoted by
8692:
8501:
8344:
7977:
7974:
7890:
7741:
7549:
7412:
7325:
7183:
7043:
6878:
6749:
6545:
6461:
6310:
6249:
6069:
5694:
5465:
5408:
5302:
4837:
4751:
4635:
4526:
4386:
4157:
3952:
3812:
3478:
3429:
3263:
3124:
2956:
2697:
2576:
2465:
2396:
2199:
2059:
1937:
1871:wavefunction
1743:
1580:
1388:
1336:
1296:
1280:
1262:showed that
1257:
1246:
1207:
1199:
1169:None of the
1168:
1156:Hermann Weyl
1151:
1139:
1129:
1090:
936:Stueckelberg
676:Jona-Lasinio
266:Vacuum state
251:Quantization
93:Gauge theory
73:Strong force
58:Field theory
11160:22 November
11154:APS Physics
11096:22 November
10793:APS Physics
10498:ELKO spinor
8727:Weyl spinor
3953:The matrix
2988:wave-vector
2064:, and thus
1319:M. SoljaÄiÄ
1076:Zinn-Justin
926:Sommerfield
851:Pomeranchuk
821:Osterwalder
816:Oppenheimer
746:ĆopuszaĆski
571:Fredenhagen
372:Technicolor
11188:1612.00416
11117:(9): 863.
10529:References
9632:See also:
9430:spin group
9033:on it. If
8805:spin group
8462:, that is
6362:isomorphic
3483:under the
2202:plane-wave
1307:M.Z. Hasan
1242:beta decay
1214:Paul Dirac
1071:Zimmermann
966:Vainshtein
711:Kontsevich
656:Iliopoulos
631:Heisenberg
456:Bogoliubov
394:Scientists
246:Propagator
133:T-symmetry
128:P-symmetry
123:C-symmetry
111:Symmetries
68:Weak force
52:Background
11133:1476-1122
10949:2073-8994
10769:119341278
10761:0031-8949
10729:: 17â22.
10675:0027-8424
10623:118685467
10615:0002-9505
10590:1006.1718
10396:−
10387:−
10336:×
10272:×
10202:≅
10144:×
10111:≅
9978:−
9969:Δ
9937:Δ
9849:−
9840:Δ
9836:⊕
9822:Δ
9778:⊕
9638:For even
9332:−
9255:−
9217:∗
8906:∈
8797:spacetime
8783:ψ
8770:rotations
8762:invariant
8725:The term
8705:†
8695:conjugate
8664:ψ
8658:μ
8654:∂
8648:μ
8641:¯
8638:σ
8629:†
8618:ψ
8567:ψ
8561:μ
8557:∂
8551:μ
8547:σ
8541:†
8530:ψ
8476:ζ
8470:η
8442:∗
8438:ζ
8434:η
8398:−
8381:and that
8366:−
8354:ω
8313:∗
8309:ζ
8305:η
8299:◻
8291:−
8268:∗
8264:ζ
8260:η
8251:→
8248:∇
8242:⋅
8236:→
8233:∇
8227:−
8213:∂
8204:−
8190:ω
8184:ζ
8176:μ
8172:∂
8166:μ
8160:¯
8157:σ
8135:ω
8129:η
8121:μ
8117:∂
8111:μ
8107:σ
7946:ψ
7915:ψ
7868:ψ
7856:′
7845:ψ
7841:↦
7830:ψ
7817:ψ
7808:−
7798:†
7780:′
7769:ψ
7765:↦
7754:ψ
7722:−
7697:−
7687:†
7669:′
7654:↦
7636:†
7608:′
7593:↦
7532:ω
7526:η
7518:μ
7514:∂
7508:μ
7504:σ
7478:ω
7472:ζ
7464:μ
7460:∂
7454:μ
7448:¯
7445:σ
7387:ψ
7363:ψ
7354:σ
7342:∗
7338:ψ
7334:ω
7294:∗
7283:ψ
7279:ω
7273:ζ
7250:ψ
7244:μ
7240:∂
7234:μ
7228:¯
7225:σ
7196:ζ
7158:∗
7147:ψ
7143:ω
7137:η
7114:ψ
7108:μ
7104:∂
7098:μ
7094:σ
7066:ϕ
7052:η
7018:∗
7007:ψ
7003:ω
6992:−
6982:†
6960:′
6946:∗
6943:′
6932:ψ
6928:ω
6922:↦
6908:∗
6897:ψ
6893:ω
6853:∗
6842:ψ
6836:∗
6819:′
6805:∗
6802:′
6791:ψ
6787:↦
6773:∗
6762:ψ
6718:ψ
6702:′
6688:′
6677:ψ
6673:↦
6653:ψ
6623:∗
6598:ω
6590:−
6580:†
6562:∗
6554:ω
6515:−
6483:σ
6473:ω
6447:ω
6438:ω
6284:−
6274:†
6218:ψ
6202:′
6188:′
6177:ψ
6173:↦
6153:ψ
6112:that is,
6083:−
6034:ψ
6022:−
6012:μ
6008:∂
6002:μ
5998:σ
5992:†
5979:−
5939:ψ
5930:ν
5926:∂
5920:ν
5913:μ
5909:Λ
5901:μ
5897:σ
5866:ψ
5857:ν
5853:∂
5847:ν
5840:μ
5820:−
5816:Λ
5803:μ
5799:σ
5771:ψ
5762:ν
5758:∂
5752:μ
5745:ν
5732:−
5728:Λ
5715:μ
5711:σ
5695:to write
5678:ν
5671:μ
5651:−
5647:Λ
5631:μ
5624:ν
5611:−
5607:Λ
5550:∈
5547:Λ
5517:−
5508:−
5499:−
5474:η
5446:−
5442:Λ
5435:η
5424:Λ
5420:η
5389:−
5379:ν
5375:σ
5369:†
5356:−
5338:ν
5331:μ
5327:Λ
5319:μ
5315:σ
5267:ψ
5258:ν
5254:∂
5248:μ
5241:ν
5228:−
5224:Λ
5211:μ
5207:σ
5176:ψ
5164:ν
5156:∂
5152:∂
5145:μ
5138:ν
5125:−
5121:Λ
5108:μ
5104:σ
5073:ψ
5061:ν
5053:∂
5049:∂
5039:μ
5036:′
5028:∂
5021:ν
5013:∂
5005:μ
5001:σ
4981:′
4967:′
4956:ψ
4947:μ
4944:′
4936:∂
4932:∂
4925:μ
4921:σ
4904:′
4890:′
4879:ψ
4873:′
4868:μ
4864:∂
4858:μ
4854:σ
4821:μ
4818:′
4808:μ
4801:ν
4788:−
4784:Λ
4768:ν
4735:ν
4725:ν
4718:μ
4714:Λ
4703:μ
4700:′
4647:∈
4604:ψ
4588:′
4574:′
4563:ψ
4559:↦
4539:ψ
4489:ψ
4480:−
4470:†
4448:′
4434:′
4423:ψ
4419:↦
4399:ψ
4355:ψ
4346:μ
4338:∂
4334:∂
4327:μ
4321:¯
4318:σ
4300:′
4286:′
4275:ψ
4266:μ
4263:′
4255:∂
4251:∂
4244:μ
4238:¯
4235:σ
4229:↦
4209:ψ
4200:μ
4192:∂
4188:∂
4181:μ
4175:¯
4172:σ
4138:−
4128:ν
4124:σ
4118:†
4105:−
4087:ν
4080:μ
4076:Λ
4068:μ
4064:σ
4052:given by
3964:∈
3921:ψ
3905:′
3891:′
3880:ψ
3876:↦
3856:ψ
3826:†
3781:ψ
3772:μ
3764:∂
3760:∂
3753:μ
3749:σ
3743:†
3730:−
3708:′
3694:′
3683:ψ
3674:μ
3671:′
3663:∂
3659:∂
3652:μ
3648:σ
3644:↦
3624:ψ
3615:μ
3607:∂
3603:∂
3596:μ
3592:σ
3539:∈
3536:Λ
3513:Λ
3505:′
3497:↦
3444:±
3438:λ
3410:λ
3379:λ
3368:λ
3347:⋅
3272:λ
3216:ψ
3210:μ
3206:∂
3200:μ
3193:¯
3190:σ
3161:ψ
3155:μ
3151:∂
3145:μ
3141:σ
3105:ω
3081:⇒
3071:ω
3068:ℏ
3044:ℏ
2935:χ
2924:→
2915:⋅
2909:→
2900:−
2879:χ
2874:μ
2864:μ
2853:χ
2843:μ
2833:μ
2826:¯
2823:σ
2804:ν
2794:ν
2790:σ
2778:χ
2768:μ
2758:μ
2754:σ
2738:ν
2728:ν
2721:¯
2718:σ
2676:χ
2665:→
2656:⋅
2650:→
2647:σ
2617:χ
2612:μ
2602:μ
2595:¯
2592:σ
2556:χ
2545:→
2536:⋅
2530:→
2527:σ
2521:−
2497:χ
2492:μ
2482:μ
2478:σ
2437:χ
2423:χ
2408:χ
2379:ℏ
2362:−
2354:⋅
2340:−
2332:χ
2318:ω
2315:−
2307:⋅
2293:−
2285:χ
2265:ψ
2251:ψ
2215:ψ
2161:ψ
2155:μ
2151:∂
2145:μ
2138:¯
2135:σ
2097:ψ
2091:μ
2087:∂
2081:μ
2077:σ
2066:chirality
2025:σ
2021:−
2010:σ
2006:−
1995:σ
1991:−
1963:μ
1956:¯
1953:σ
1917:ψ
1912:μ
1908:∂
1902:μ
1895:¯
1892:σ
1857:ψ
1816:μ
1786:μ
1715:σ
1703:σ
1691:σ
1650:σ
1638:σ
1626:σ
1614:σ
1597:μ
1593:σ
1554:∂
1549:ψ
1546:∂
1534:σ
1521:∂
1516:ψ
1513:∂
1501:σ
1488:∂
1483:ψ
1480:∂
1468:σ
1455:∂
1450:ψ
1447:∂
1368:ψ
1363:μ
1359:∂
1353:μ
1349:σ
1247:In 1937,
1006:Wetterich
991:Weisskopf
941:Sudarshan
891:Schwinger
806:Nishijima
771:Maldacena
736:Leutwyler
701:Kinoshita
601:Goldstone
591:Gell-Mann
506:Doplicher
283:Equations
11259:Category
11213:27777402
11141:26288972
10931:Symmetry
10693:16587474
9863:, where
9600:Given a
8743:fermions
8739:manifold
7379:form of
3414:⟩
3372:⟩
3254:Helicity
2963:momentum
2959:massless
2062:helicity
1804:and the
1409:for the
1333:Equation
1272:helicity
1238:neutrino
1218:spin-1/2
1183:neutrino
1162:and the
1148:spin-1/2
1021:Wightman
986:Weinberg
976:Virasoro
956:Tomonaga
951:Thirring
946:Symanzik
906:Semenoff
881:Schrader
846:Polyakov
766:Majorana
706:Klebanov
661:Ivanenko
651:'t Hooft
621:Guralnik
566:Fröhlich
561:Fritzsch
556:Frampton
471:Buchholz
416:Bargmann
406:Anderson
206:Crossing
11221:1115349
11193:Bibcode
10822:Bibcode
10741:Bibcode
10653:Bibcode
10595:Bibcode
9608:is the
9512:of the
8735:spinors
6637:is the
6364:to the
4007:by the
4003:of the
3840:is the
3011:by the
1875:spinors
1869:is the
1234:fermion
1204:History
1173:in the
1132:physics
1031:Wilczek
996:Wentzel
971:Veltman
916:Shirkov
911:Shifman
901:Seiberg
886:Schwarz
866:Rubakov
791:Naimark
741:Lipatov
731:Lehmann
696:Kendall
586:Gelfand
581:Glashow
541:Feynman
521:Faddeev
516:Englert
486:Coleman
476:Cachazo
461:Brodsky
446:Bjorken
436:Berezin
426:Belavin
186:Anomaly
44:History
11219:
11211:
11139:
11131:
11074:
11053:
11030:
11009:
10990:
10947:
10909:
10882:
10857:
10767:
10759:
10691:
10684:522457
10681:
10673:
10621:
10613:
10549:
10184:where
9570:
9355:
9066:are a
8774:spinor
8766:boosts
8675:
8578:
8479:
8407:
7550:where
7390:
6610:where
6462:where
6400:
5466:where
4509:
3813:where
3565:
3528:where
3457:
2397:where
2178:
2042:
1938:where
1746:vector
1581:where
1321:team (
1264:parity
1236:, the
1230:parity
1138:, the
1086:Zumino
1051:Yukawa
1041:Witten
1036:Wilson
1026:Wigner
961:Tyutin
921:Skyrme
871:Ruelle
841:Plefka
836:Peskin
826:Parisi
786:MĂžller
776:Migdal
761:Maiani
756:LĂŒders
721:Landau
716:Kuraev
691:KÀllén
681:Jordan
666:Jackiw
606:Gribov
496:DeWitt
491:Dashen
481:Callan
451:Bleuer
421:Becchi
411:Anselm
11217:S2CID
11183:arXiv
10765:S2CID
10731:arXiv
10619:S2CID
10585:arXiv
10504:Notes
9559:over
9508:is a
9196:and
8957:is a
7208:) is
6250:with
3430:Here
1744:is a
1160:Dirac
1142:is a
1081:Zuber
931:Stora
896:Segal
876:Salam
861:Proca
856:Popov
831:Pauli
811:Oehme
801:Neveu
796:Nambu
781:Mills
671:Jaffe
646:Hagen
641:Higgs
616:Gupta
611:Gross
596:Glimm
576:Furry
546:Fierz
536:Fermi
531:Fayet
526:Fadin
511:Dyson
501:Dirac
466:Brout
441:Bethe
401:Adler
180:Tools
11209:PMID
11162:2018
11137:PMID
11129:ISSN
11098:2018
11072:ISBN
11051:ISBN
11028:ISBN
11007:ISBN
10988:ISBN
10945:ISSN
10907:ISBN
10880:ISBN
10855:ISBN
10757:ISSN
10727:T121
10689:PMID
10671:ISSN
10611:ISSN
10547:ISBN
10517:the
10417:The
8924:the
8768:and
8019:and
5482:diag
3015:as:
2577:or
2200:The
2123:and
1849:and
1808:for
1778:for
1309:'s (
1208:The
1046:Yang
1016:Wick
1011:Weyl
1001:Wess
981:Ward
686:Jost
636:Hepp
626:Haag
551:Fock
431:Bell
11201:doi
11119:doi
10935:doi
10830:doi
10818:105
10749:doi
10679:PMC
10661:doi
10603:doi
10259:of
9992:).
9908:in
9070:on
8753:or
6360:is
1130:In
1066:Zee
751:Low
726:Lee
11261::
11215:.
11207:.
11199:.
11191:.
11179:15
11177:.
11173:.
11152:.
11135:.
11127:.
11115:14
11113:.
11109:.
11089:.
10943:.
10921:^
10828:.
10816:.
10810:.
10791:.
10777:^
10763:.
10755:.
10747:.
10739:.
10725:.
10721:.
10701:^
10687:.
10677:.
10669:.
10659:.
10649:15
10647:.
10643:.
10617:.
10609:.
10601:.
10593:.
10581:79
10579:.
10575:.
10561:^
9818:=:
9624:.
9589:.
9432:.
3331::
1329:.
1197:.
1166:.
11223:.
11203::
11195::
11185::
11164:.
11143:.
11121::
11100:.
11080:.
11015:.
10996:.
10951:.
10937::
10888:.
10863:.
10838:.
10832::
10824::
10771:.
10751::
10743::
10733::
10695:.
10663::
10655::
10625:.
10605::
10597::
10587::
10555:.
10521:.
10481:)
10478:1
10475:(
10471:U
10447:C
10441:n
10438:i
10435:p
10432:s
10402:)
10399:u
10393:,
10390:s
10384:(
10381:=
10378:)
10375:u
10372:,
10369:s
10366:(
10344:1
10340:S
10333:)
10330:q
10327:,
10324:p
10321:(
10317:n
10314:i
10311:p
10308:S
10283:2
10278:Z
10247:)
10244:1
10241:(
10237:U
10216:)
10213:1
10210:(
10206:U
10197:1
10193:S
10167:1
10163:S
10155:2
10150:Z
10140:)
10137:q
10134:,
10131:p
10128:(
10124:n
10121:i
10118:p
10115:S
10108:)
10105:q
10102:,
10099:p
10096:(
10090:C
10084:n
10081:i
10078:p
10075:S
10050:.
10047:)
10044:q
10041:,
10038:p
10035:(
10029:C
10023:n
10020:i
10017:p
10014:S
9973:n
9946:+
9941:n
9916:n
9890:2
9886:/
9882:n
9878:2
9874:=
9871:N
9844:n
9831:+
9826:n
9815:)
9810:2
9806:/
9802:N
9797:C
9792:(
9788:d
9785:n
9782:E
9775:)
9770:2
9766:/
9762:N
9757:C
9752:(
9748:d
9745:n
9742:E
9721:)
9718:n
9715:(
9712:l
9708:C
9687:)
9684:n
9681:(
9676:0
9672:l
9667:C
9646:n
9573:.
9567:M
9543:)
9540:q
9537:,
9534:p
9531:(
9527:O
9524:S
9496:)
9493:q
9490:,
9487:p
9484:(
9480:n
9477:i
9474:p
9471:S
9446:,
9443:M
9412:)
9409:3
9406:,
9403:1
9400:(
9397:=
9394:)
9391:q
9388:,
9385:p
9382:(
9358:.
9350:j
9346:w
9340:m
9336:w
9329:=
9324:m
9320:w
9314:j
9310:w
9281:)
9275:1
9272:+
9269:j
9266:2
9262:e
9258:i
9250:j
9247:2
9243:e
9238:(
9231:2
9227:1
9222:=
9212:j
9208:w
9180:)
9174:1
9171:+
9168:j
9165:2
9161:e
9157:i
9154:+
9149:j
9146:2
9142:e
9137:(
9130:2
9126:1
9121:=
9116:j
9112:w
9088:M
9083:x
9079:T
9054:}
9049:i
9045:e
9041:{
9021:)
9018:q
9015:,
9012:p
9009:(
9005:l
9002:C
8977:)
8974:q
8971:,
8968:p
8965:(
8945:M
8940:x
8936:T
8912:,
8909:M
8903:x
8883:M
8880:T
8857:)
8854:q
8851:,
8848:p
8845:(
8825:M
8678:.
8669:L
8623:L
8614:i
8611:=
8606:L
8581:,
8572:R
8535:R
8526:i
8523:=
8518:L
8482:.
8473:=
8450:1
8447:=
8410:.
8404:K
8401:i
8395:=
8392:i
8389:K
8369:1
8363:=
8358:2
8329:)
8323:2
8319:m
8302:+
8295:(
8288:=
8284:)
8278:2
8274:m
8257:+
8222:2
8217:t
8208:(
8201:=
8197:)
8193:K
8187:m
8181:+
8151:i
8147:(
8142:)
8138:K
8132:m
8126:+
8103:i
8099:(
8095:=
8089:L
8084:D
8077:R
8072:D
8045:L
8040:D
8033:R
8028:D
8004:R
7999:D
7992:L
7987:D
7960:0
7957:=
7951:R
7939:R
7934:D
7929:0
7926:=
7920:L
7908:L
7903:D
7873:R
7864:S
7861:=
7850:R
7835:R
7822:L
7811:1
7803:)
7794:S
7790:(
7785:=
7774:L
7759:L
7725:1
7718:S
7711:R
7706:D
7700:1
7692:)
7683:S
7679:(
7674:=
7663:R
7658:D
7648:R
7643:D
7632:S
7625:L
7620:D
7616:S
7613:=
7602:L
7597:D
7587:L
7582:D
7558:K
7535:K
7529:m
7523:+
7500:i
7497:=
7491:R
7486:D
7481:K
7475:m
7469:+
7439:i
7436:=
7430:L
7425:D
7393:.
7358:2
7350:i
7347:=
7311:0
7308:=
7305:)
7302:x
7299:(
7288:L
7276:m
7270:+
7267:)
7264:x
7261:(
7255:L
7219:i
7169:)
7166:x
7163:(
7152:R
7140:m
7134:+
7131:)
7128:x
7125:(
7119:R
7090:i
7063:i
7059:e
7055:=
7029:)
7026:x
7023:(
7012:R
7000:m
6995:1
6987:)
6978:S
6974:(
6969:=
6965:)
6956:x
6952:(
6937:R
6925:m
6919:)
6916:x
6913:(
6902:R
6890:m
6864:)
6861:x
6858:(
6847:R
6832:S
6828:=
6824:)
6815:x
6811:(
6796:R
6784:)
6781:x
6778:(
6767:R
6735:)
6732:x
6729:(
6723:R
6714:S
6711:=
6707:)
6698:x
6694:(
6682:R
6670:)
6667:x
6664:(
6658:R
6619:S
6593:1
6585:)
6576:S
6572:(
6567:=
6558:S
6529:]
6523:0
6518:1
6508:1
6503:0
6497:[
6492:=
6487:2
6479:i
6476:=
6444:=
6441:S
6432:T
6427:S
6403:.
6397:)
6393:C
6389:,
6386:2
6383:(
6379:p
6376:S
6348:)
6344:C
6340:,
6337:2
6334:(
6330:L
6327:S
6292:.
6287:1
6279:)
6270:S
6266:(
6261:=
6258:L
6235:)
6232:x
6229:(
6223:L
6214:L
6211:=
6207:)
6198:x
6194:(
6182:L
6170:)
6167:x
6164:(
6158:L
6129:.
6126:S
6123:=
6120:R
6100:,
6097:1
6094:=
6091:R
6086:1
6079:S
6051:)
6048:x
6045:(
6039:R
6030:R
6025:1
6018:S
5987:)
5982:1
5975:S
5971:(
5966:=
5956:)
5953:x
5950:(
5944:R
5935:R
5893:=
5883:)
5880:x
5877:(
5871:R
5862:R
5835:)
5828:T
5823:1
5812:(
5795:=
5788:)
5785:x
5782:(
5776:R
5767:R
5740:)
5735:1
5724:(
5666:)
5659:T
5654:1
5643:(
5636:=
5619:)
5614:1
5603:(
5576:.
5573:)
5570:3
5567:,
5564:1
5561:(
5557:O
5554:S
5523:)
5520:1
5514:,
5511:1
5505:,
5502:1
5496:,
5493:1
5490:+
5487:(
5477:=
5449:1
5438:=
5429:T
5392:1
5385:S
5364:)
5359:1
5352:S
5348:(
5343:=
5284:)
5281:x
5278:(
5272:R
5263:R
5236:)
5231:1
5220:(
5203:=
5193:)
5190:x
5187:(
5181:R
5172:R
5160:x
5133:)
5128:1
5117:(
5100:=
5090:)
5087:x
5084:(
5078:R
5069:R
5057:x
5032:x
5017:x
4997:=
4986:)
4977:x
4973:(
4961:R
4940:x
4917:=
4909:)
4900:x
4896:(
4884:R
4814:x
4796:)
4791:1
4780:(
4773:=
4764:x
4731:x
4708:=
4696:x
4672:)
4668:C
4664:,
4661:2
4658:(
4654:L
4651:S
4644:R
4621:)
4618:x
4615:(
4609:R
4600:R
4597:=
4593:)
4584:x
4580:(
4568:R
4556:)
4553:x
4550:(
4544:R
4512:.
4506:)
4503:x
4500:(
4494:L
4483:1
4475:)
4466:S
4462:(
4457:=
4453:)
4444:x
4440:(
4428:L
4416:)
4413:x
4410:(
4404:L
4372:)
4369:x
4366:(
4360:L
4342:x
4312:S
4309:=
4305:)
4296:x
4292:(
4280:L
4259:x
4226:)
4223:x
4220:(
4214:L
4196:x
4141:1
4134:S
4113:)
4108:1
4101:S
4097:(
4092:=
4040:)
4036:C
4032:,
4029:2
4026:(
4022:L
4019:S
3987:)
3983:C
3979:,
3976:2
3973:(
3970:L
3967:S
3961:S
3938:)
3935:x
3932:(
3926:R
3917:S
3914:=
3910:)
3901:x
3897:(
3885:R
3873:)
3870:x
3867:(
3861:R
3822:S
3798:)
3795:x
3792:(
3786:R
3768:x
3738:)
3733:1
3726:S
3722:(
3717:=
3713:)
3704:x
3700:(
3688:R
3667:x
3641:)
3638:x
3635:(
3629:R
3611:x
3568:.
3562:)
3559:3
3556:,
3553:1
3550:(
3546:O
3543:S
3516:x
3510:=
3501:x
3494:x
3460:.
3452:2
3449:1
3441:=
3407:,
3403:p
3398:|
3393:|
3388:p
3383:|
3376:=
3365:,
3361:p
3356:|
3351:J
3343:p
3318:p
3296:J
3234:0
3231:=
3221:L
3179:0
3176:=
3166:R
3108:c
3100:=
3096:|
3091:k
3086:|
3075:c
3062:=
3058:|
3053:k
3048:|
3041:=
3037:|
3032:p
3027:|
2998:k
2973:p
2953:,
2941:0
2938:=
2931:)
2921:p
2906:p
2895:2
2891:E
2886:(
2882:=
2870:p
2860:p
2856:=
2849:)
2839:p
2815:(
2810:)
2800:p
2785:(
2781:=
2774:)
2764:p
2749:(
2744:)
2734:p
2710:(
2694:.
2682:0
2679:=
2672:)
2662:p
2641:+
2638:E
2633:2
2629:I
2624:(
2620:=
2608:p
2562:0
2559:=
2552:)
2542:p
2518:E
2513:2
2509:I
2504:(
2500:=
2488:p
2449:)
2441:2
2427:1
2416:(
2411:=
2393:,
2375:/
2371:)
2368:t
2365:E
2358:r
2350:p
2346:(
2343:i
2336:e
2329:=
2324:)
2321:t
2311:r
2303:k
2299:(
2296:i
2289:e
2282:=
2277:)
2269:2
2255:1
2244:(
2239:=
2235:)
2231:t
2228:,
2224:r
2219:(
2181:.
2175:0
2172:=
2166:L
2111:0
2108:=
2102:R
2045:.
2037:)
2029:z
2014:y
1999:x
1984:2
1980:I
1973:(
1968:=
1923:0
1920:=
1837:,
1834:3
1831:,
1828:2
1825:,
1822:1
1819:=
1792:0
1789:=
1764:2
1760:I
1727:)
1719:z
1707:y
1695:x
1683:2
1679:I
1672:(
1667:=
1662:)
1654:3
1642:2
1630:1
1618:0
1607:(
1602:=
1566:0
1563:=
1557:z
1538:z
1530:+
1524:y
1505:y
1497:+
1491:x
1472:x
1464:+
1458:t
1439:c
1436:1
1429:2
1425:I
1397:c
1374:0
1371:=
1119:e
1112:t
1105:v
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
