6440:
5577:
6435:{\displaystyle {\begin{aligned}Z&=\exp \left(-i\ g\int \mathrm {d} ^{4}x\ {\frac {\delta }{i\ \delta \ {\bar {\varepsilon }}^{a}(x)}}\ f^{abc}\partial _{\mu }\ {\frac {i\ \delta }{\delta \ j_{\mu }^{b}(x)}}\ {\frac {i\ \delta }{\delta \ \varepsilon ^{c}(x)}}\right)\\&\qquad \times \exp \left(-i\ g\int \mathrm {d} ^{4}x\ f^{abc}\partial _{\mu }{\frac {i\ \delta }{\delta \ j_{\nu }^{a}(x)}}{\frac {i\ \delta }{\delta \ j_{\mu }^{b}(x)}}\ {\frac {i\ \delta }{\delta \ j^{c\nu }(x)}}\right)\\&\qquad \qquad \times \exp \left(-i\ {\frac {g^{2}}{4}}\int \mathrm {d} ^{4}x\ f^{abc}\ f^{ars}{\frac {i\ \delta }{\delta \ j_{\mu }^{b}(x)}}\ {\frac {i\ \delta }{\delta \ j_{\nu }^{c}(x)}}\ {\frac {\ i\delta }{\delta \ j^{r\mu }(x)}}{\frac {i\ \delta }{\delta \ j^{s\nu }(x)}}\right)\\&\qquad \qquad \qquad \times Z_{0}\end{aligned}}}
1663:
1672:
60:
1384:... I was at Brookhaven also ... and was assigned to the same office as Yang. Yang, who has demonstrated on a number of occasions his generosity to physicists beginning their careers, told me about his idea of generalizing gauge invariance and we discussed it at some length ... I was able to contribute something to the discussions, especially with regard to the quantization procedures, and to a small degree in working out the formalism; however, the key ideas were Yang's."
7023:). In this case, large computational resources are needed to be sure the correct limit of infinite volume (smaller lattice spacing) is obtained. This is the limit the results must be compared with. Smaller spacing and larger coupling are not independent of each other, and larger computational resources are needed for each. As of today, the situation appears somewhat satisfactory for the hadronic spectrum and the computation of the gluon and ghost propagators, but the
8458:
1683:
7015:. In the opposite limit, the infrared limit, the situation is the opposite, as the coupling is too large for perturbation theory to be reliable. Most of the difficulties that research meets is just managing the theory at low energies. That is the interesting case, being inherent to the description of hadronic matter and, more generally, to all the observed bound states of gluons and quarks and their confinement (see
6803:
1692:
5104:
6451:
5565:
5544:
4568:
1402:
to provide an explanation for isospin conservation in collisions involving the strong interactions. Yang's presentation of the work at
Princeton in February 1954 was challenged by Pauli, asking about the mass in the field developed with the gauge invariance idea. Pauli knew that this might be an
3417:
1626:
or the quantization of it. Because Pauli found that his theory "leads to some rather unphysical shadow particles", he refrained from publishing his results formally. Although Pauli did not publish his six-dimensional theory, he gave two seminar lectures about it in Zürich in
November 1953.
1962:
7781:
6798:{\displaystyle Z_{0}=\exp \left(-\int \mathrm {d} ^{4}x\ \mathrm {d} ^{4}y\ {\bar {\varepsilon }}^{a}(x)\ C^{ab}(x-y)\ \varepsilon ^{b}(y)\right)\exp \left({\tfrac {1}{2}}\int \mathrm {d} ^{4}x\ \mathrm {d} ^{4}y\ j_{\mu }^{a}(x)\ D^{ab\mu \nu }(x-y)\ j_{\nu }^{b}(y)\right)\ }
2702:
2191:
5224:
7074:
consists, especially, in a proof of the conjecture that the lowest excitations of a pure Yang–Mills theory (i.e. without matter fields) have a finite mass-gap with regard to the vacuum state. Another open problem, connected with this conjecture, is a proof of the
1642:
chose not to publish their work. Shortly after Yang and Mills published their paper in
October 1954, Salam encouraged Shaw to publish his work to mark his contribution. Shaw declined, and instead it only forms a chapter of his PhD thesis published in 1956.
3832:
1403:
issue as he had worked on applying gauge invariance but chose not to publish it, viewing the massless excitations of the theory to be "unphysical 'shadow particles'". Yang and Mills published in
October 1954; near the end of the paper, they admit:
5338:
4080:
6981:
are zero and so there is no coupling. In the non-abelian case, the ghost field appears as a useful way to rewrite the quantum field theory without physical consequences on the observables of the theory such as cross sections or decay rates.
7311:
5099:{\displaystyle {\begin{aligned}Z&=\int \ \exp {\Bigl \{}i\ S_{F}\ \left+i\ S_{gf}\left+i\ S_{g}\left{\Bigr \}}\\&\exp \left\{i\int \mathrm {d} ^{4}x\ j_{\mu }^{a}(x)A^{a\mu }(x)+i\int \mathrm {d} ^{4}x\ \left\right\}\end{aligned}}}
3095:
5327:
2824:
The field has the property of being self-interacting and the equations of motion that one obtains are said to be semilinear, as nonlinearities are both with and without derivatives. This means that one can manage this theory only by
6940:
3258:
2401:
1367:
Yang's core idea was to look for a conserved quantity in nuclear physics comparable to electric charge and use it to develop a corresponding gauge theory comparable to electrodynamics. He settled on conservation of
1833:
3629:
2549:
3243:
2819:
3733:
3168:
2968:
3899:
2260:
2022:
5582:
4573:
7000:. The relevance of this result is due to the fact that a Yang–Mills theory that describes strong interaction and asymptotic freedom permits proper treatment of experimental results coming from
1372:, a quantum number that distinguishes a neutron from a proton, but he made no progress on a theory. Taking a break from Princeton in the summer of 1953, Yang met a collaborator who could help:
5115:
1582:
has not been theoretically proven, though it is a consistent experimental observation. This shows why QCD confinement at low energy is a mathematical problem of great relevance, and why the
7082:
In physics the survey of Yang–Mills theories does not usually start from perturbation analysis or analytical methods, but more recently from systematic application of numerical methods to
7051:. Renormalizability is obtained even if the gauge bosons described by this theory are massive, as in the electroweak theory, provided the mass is only an "acquired" one, generated by the
2539:
4529:
3742:
5539:{\displaystyle \ S_{g}=-\int \mathrm {d} ^{4}x\ \left({\bar {c}}^{a}\ \partial _{\mu }\partial ^{\mu }c^{a}+g\ {\bar {c}}^{a}\ f^{abc}\ \partial _{\mu }\ A^{b\mu }\ c^{c}\right)\ }
7007:
To obtain the behavior of the Yang–Mills theory at high energies, and so to prove asymptotic freedom, one applies perturbation theory assuming a small coupling. This is verified
2892:
6839:
For quantum electrodynamics the ghost field decouples because the gauge group is abelian. This can be seen from the coupling between the gauge field and the ghost field that is
3948:
3940:
2449:
2296:
4251:
4170:
1578:
at lower energies in quantum chromodynamics is not completely understood due to the difficulties of managing such a theory with a strong coupling. This may be the reason why
7058:
The mathematics of the Yang–Mills theory is a very active field of research, yielding e.g. invariants of differentiable structures on four-dimensional manifolds via work of
1320:
6979:
4538:. This problem was already known for quantum electrodynamics but here becomes more severe due to non-abelian properties of the gauge group. A way out has been given by
1998:
7888:
1245:
proved that every conserved physical quantity has a matching symmetry, and culminated in 1928 when he published his book applying the geometrical theory of symmetry (
2978:
1353:
1290:
1146:
8311:
8121:
5235:
1425:
8203:
8162:
7141:
4267:
the coupling is dimensionless and both the field and the square of the coupling have the same dimensions of the field and the coupling of a massless quartic
7821:
7106:
229:
39:
Yang–Mills theory and the mass gap. Quantum particles described by the theory have mass but the classical waves of the field travel at the speed of light.
3412:{\displaystyle \ \left(D_{\mu }\ F_{\nu \kappa }\right)^{a}+\left(D_{\kappa }\ F_{\mu \nu }\right)^{a}+\left(D_{\nu }\ F_{\kappa \mu }\right)^{a}=0\ }
269:
7941:
6842:
2308:
7035:
resonance is not seen in any of such lattice computations and contrasting interpretations have been put forward. This is a hotly debated issue.
7966:
8070:
7184:
7169:
7071:
1583:
8301:
7696:
7294:
8193:
7881:
1957:{\displaystyle \ {\mathcal {L}}_{\mathrm {gf} }=-{\tfrac {1}{2}}\operatorname {tr} (F^{2})=-{\tfrac {1}{4}}F^{a\mu \nu }F_{\mu \nu }^{a}\ }
42:
1263:
in 1922, three years before working on his equation, connected Weyl's group concept to electron charge. Schrödinger showed that the group
1004:
8049:
7189:
7931:
7121:
7473:
1139:
2697:{\displaystyle \ F_{\mu \nu }^{a}=\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+g\ f^{abc}\ A_{\mu }^{b}\ A_{\nu }^{c}\ }
8461:
8141:
7745:
7719:
7324:
774:
3431:
8391:
8198:
7961:
3176:
1634:
also developed a non-Abelian gauge theory for nuclear forces. However, the theory needed massless particles in order to maintain
1575:
7994:
2832:
Note that the transition between "upper" ("contravariant") and "lower" ("covariant") vector or tensor components is trivial for
8351:
8254:
8234:
7874:
72:
2716:
8157:
7916:
7911:
7159:
3637:
3103:
2897:
2186:{\displaystyle \ \operatorname {tr} \left(T^{a}\ T^{b}\right)={\tfrac {1}{2}}\delta ^{ab}\ ,\qquad \left=i\ f^{abc}\ T^{c}~.}
1794:
1381:
1355:
group transformations played a central role. Many physicists thought there must be an analog for the dynamics of nucleons.
337:
3840:
2207:
7813:
5219:{\displaystyle S_{F}=-{\tfrac {1}{2}}\int \mathrm {d} ^{4}x\ \operatorname {tr} \left(F^{\mu \nu }\ F_{\mu \nu }\right)\ }
1603:
1473:
1132:
186:
146:
8306:
8430:
7956:
874:
804:
176:
7552:
Yndurain, F.J.; Garcia-Martin, R.; Pelaez, J.R. (2007). "Experimental status of the ππ isoscalar S wave at low energy:
4559:
332:
8425:
8219:
7808:
1561:
274:
7999:
4555:
1568:. In the current epoch the strong interaction is not unified with the electroweak interaction, but from the observed
1450:. This prompted a significant restart of Yang–Mills theory studies that proved successful in the formulation of both
7844:
7848:
7116:
7111:
7063:
1662:
1237:
All known fundamental interactions can be described in terms of gauge theories, but working this out took decades.
984:
181:
8167:
1671:
322:
7829:
7067:
7001:
4284:
1825:
1619:
1587:
7179:
7096:
1811:
1652:
8403:
7344:
7047:, in 1972, worked out their renormalization, relying on a formulation of the problem worked out by his advisor
7031:
spectra are yet a questioned matter in view of the experimental observation of such exotic states. Indeed, the
5561:
fixes the gauge's choice for the quantization. Feynman's rules obtained from this functional are the following
3827:{\displaystyle \ {\tilde {F}}^{\mu \nu }={\tfrac {1}{2}}\varepsilon ^{\mu \nu \rho \sigma }F_{\rho \sigma }\ ,}
2478:
1373:
1167:
809:
439:
284:
249:
8275:
8100:
8075:
7126:
5571:
These rules for
Feynman's diagrams can be obtained when the generating functional given above is rewritten as
4551:
4297:
1611:
799:
8480:
8131:
8014:
7711:
7491:
7450:
6829:
4534:
but this integral has no meaning as it is because the potential vector can be arbitrarily chosen due to the
1631:
1618:, and others to a higher-dimensional internal space. However, there is no evidence that Pauli developed the
1451:
1393:
1323:
1084:
554:
362:
357:
219:
8442:
8366:
8361:
8296:
8224:
8126:
8029:
8019:
8004:
7921:
7012:
4075:{\displaystyle \ \partial ^{\mu }F_{\mu \nu }^{a}+g\ f^{abc}\ A^{b\mu }\ F_{\mu \nu }^{c}=-J_{\nu }^{a}~.}
1824:
Yang–Mills theories are special examples of gauge theories with a non-abelian symmetry group given by the
1455:
1209:
954:
934:
434:
367:
279:
244:
8177:
8080:
7989:
2839:
8415:
8410:
8249:
8244:
8034:
6821:
6813:
2204:
of the Lie algebra (totally antisymmetric if the generators of the Lie algebra are normalised such that
1447:
1197:
1175:
844:
704:
400:
264:
1544:
field. The dynamics of the photon field and its interactions with matter are, in turn, governed by the
7803:
3907:
2416:
2265:
8356:
8090:
7926:
7897:
7634:
7580:
7510:
7362:
7154:
7131:
7083:
7020:
4175:
4095:
2299:
1260:
1250:
1089:
869:
405:
259:
171:
51:
8420:
8386:
8321:
8239:
8229:
8024:
7489:
Caprini, I.; Colangelo, G.; Leutwyler, H. (2006). "Mass and width of the lowest resonance in QCD".
7164:
6997:
4268:
3736:
2826:
1099:
814:
544:
429:
7019:). The most used method to study the theory in this limit is to try to solve it on computers (see
1295:
719:
594:
8054:
7763:
7596:
7570:
7534:
7500:
7396:
6986:
6945:
2201:
1607:
1553:
1170:
in 1953, as well as a generic term for the class of similar theories. The Yang–Mills theory is a
1109:
1034:
964:
849:
834:
764:
729:
484:
444:
234:
196:
151:
139:
116:
111:
7729:
7614:
7210:
7044:
679:
6836:. The theory is renormalizable and corrections are finite at any order of perturbation theory.
1970:
8485:
8105:
7741:
7715:
7692:
7561:
7526:
7469:
7320:
7290:
7258:
7076:
6990:
3090:{\displaystyle \ \partial ^{\mu }F_{\mu \nu }^{a}+g\ f^{abc}\ A^{\mu b}\ F_{\mu \nu }^{c}=0~.}
2460:
1682:
1579:
1439:
1435:
1190:
1186:
1064:
994:
879:
784:
724:
659:
629:
599:
459:
454:
299:
224:
214:
106:
1789:(bottom left). A visual representation of the field strength of a BPST instanton with center
1322:
in electromagnetic fields that matched the conservation of electric charge. As the theory of
8376:
7737:
7650:
7642:
7618:
7588:
7518:
7370:
7250:
7048:
4272:
3249:
2894:), whereas for μ and ν it is nontrivial, corresponding e.g. to the usual Lorentz signature,
1761:
coefficient (top right). These coefficients determine the restriction of the BPST instanton
1635:
1569:
1557:
1104:
1019:
999:
969:
919:
744:
739:
734:
619:
604:
534:
239:
166:
156:
91:
7447:
The problem of particle types and other contributions to the theory of elementary particles
7423:
8398:
8136:
8085:
7353:
7059:
7052:
5550:
5322:{\displaystyle S_{gf}=-{\frac {1}{2\xi }}\int \mathrm {d} ^{4}x\ (\partial \cdot A)^{2}\ }
4254:
3423:
2410:
1329:
1266:
1049:
1044:
1014:
1009:
979:
939:
909:
889:
794:
689:
684:
649:
634:
609:
589:
569:
499:
489:
410:
372:
289:
254:
86:
64:
59:
7395:
Straumann, N. (2000). "On Pauli's invention of non-abelian Kaluza-Klein Theory in 1953".
1189:. A Yang–Mills theory seeks to describe the behavior of elementary particles using these
7638:
7584:
7514:
7366:
1434:
The idea was set aside until 1960, when the concept of particles acquiring mass through
8341:
8316:
8095:
7419:
7340:
4539:
1715:
1599:
1549:
1477:
1443:
1410:
1356:
1221:
1163:
1074:
1024:
944:
929:
914:
894:
864:
859:
854:
839:
829:
824:
789:
779:
709:
644:
614:
549:
514:
504:
479:
350:
317:
191:
8474:
8437:
8371:
8346:
7946:
7680:
7646:
7600:
7101:
4535:
1615:
1389:
1079:
1069:
1059:
1054:
974:
959:
759:
754:
584:
579:
539:
509:
474:
464:
395:
390:
7622:
7538:
1638:. Since no such massless particles were known at the time, Shaw and his supervisor
1572:
constants it is believed they all converge to a single value at very high energies.
8381:
8291:
8009:
7951:
7936:
7238:
7136:
7028:
7008:
5549:
for the ghost. This is the expression commonly used to derive
Feynman's rules (see
4543:
1744:
1246:
1242:
1238:
1213:
1171:
1114:
1039:
924:
899:
884:
819:
699:
694:
574:
564:
559:
524:
519:
494:
449:
327:
294:
121:
101:
7522:
8039:
7777:
2452:
2001:
1639:
1623:
1254:
1094:
1029:
989:
949:
904:
769:
674:
669:
654:
639:
624:
7592:
7043:
Yang–Mills theories met with general acceptance in the physics community after
4085:
Note that the currents must properly change under gauge group transformations.
7174:
7149:
6833:
2708:
1201:
749:
664:
529:
469:
161:
96:
7262:
6935:{\displaystyle \ {\bar {c}}^{a}\ f^{abc}\ \partial _{\mu }A^{b\mu }\ c^{c}~.}
4088:
We give here some comments about the physical dimensions of the coupling. In
8172:
7688:
7254:
4283:
A method of quantizing the Yang–Mills theory is by functional methods, i.e.
2013:
1807:
1691:
1388:
In the summer 1953, Yang and Mills extended the concept of gauge theory for
1193:
7530:
7375:
7348:
4554:) that has the property of being unphysical since, although it agrees with
2396:{\displaystyle \ D_{\mu }=I\ \partial _{\mu }-i\ g\ T^{a}\ A_{\mu }^{a}\ ,}
2972:
From the given
Lagrangian one can derive the equations of motion given by
17:
8270:
7866:
7024:
1431:
This problem of unphysical massless excitation blocked further progress.
714:
7852:
7768:
5564:
7575:
7505:
7016:
6996:
is small (so small nonlinearities), as for high energies, and applying
1369:
7401:
1241:'s pioneering work on this project started in 1915 when his colleague
7655:
1541:
7062:. Furthermore, the field of Yang–Mills theories was included in the
6985:
One of the most important results obtained for Yang–Mills theory is
7289:(Impression 3 ed.). Oxford, UK: Oxford University Press.
1458:(QCD). The electroweak interaction is described by the gauge group
31:
1464:
1398:
6808:
being the generating functional of the free theory. Expanding in
1782:
to this slice. The corresponding field strength centered around
1468:
Yang–Mills theory. The massless gauge bosons of the electroweak
1253:
the "gauge symmetry", by analogy to distance standardization in
7870:
7239:"Gauge theory: Historical origins and some modern developments"
7349:"Conservation of isotopic spin and isotopic gauge invariance"
7466:
Cosmic Anger: Abdus Salam – the first Muslim Nobel scientist
7237:
O’Raifeartaigh, Lochlainn; Straumann, Norbert (2000-01-01).
5563:
3624:{\displaystyle \ \left\right]+\left\right]+\left\right]=0\ }
1843:
1630:
In
January 1954 Ronald Shaw, a graduate student at the
1249:) to quantum mechanics. Weyl named the relevant symmetry in
7762:
Svetlichny, George (1999). "Preparation for Gauge Theory".
3238:{\displaystyle \ \left(D^{\mu }F_{\mu \nu }\right)^{a}=0~.}
1380:"During the academic year 1953–1954, Yang was a visitor to
1162:
is a quantum field theory for nuclear binding devised by
2814:{\displaystyle \ \left=-i\ g\ T^{a}\ F_{\mu \nu }^{a}~.}
6828:-point functions the corresponding process amplitudes,
3728:{\displaystyle \ \left=D_{\mu }\ F_{\nu \kappa }^{a}~.}
3163:{\displaystyle \ F_{\mu \nu }=T^{a}F_{\mu \nu }^{a}\ ,}
2963:{\displaystyle \ \eta _{\mu \nu }={\rm {diag}}(+---)~.}
1556:
with the unified electroweak interaction (unifying the
1427:
quantum, to which we do not have a satisfactory answer.
1220:). Thus it forms the basis of the understanding of the
6649:
5136:
4354:
4130:
3775:
2072:
1906:
1866:
7319:. Springer Science & Business Media. p. 63.
6948:
6845:
6454:
5580:
5341:
5238:
5118:
4571:
4300:
4178:
4098:
3951:
3910:
3894:{\displaystyle \ D_{\mu }{\tilde {F}}^{\mu \nu }=0~.}
3843:
3745:
3640:
3434:
3261:
3179:
3106:
2981:
2900:
2842:
2719:
2552:
2481:
2419:
2311:
2268:
2255:{\displaystyle \ \operatorname {tr} (T^{a}\ T^{b})\ }
2210:
2025:
1973:
1836:
1413:
1332:
1298:
1269:
7623:"Regularization and renormalization of gauge fields"
7468:. Oxford, UK: Oxford University Press. p. 117.
8334:
8284:
8263:
8212:
8186:
8150:
8114:
8063:
7982:
7975:
7904:
7160:
Field theoretical formulation of the standard model
6989:. This result can be obtained by assuming that the
4558:, it is a complex scalar field, which violates the
4257:for dimensions greater than four. Furthermore, for
1438:in massless theories was put forward, initially by
6973:
6942:For the abelian case, all the structure constants
6934:
6797:
6434:
5538:
5321:
5218:
5098:
4523:
4245:
4164:
4074:
3934:
3893:
3826:
3727:
3623:
3411:
3237:
3162:
3089:
2962:
2886:
2813:
2696:
2533:
2443:
2395:
2290:
2254:
2185:
1992:
1956:
1806:(bottom right). The BPST instanton is a classical
1419:
1363:Yang and Mills find the nuclear force gauge theory
1359:in particular was obsessed with this possibility.
1347:
1314:
1284:
7079:property in the presence of additional fermions.
6820:-point functions with perturbation theory. Using
4856:
4690:
7424:"Ronald Shaw 1929–2016 by Michael Atiyah (1954)"
7313:Mathematical Conversations: Selections from the
4562:. So, we can write the generating functional as
1407:We next come to the question of the mass of the
7287:The Quantum Story: A history in 40 moments
7232:
7230:
1405:
3834:then the Bianchi identity can be rewritten as
7882:
4287:. One introduces a generating functional for
1548:gauge theory of quantum electrodynamics. The
1396:, to non-abelian groups, selecting the group
1196:and is at the core of the unification of the
1140:
8:
2463:. In four dimensions, the coupling constant
7708:Gauge Theory of Elementary Particle Physics
7390:
7388:
7386:
7280:
7278:
7276:
7274:
7272:
4253:This implies that Yang–Mills theory is not
1610:, extending the five-dimensional theory of
1594:Parallel work on non-Abelian gauge theories
7979:
7889:
7875:
7867:
7310:Gray, Jeremy; Wilson, Robin (2012-12-06).
1147:
1133:
58:
47:
7782:"Gauge theory – Past, Present and Future"
7767:
7654:
7574:
7504:
7400:
7374:
6956:
6947:
6920:
6904:
6894:
6875:
6862:
6851:
6850:
6844:
6772:
6767:
6730:
6708:
6703:
6687:
6682:
6669:
6664:
6648:
6617:
6586:
6564:
6553:
6552:
6539:
6534:
6521:
6516:
6475:
6474:
6459:
6453:
6408:
6407:
6392:
6349:
6325:
6304:
6280:
6259:
6254:
6230:
6209:
6204:
6180:
6168:
6149:
6133:
6128:
6113:
6107:
6049:
6025:
6004:
5999:
5975:
5957:
5952:
5928:
5922:
5906:
5890:
5885:
5824:
5800:
5779:
5774:
5750:
5741:
5725:
5700:
5689:
5688:
5669:
5657:
5652:
5598:
5597:
5581:
5579:
5522:
5506:
5493:
5474:
5461:
5450:
5449:
5433:
5423:
5413:
5400:
5389:
5388:
5370:
5365:
5349:
5340:
5310:
5282:
5277:
5258:
5243:
5237:
5199:
5183:
5156:
5151:
5135:
5123:
5117:
5067:
5045:
5034:
5033:
5014:
4992:
4981:
4980:
4962:
4957:
4929:
4910:
4905:
4889:
4884:
4855:
4854:
4832:
4831:
4811:
4810:
4787:
4749:
4705:
4689:
4688:
4665:
4648:
4647:
4639:
4622:
4589:
4588:
4572:
4570:
4492:
4470:
4465:
4449:
4444:
4417:
4401:
4374:
4369:
4353:
4325:
4299:
4211:
4190:
4177:
4129:
4124:
4097:
4060:
4055:
4039:
4031:
4015:
3996:
3977:
3969:
3959:
3950:
3923:
3918:
3909:
3870:
3859:
3858:
3851:
3842:
3809:
3790:
3774:
3762:
3751:
3750:
3744:
3713:
3705:
3692:
3674:
3666:
3653:
3639:
3596:
3583:
3565:
3537:
3524:
3506:
3478:
3465:
3447:
3433:
3394:
3380:
3367:
3348:
3334:
3321:
3302:
3288:
3275:
3260:
3217:
3203:
3193:
3178:
3148:
3140:
3130:
3114:
3105:
3069:
3061:
3045:
3026:
3007:
2999:
2989:
2980:
2921:
2920:
2908:
2899:
2869:
2850:
2841:
2799:
2791:
2778:
2745:
2732:
2718:
2685:
2680:
2667:
2662:
2643:
2624:
2619:
2609:
2596:
2591:
2581:
2568:
2560:
2551:
2534:{\displaystyle \ a,b,c=1\ldots n^{2}-1~.}
2513:
2480:
2432:
2427:
2418:
2381:
2376:
2363:
2338:
2319:
2310:
2276:
2267:
2240:
2227:
2209:
2171:
2152:
2128:
2112:
2087:
2071:
2057:
2044:
2024:
1981:
1972:
1945:
1937:
1921:
1905:
1890:
1865:
1849:
1848:
1842:
1841:
1835:
1412:
1331:
1303:
1297:
1268:
4524:{\displaystyle \ Z=\int \ \exp \left\ ,}
7202:
7107:Deformed Hermitian Yang–Mills equations
3942:enters into the equations of motion as
2413:(matching the size of the generators),
1602:formulated a six-dimensional theory of
1326:developed in the 1930's and 1940's the
50:
7967:Two-dimensional conformal field theory
7414:
7412:
1598:In 1953, in a private correspondence,
7:
7145:= 4 supersymmetric Yang–Mills theory
7559:(600) pole and scattering length".
2887:{\displaystyle \ f^{abc}=f_{abc}\ }
2016:or field-strength form) satisfying
1228:History and qualitative description
43:(more unsolved problems in physics)
7734:50 Years of Yang–Mills theory
6891:
6683:
6665:
6535:
6517:
6129:
5919:
5886:
5738:
5653:
5490:
5420:
5410:
5366:
5297:
5278:
5152:
4958:
4885:
4807:
4798:
4763:
4719:
4666:
4640:
4623:
4445:
4370:
4326:
4172:and so the coupling must scale as
3956:
2986:
2931:
2928:
2925:
2922:
2606:
2578:
2335:
1853:
1850:
27:In physics, a quantum field theory
25:
8462:Template:Quantum mechanics topics
7185:Yang–Mills existence and mass gap
7170:Two-dimensional Yang–Mills theory
1584:Yang–Mills existence and mass gap
8457:
8456:
7706:Cheng, T.-P.; Li, L.-F. (1983).
6816:, we are able to obtain all the
4092:dimensions, the field scales as
3935:{\displaystyle \ J_{\mu }^{a}\ }
2444:{\displaystyle \ A_{\mu }^{a}\ }
2291:{\displaystyle \ \delta ^{ab}\ }
1690:
1681:
1670:
1661:
1540:) as well as the still-massless
7845:"The Millennium Prize Problems"
7445:Shaw, Ronald (September 1956).
7211:"Yang-Mills & The Mass Gap"
6384:
6383:
6382:
6083:
6082:
5854:
4271:. So, these theories share the
4246:{\displaystyle \ \left=\left~.}
4165:{\displaystyle \ \left=\left\ }
2102:
1233:Gauge theory in electrodynamics
7122:Hermitian Yang–Mills equations
6856:
6784:
6778:
6757:
6745:
6720:
6714:
6629:
6623:
6607:
6595:
6576:
6570:
6558:
6492:
6480:
6465:
6425:
6413:
6398:
6364:
6358:
6319:
6313:
6271:
6265:
6221:
6215:
6064:
6058:
6016:
6010:
5969:
5963:
5836:
5830:
5791:
5785:
5712:
5706:
5694:
5615:
5603:
5588:
5455:
5394:
5307:
5294:
5079:
5073:
5057:
5051:
5039:
5026:
5020:
5004:
4998:
4986:
4944:
4938:
4922:
4916:
4837:
4816:
4676:
4662:
4659:
4653:
4636:
4633:
4619:
4606:
4594:
4579:
4507:
4501:
4482:
4476:
4333:
4322:
4313:
4307:
3864:
3756:
2951:
2936:
2246:
2220:
1896:
1883:
1382:Brookhaven National Laboratory
1342:
1336:
1279:
1273:
1:
7523:10.1103/PhysRevLett.96.132001
7453:. ch. 3, pp. 34–46.
7428:Trinity College Annual Record
1564:) through the symmetry group
1474:spontaneous symmetry breaking
1376:. As Mills himself describes:
187:Spontaneous symmetry breaking
147:Symmetry in quantum mechanics
7647:10.1016/0550-3213(72)90279-9
1315:{\displaystyle e^{i\theta }}
8426:Quantum information science
7809:Encyclopedia of Mathematics
6974:{\displaystyle \ f^{abc}\ }
4546:with the introduction of a
3422:which is equivalent to the
2829:with small nonlinearities.
2467:is a pure number and for a
1562:electromagnetic interaction
34:Unsolved problem in physics
8502:
7849:Clay Mathematics Institute
7593:10.1103/PhysRevD.76.074034
7315:Mathematical Intelligencer
7215:Clay Mathematics Institute
7190:Yang–Mills–Higgs equations
7117:Gauge theory (mathematics)
7112:Gauge covariant derivative
7064:Clay Mathematics Institute
5557:for the ghost field while
3170:these can be rewritten as
1650:
1604:Einstein's field equations
182:Explicit symmetry breaking
8451:
7329:– via Google Books.
7243:Reviews of Modern Physics
7068:Millennium Prize Problems
7002:deep inelastic scattering
5332:for the gauge fixing and
1993:{\displaystyle \ T^{a}\ }
1480:of the weak interaction (
338:Bargmann–Wigner equations
4275:at the classical level.
1588:Millennium Prize Problem
1478:the three massive bosons
1185:, or more generally any
8122:2D free massless scalar
8015:Quantum electrodynamics
7942:QFT in curved spacetime
7712:Oxford University Press
7492:Physical Review Letters
7464:Fraser, Gordon (2008).
7451:University of Cambridge
7255:10.1103/RevModPhys.72.1
4560:spin–statistics theorem
2008:, corresponding to the
1632:University of Cambridge
1570:running of the coupling
1452:electroweak unification
1394:quantum electrodynamics
1324:quantum electrodynamics
1292:produced a phase shift
363:Electroweak interaction
358:Quantum electrodynamics
333:Wheeler–DeWitt equation
220:Background field method
8443:Quantum thermodynamics
8367:On shell and off shell
8362:Loop quantum cosmology
8204:N = 4 super Yang–Mills
8163:N = 1 super Yang–Mills
8030:Scalar electrodynamics
8020:Quantum chromodynamics
7922:Conformal field theory
7898:Quantum field theories
7376:10.1103/PhysRev.96.191
7285:Baggott, J.E. (2013).
7084:lattice gauge theories
6975:
6936:
6814:functional derivatives
6799:
6436:
5568:
5540:
5323:
5220:
5100:
4556:Fermi–Dirac statistics
4525:
4247:
4166:
4076:
3936:
3895:
3828:
3729:
3625:
3413:
3239:
3164:
3091:
2964:
2888:
2815:
2707:can be derived by the
2698:
2535:
2445:
2397:
2292:
2256:
2187:
1994:
1958:
1456:quantum chromodynamics
1429:
1421:
1386:
1349:
1316:
1286:
1210:quantum chromodynamics
368:Quantum chromodynamics
245:Effective field theory
8416:Quantum hydrodynamics
8411:Quantum hadrodynamics
8035:Scalar chromodynamics
6976:
6937:
6822:LSZ reduction formula
6800:
6437:
5567:
5541:
5324:
5221:
5101:
4526:
4248:
4167:
4077:
3937:
3896:
3829:
3730:
3626:
3414:
3240:
3165:
3092:
2965:
2889:
2816:
2699:
2536:
2446:
2398:
2293:
2257:
2188:
1995:
1959:
1647:Mathematical overview
1448:Giovanni Jona-Lasinio
1422:
1378:
1350:
1317:
1287:
1224:of particle physics.
1198:electromagnetic force
1176:special unitary group
323:Klein–Gordon equation
265:LSZ reduction formula
8387:Quantum fluctuations
8357:Loop quantum gravity
7927:Lattice field theory
7685:Gauge Field Theories
7180:Yang–Mills equations
7155:Quantum gauge theory
7132:Lattice gauge theory
7097:Aharonov–Bohm effect
7021:lattice gauge theory
6946:
6843:
6452:
5578:
5339:
5236:
5116:
4569:
4298:
4291:-point functions as
4176:
4096:
3949:
3908:
3841:
3743:
3638:
3432:
3259:
3177:
3104:
2979:
2898:
2840:
2717:
2550:
2479:
2417:
2309:
2300:covariant derivative
2266:
2208:
2023:
1971:
1967:with the generators
1834:
1812:Yang–Mills equations
1653:Yang–Mills equations
1566:SU(3) × SU(2) × U(1)
1411:
1348:{\displaystyle U(1)}
1330:
1296:
1285:{\displaystyle U(1)}
1267:
1212:, the theory of the
406:Theory of everything
260:Lattice field theory
230:Correlation function
52:Quantum field theory
8421:Quantum information
8025:Quartic interaction
7822:"Yang–Mills theory"
7639:1972NuPhB..44..189T
7585:2007PhRvD..76g4034G
7515:2006PhRvL..96m2001C
7367:1954PhRv...96..191Y
7165:Symmetry in physics
7127:Kaluza–Klein theory
6998:perturbation theory
6777:
6713:
6264:
6214:
6009:
5962:
5784:
4915:
4552:Faddeev–Popov ghost
4475:
4269:scalar field theory
4065:
4044:
3982:
3928:
3718:
3679:
3153:
3074:
3012:
2827:perturbation theory
2804:
2690:
2672:
2629:
2601:
2573:
2437:
2386:
2262:is proportional to
2202:structure constants
1950:
385:Incomplete theories
8307:Nambu–Jona-Lasinio
8235:Higher dimensional
8142:Wess–Zumino–Witten
7932:Noncommutative QFT
7804:"Yang-Mills field"
6987:asymptotic freedom
6971:
6932:
6812:and computing the
6795:
6763:
6699:
6658:
6432:
6430:
6250:
6200:
5995:
5948:
5770:
5569:
5536:
5319:
5216:
5145:
5096:
5094:
4901:
4521:
4461:
4363:
4243:
4162:
4147:
4072:
4051:
4027:
3965:
3932:
3914:
3891:
3824:
3784:
3725:
3701:
3662:
3621:
3409:
3235:
3160:
3136:
3087:
3057:
2995:
2960:
2884:
2811:
2787:
2694:
2676:
2658:
2615:
2587:
2556:
2531:
2441:
2423:
2393:
2372:
2288:
2252:
2183:
2081:
1990:
1954:
1933:
1915:
1875:
1608:general relativity
1554:strong interaction
1462:, while QCD is an
1417:
1345:
1312:
1282:
270:Partition function
197:Topological charge
117:General relativity
112:Special relativity
8468:
8467:
8330:
8329:
7730:'t Hooft, G.
7698:978-3-527-40835-1
7627:Nuclear Physics B
7615:'t Hooft, G.
7562:Physical Review D
7296:978-0-19-956684-6
7072:the prize-problem
7013:ultraviolet limit
6991:coupling constant
6970:
6951:
6928:
6915:
6889:
6870:
6859:
6848:
6794:
6762:
6725:
6698:
6680:
6657:
6612:
6581:
6561:
6550:
6532:
6483:
6416:
6368:
6344:
6333:
6323:
6299:
6285:
6279:
6275:
6249:
6238:
6229:
6225:
6199:
6188:
6163:
6144:
6122:
6106:
6068:
6044:
6033:
6024:
6020:
5994:
5983:
5973:
5947:
5936:
5901:
5877:
5840:
5819:
5808:
5799:
5795:
5769:
5758:
5749:
5720:
5716:
5697:
5686:
5680:
5668:
5644:
5606:
5535:
5517:
5501:
5488:
5469:
5458:
5447:
5408:
5397:
5381:
5344:
5318:
5293:
5271:
5215:
5194:
5167:
5144:
5062:
5042:
5009:
4989:
4973:
4900:
4840:
4819:
4782:
4744:
4713:
4700:
4681:
4672:
4656:
4646:
4629:
4597:
4517:
4487:
4460:
4439:
4412:
4385:
4362:
4338:
4303:
4239:
4181:
4161:
4146:
4101:
4068:
4026:
4010:
3991:
3954:
3931:
3913:
3887:
3867:
3846:
3820:
3783:
3759:
3748:
3721:
3700:
3643:
3620:
3437:
3408:
3375:
3329:
3283:
3264:
3231:
3182:
3156:
3109:
3083:
3056:
3040:
3021:
2984:
2956:
2903:
2883:
2845:
2807:
2786:
2773:
2767:
2722:
2693:
2675:
2657:
2638:
2555:
2527:
2484:
2461:coupling constant
2440:
2422:
2389:
2371:
2358:
2352:
2333:
2314:
2287:
2271:
2251:
2235:
2213:
2179:
2166:
2147:
2123:
2098:
2080:
2052:
2028:
2012:-quantities (the
1989:
1976:
1953:
1914:
1874:
1839:
1714:coefficient of a
1440:Jeffrey Goldstone
1436:symmetry breaking
1420:{\displaystyle b}
1261:Erwin Schrödinger
1251:Noether's theorem
1187:compact Lie group
1160:Yang–Mills theory
1157:
1156:
250:Expectation value
225:BRST quantization
172:Poincaré symmetry
127:Yang–Mills theory
107:Quantum mechanics
16:(Redirected from
8493:
8460:
8459:
8377:Quantum dynamics
8050:Yang–Mills–Higgs
8005:Non-linear sigma
7995:Euler–Heisenberg
7980:
7891:
7884:
7877:
7868:
7863:
7861:
7860:
7851:. Archived from
7840:
7838:
7837:
7828:. Archived from
7817:
7791:
7789:
7788:
7773:
7771:
7751:
7738:World Scientific
7725:
7702:
7687:(3rd ed.).
7661:
7660:
7658:
7611:
7605:
7604:
7578:
7555:
7549:
7543:
7542:
7508:
7486:
7480:
7479:
7461:
7455:
7454:
7449:(Ph.D. thesis).
7442:
7436:
7435:
7416:
7407:
7406:
7404:
7392:
7381:
7380:
7378:
7337:
7331:
7330:
7307:
7301:
7300:
7282:
7267:
7266:
7234:
7225:
7224:
7222:
7221:
7207:
7144:
7049:Martinus Veltman
7034:
6995:
6980:
6978:
6977:
6972:
6968:
6967:
6966:
6949:
6941:
6939:
6938:
6933:
6926:
6925:
6924:
6913:
6912:
6911:
6899:
6898:
6887:
6886:
6885:
6868:
6867:
6866:
6861:
6860:
6852:
6846:
6827:
6824:we get from the
6819:
6811:
6804:
6802:
6801:
6796:
6792:
6791:
6787:
6776:
6771:
6760:
6744:
6743:
6723:
6712:
6707:
6696:
6692:
6691:
6686:
6678:
6674:
6673:
6668:
6659:
6650:
6636:
6632:
6622:
6621:
6610:
6594:
6593:
6579:
6569:
6568:
6563:
6562:
6554:
6548:
6544:
6543:
6538:
6530:
6526:
6525:
6520:
6485:
6484:
6476:
6464:
6463:
6441:
6439:
6438:
6433:
6431:
6418:
6417:
6409:
6397:
6396:
6378:
6374:
6370:
6369:
6367:
6357:
6356:
6342:
6337:
6331:
6326:
6324:
6322:
6312:
6311:
6297:
6292:
6283:
6281:
6277:
6276:
6274:
6263:
6258:
6247:
6242:
6236:
6231:
6227:
6226:
6224:
6213:
6208:
6197:
6192:
6186:
6181:
6179:
6178:
6161:
6160:
6159:
6142:
6138:
6137:
6132:
6123:
6118:
6117:
6108:
6104:
6078:
6074:
6070:
6069:
6067:
6057:
6056:
6042:
6037:
6031:
6026:
6022:
6021:
6019:
6008:
6003:
5992:
5987:
5981:
5976:
5974:
5972:
5961:
5956:
5945:
5940:
5934:
5929:
5927:
5926:
5917:
5916:
5899:
5895:
5894:
5889:
5875:
5850:
5846:
5842:
5841:
5839:
5829:
5828:
5817:
5812:
5806:
5801:
5797:
5796:
5794:
5783:
5778:
5767:
5762:
5756:
5751:
5747:
5746:
5745:
5736:
5735:
5718:
5717:
5715:
5705:
5704:
5699:
5698:
5690:
5684:
5678:
5670:
5666:
5662:
5661:
5656:
5642:
5608:
5607:
5599:
5560:
5556:
5553:). Here we have
5545:
5543:
5542:
5537:
5533:
5532:
5528:
5527:
5526:
5515:
5514:
5513:
5499:
5498:
5497:
5486:
5485:
5484:
5467:
5466:
5465:
5460:
5459:
5451:
5445:
5438:
5437:
5428:
5427:
5418:
5417:
5406:
5405:
5404:
5399:
5398:
5390:
5379:
5375:
5374:
5369:
5354:
5353:
5342:
5328:
5326:
5325:
5320:
5316:
5315:
5314:
5291:
5287:
5286:
5281:
5272:
5270:
5259:
5251:
5250:
5225:
5223:
5222:
5217:
5213:
5212:
5208:
5207:
5206:
5192:
5191:
5190:
5165:
5161:
5160:
5155:
5146:
5137:
5128:
5127:
5105:
5103:
5102:
5097:
5095:
5091:
5087:
5086:
5082:
5072:
5071:
5060:
5050:
5049:
5044:
5043:
5035:
5019:
5018:
5007:
4997:
4996:
4991:
4990:
4982:
4971:
4967:
4966:
4961:
4937:
4936:
4914:
4909:
4898:
4894:
4893:
4888:
4864:
4860:
4859:
4853:
4849:
4842:
4841:
4833:
4821:
4820:
4812:
4792:
4791:
4780:
4773:
4769:
4757:
4756:
4742:
4735:
4731:
4711:
4710:
4709:
4698:
4694:
4693:
4679:
4670:
4669:
4658:
4657:
4649:
4644:
4643:
4627:
4626:
4599:
4598:
4590:
4530:
4528:
4527:
4522:
4515:
4514:
4510:
4500:
4499:
4485:
4474:
4469:
4458:
4454:
4453:
4448:
4437:
4430:
4426:
4425:
4424:
4410:
4409:
4408:
4383:
4379:
4378:
4373:
4364:
4355:
4336:
4329:
4301:
4290:
4273:scale invariance
4266:
4264:
4252:
4250:
4249:
4244:
4237:
4236:
4232:
4231:
4230:
4226:
4199:
4195:
4194:
4179:
4171:
4169:
4168:
4163:
4159:
4158:
4154:
4153:
4152:
4148:
4142:
4131:
4112:
4099:
4091:
4081:
4079:
4078:
4073:
4066:
4064:
4059:
4043:
4038:
4024:
4023:
4022:
4008:
4007:
4006:
3989:
3981:
3976:
3964:
3963:
3952:
3941:
3939:
3938:
3933:
3929:
3927:
3922:
3911:
3900:
3898:
3897:
3892:
3885:
3878:
3877:
3869:
3868:
3860:
3856:
3855:
3844:
3833:
3831:
3830:
3825:
3818:
3817:
3816:
3804:
3803:
3785:
3776:
3770:
3769:
3761:
3760:
3752:
3746:
3739:strength tensor
3734:
3732:
3731:
3726:
3719:
3717:
3712:
3698:
3697:
3696:
3684:
3680:
3678:
3673:
3658:
3657:
3641:
3630:
3628:
3627:
3622:
3618:
3611:
3607:
3606:
3602:
3601:
3600:
3588:
3587:
3570:
3569:
3552:
3548:
3547:
3543:
3542:
3541:
3529:
3528:
3511:
3510:
3493:
3489:
3488:
3484:
3483:
3482:
3470:
3469:
3452:
3451:
3435:
3418:
3416:
3415:
3410:
3406:
3399:
3398:
3393:
3389:
3388:
3387:
3373:
3372:
3371:
3353:
3352:
3347:
3343:
3342:
3341:
3327:
3326:
3325:
3307:
3306:
3301:
3297:
3296:
3295:
3281:
3280:
3279:
3262:
3250:Bianchi identity
3244:
3242:
3241:
3236:
3229:
3222:
3221:
3216:
3212:
3211:
3210:
3198:
3197:
3180:
3169:
3167:
3166:
3161:
3154:
3152:
3147:
3135:
3134:
3122:
3121:
3107:
3096:
3094:
3093:
3088:
3081:
3073:
3068:
3054:
3053:
3052:
3038:
3037:
3036:
3019:
3011:
3006:
2994:
2993:
2982:
2969:
2967:
2966:
2961:
2954:
2935:
2934:
2916:
2915:
2901:
2893:
2891:
2890:
2885:
2881:
2880:
2879:
2861:
2860:
2843:
2820:
2818:
2817:
2812:
2805:
2803:
2798:
2784:
2783:
2782:
2771:
2765:
2755:
2751:
2750:
2749:
2737:
2736:
2720:
2703:
2701:
2700:
2695:
2691:
2689:
2684:
2673:
2671:
2666:
2655:
2654:
2653:
2636:
2628:
2623:
2614:
2613:
2600:
2595:
2586:
2585:
2572:
2567:
2553:
2540:
2538:
2537:
2532:
2525:
2518:
2517:
2482:
2474:
2466:
2458:
2450:
2448:
2447:
2442:
2438:
2436:
2431:
2420:
2408:
2402:
2400:
2399:
2394:
2387:
2385:
2380:
2369:
2368:
2367:
2356:
2350:
2343:
2342:
2331:
2324:
2323:
2312:
2297:
2295:
2294:
2289:
2285:
2284:
2283:
2269:
2261:
2259:
2258:
2253:
2249:
2245:
2244:
2233:
2232:
2231:
2211:
2199:
2192:
2190:
2189:
2184:
2177:
2176:
2175:
2164:
2163:
2162:
2145:
2138:
2134:
2133:
2132:
2121:
2117:
2116:
2096:
2095:
2094:
2082:
2073:
2067:
2063:
2062:
2061:
2050:
2049:
2048:
2026:
2011:
2007:
1999:
1997:
1996:
1991:
1987:
1986:
1985:
1974:
1963:
1961:
1960:
1955:
1951:
1949:
1944:
1932:
1931:
1916:
1907:
1895:
1894:
1876:
1867:
1858:
1857:
1856:
1847:
1846:
1837:
1817:
1810:solution to the
1805:
1801:
1795:compactification
1792:
1788:
1781:
1780:
1764:
1760:
1747:(top left). The
1742:
1733:
1729:
1713:
1694:
1685:
1674:
1665:
1636:gauge invariance
1567:
1547:
1539:
1538:
1537:
1536:
1529:
1528:
1519:
1518:
1517:
1516:
1509:
1508:
1499:
1498:
1497:
1496:
1489:
1488:
1471:
1467:
1461:
1426:
1424:
1423:
1418:
1401:
1354:
1352:
1351:
1346:
1321:
1319:
1318:
1313:
1311:
1310:
1291:
1289:
1288:
1283:
1219:
1207:
1184:
1149:
1142:
1135:
240:Effective action
167:Lorentz symmetry
92:Electromagnetism
62:
48:
35:
21:
8501:
8500:
8496:
8495:
8494:
8492:
8491:
8490:
8471:
8470:
8469:
8464:
8447:
8399:Quantum gravity
8326:
8285:Particle theory
8280:
8259:
8208:
8182:
8146:
8110:
8064:Low dimensional
8059:
8000:Ginzburg–Landau
7971:
7962:Topological QFT
7900:
7895:
7858:
7856:
7843:
7835:
7833:
7820:
7802:
7799:
7794:
7786:
7784:
7776:
7769:math-ph/9902027
7761:
7748:
7728:
7722:
7705:
7699:
7679:
7670:
7668:Further reading
7665:
7664:
7613:
7612:
7608:
7558:
7553:
7551:
7550:
7546:
7488:
7487:
7483:
7476:
7463:
7462:
7458:
7444:
7443:
7439:
7418:
7417:
7410:
7394:
7393:
7384:
7354:Physical Review
7339:
7338:
7334:
7327:
7309:
7308:
7304:
7297:
7284:
7283:
7270:
7236:
7235:
7228:
7219:
7217:
7209:
7208:
7204:
7199:
7194:
7142:
7092:
7060:Simon Donaldson
7053:Higgs mechanism
7045:Gerard 't Hooft
7041:
7032:
6993:
6952:
6944:
6943:
6916:
6900:
6890:
6871:
6849:
6841:
6840:
6825:
6817:
6809:
6726:
6681:
6663:
6647:
6643:
6613:
6582:
6551:
6533:
6515:
6508:
6504:
6455:
6450:
6449:
6429:
6428:
6388:
6376:
6375:
6345:
6338:
6327:
6300:
6293:
6282:
6243:
6232:
6193:
6182:
6164:
6145:
6127:
6109:
6097:
6093:
6076:
6075:
6045:
6038:
6027:
5988:
5977:
5941:
5930:
5918:
5902:
5884:
5868:
5864:
5848:
5847:
5820:
5813:
5802:
5763:
5752:
5737:
5721:
5687:
5674:
5651:
5635:
5631:
5618:
5576:
5575:
5558:
5554:
5551:Feynman diagram
5518:
5502:
5489:
5470:
5448:
5429:
5419:
5409:
5387:
5386:
5382:
5364:
5345:
5337:
5336:
5306:
5276:
5263:
5239:
5234:
5233:
5229:for the field,
5195:
5179:
5178:
5174:
5150:
5119:
5114:
5113:
5093:
5092:
5063:
5032:
5010:
4979:
4978:
4974:
4956:
4925:
4883:
4876:
4872:
4862:
4861:
4797:
4793:
4783:
4762:
4758:
4745:
4718:
4714:
4701:
4609:
4567:
4566:
4488:
4443:
4413:
4397:
4396:
4392:
4368:
4349:
4345:
4296:
4295:
4288:
4281:
4259:
4258:
4216:
4212:
4207:
4203:
4186:
4182:
4174:
4173:
4132:
4125:
4120:
4116:
4102:
4094:
4093:
4089:
4011:
3992:
3955:
3947:
3946:
3906:
3905:
3857:
3847:
3839:
3838:
3805:
3786:
3749:
3741:
3740:
3688:
3649:
3648:
3644:
3636:
3635:
3592:
3579:
3578:
3574:
3561:
3560:
3556:
3533:
3520:
3519:
3515:
3502:
3501:
3497:
3474:
3461:
3460:
3456:
3443:
3442:
3438:
3430:
3429:
3424:Jacobi identity
3376:
3363:
3362:
3358:
3357:
3330:
3317:
3316:
3312:
3311:
3284:
3271:
3270:
3266:
3265:
3257:
3256:
3199:
3189:
3188:
3184:
3183:
3175:
3174:
3126:
3110:
3102:
3101:
3041:
3022:
2985:
2977:
2976:
2904:
2896:
2895:
2865:
2846:
2838:
2837:
2774:
2741:
2728:
2727:
2723:
2715:
2714:
2639:
2605:
2577:
2548:
2547:
2509:
2477:
2476:
2468:
2464:
2456:
2455:potential, and
2415:
2414:
2411:identity matrix
2406:
2359:
2334:
2315:
2307:
2306:
2272:
2264:
2263:
2236:
2223:
2206:
2205:
2197:
2167:
2148:
2124:
2108:
2107:
2103:
2083:
2053:
2040:
2039:
2035:
2021:
2020:
2009:
2005:
1977:
1969:
1968:
1917:
1886:
1840:
1832:
1831:
1822:
1821:
1820:
1819:
1815:
1803:
1797:
1790:
1783:
1767:
1766:
1762:
1759:
1748:
1741:
1735:
1731:
1719:
1712:
1701:
1697:
1696:
1695:
1687:
1686:
1677:
1676:
1675:
1667:
1666:
1655:
1649:
1596:
1565:
1545:
1535:
1533:
1532:
1531:
1527:
1525:
1524:
1523:
1522:
1521:
1515:
1513:
1512:
1511:
1507:
1505:
1504:
1503:
1502:
1501:
1495:
1493:
1492:
1491:
1487:
1485:
1484:
1483:
1482:
1481:
1469:
1463:
1459:
1409:
1408:
1397:
1365:
1328:
1327:
1299:
1294:
1293:
1265:
1264:
1255:railroad gauges
1235:
1230:
1217:
1205:
1178:
1153:
1124:
1123:
1122:
1120:
424:
416:
415:
411:Quantum gravity
386:
378:
377:
373:Higgs mechanism
353:
343:
342:
328:Proca equations
313:
305:
304:
290:Renormalization
255:Feynman diagram
210:
202:
201:
142:
132:
131:
82:
67:
65:Feynman diagram
46:
45:
40:
37:
33:
28:
23:
22:
15:
12:
11:
5:
8499:
8497:
8489:
8488:
8483:
8481:Gauge theories
8473:
8472:
8466:
8465:
8452:
8449:
8448:
8446:
8445:
8440:
8435:
8434:
8433:
8423:
8418:
8413:
8408:
8407:
8406:
8396:
8395:
8394:
8384:
8379:
8374:
8369:
8364:
8359:
8354:
8349:
8344:
8342:Casimir effect
8338:
8336:
8332:
8331:
8328:
8327:
8325:
8324:
8319:
8317:Standard Model
8314:
8309:
8304:
8299:
8294:
8288:
8286:
8282:
8281:
8279:
8278:
8273:
8267:
8265:
8261:
8260:
8258:
8257:
8252:
8247:
8242:
8237:
8232:
8227:
8222:
8216:
8214:
8210:
8209:
8207:
8206:
8201:
8196:
8190:
8188:
8187:Superconformal
8184:
8183:
8181:
8180:
8175:
8170:
8168:Seiberg–Witten
8165:
8160:
8154:
8152:
8151:Supersymmetric
8148:
8147:
8145:
8144:
8139:
8134:
8129:
8124:
8118:
8116:
8112:
8111:
8109:
8108:
8103:
8098:
8093:
8088:
8083:
8078:
8073:
8067:
8065:
8061:
8060:
8058:
8057:
8052:
8047:
8042:
8037:
8032:
8027:
8022:
8017:
8012:
8007:
8002:
7997:
7992:
7986:
7984:
7977:
7973:
7972:
7970:
7969:
7964:
7959:
7954:
7949:
7944:
7939:
7934:
7929:
7924:
7919:
7914:
7908:
7906:
7902:
7901:
7896:
7894:
7893:
7886:
7879:
7871:
7865:
7864:
7841:
7826:DispersiveWiki
7818:
7798:
7797:External links
7795:
7793:
7792:
7774:
7758:
7757:
7753:
7752:
7746:
7732:, ed. (2005).
7726:
7720:
7703:
7697:
7676:
7675:
7671:
7669:
7666:
7663:
7662:
7633:(1): 189–213.
7606:
7576:hep-ph/0701025
7556:
7544:
7506:hep-ph/0512364
7499:(13): 132001.
7481:
7475:978-0199208463
7474:
7456:
7437:
7408:
7382:
7361:(1): 191–195.
7332:
7325:
7302:
7295:
7268:
7226:
7201:
7200:
7198:
7195:
7193:
7192:
7187:
7182:
7177:
7172:
7167:
7162:
7157:
7152:
7147:
7139:
7134:
7129:
7124:
7119:
7114:
7109:
7104:
7099:
7093:
7091:
7088:
7040:
7037:
6965:
6962:
6959:
6955:
6931:
6923:
6919:
6910:
6907:
6903:
6897:
6893:
6884:
6881:
6878:
6874:
6865:
6858:
6855:
6830:cross sections
6806:
6805:
6790:
6786:
6783:
6780:
6775:
6770:
6766:
6759:
6756:
6753:
6750:
6747:
6742:
6739:
6736:
6733:
6729:
6722:
6719:
6716:
6711:
6706:
6702:
6695:
6690:
6685:
6677:
6672:
6667:
6662:
6656:
6653:
6646:
6642:
6639:
6635:
6631:
6628:
6625:
6620:
6616:
6609:
6606:
6603:
6600:
6597:
6592:
6589:
6585:
6578:
6575:
6572:
6567:
6560:
6557:
6547:
6542:
6537:
6529:
6524:
6519:
6514:
6511:
6507:
6503:
6500:
6497:
6494:
6491:
6488:
6482:
6479:
6473:
6470:
6467:
6462:
6458:
6443:
6442:
6427:
6424:
6421:
6415:
6412:
6406:
6403:
6400:
6395:
6391:
6387:
6381:
6379:
6377:
6373:
6366:
6363:
6360:
6355:
6352:
6348:
6341:
6336:
6330:
6321:
6318:
6315:
6310:
6307:
6303:
6296:
6291:
6288:
6273:
6270:
6267:
6262:
6257:
6253:
6246:
6241:
6235:
6223:
6220:
6217:
6212:
6207:
6203:
6196:
6191:
6185:
6177:
6174:
6171:
6167:
6158:
6155:
6152:
6148:
6141:
6136:
6131:
6126:
6121:
6116:
6112:
6103:
6100:
6096:
6092:
6089:
6086:
6081:
6079:
6077:
6073:
6066:
6063:
6060:
6055:
6052:
6048:
6041:
6036:
6030:
6018:
6015:
6012:
6007:
6002:
5998:
5991:
5986:
5980:
5971:
5968:
5965:
5960:
5955:
5951:
5944:
5939:
5933:
5925:
5921:
5915:
5912:
5909:
5905:
5898:
5893:
5888:
5883:
5880:
5874:
5871:
5867:
5863:
5860:
5857:
5853:
5851:
5849:
5845:
5838:
5835:
5832:
5827:
5823:
5816:
5811:
5805:
5793:
5790:
5787:
5782:
5777:
5773:
5766:
5761:
5755:
5744:
5740:
5734:
5731:
5728:
5724:
5714:
5711:
5708:
5703:
5696:
5693:
5683:
5677:
5673:
5665:
5660:
5655:
5650:
5647:
5641:
5638:
5634:
5630:
5627:
5624:
5621:
5619:
5617:
5614:
5611:
5605:
5602:
5596:
5593:
5590:
5587:
5584:
5583:
5547:
5546:
5531:
5525:
5521:
5512:
5509:
5505:
5496:
5492:
5483:
5480:
5477:
5473:
5464:
5457:
5454:
5444:
5441:
5436:
5432:
5426:
5422:
5416:
5412:
5403:
5396:
5393:
5385:
5378:
5373:
5368:
5363:
5360:
5357:
5352:
5348:
5330:
5329:
5313:
5309:
5305:
5302:
5299:
5296:
5290:
5285:
5280:
5275:
5269:
5266:
5262:
5257:
5254:
5249:
5246:
5242:
5227:
5226:
5211:
5205:
5202:
5198:
5189:
5186:
5182:
5177:
5173:
5170:
5164:
5159:
5154:
5149:
5143:
5140:
5134:
5131:
5126:
5122:
5107:
5106:
5090:
5085:
5081:
5078:
5075:
5070:
5066:
5059:
5056:
5053:
5048:
5041:
5038:
5031:
5028:
5025:
5022:
5017:
5013:
5006:
5003:
5000:
4995:
4988:
4985:
4977:
4970:
4965:
4960:
4955:
4952:
4949:
4946:
4943:
4940:
4935:
4932:
4928:
4924:
4921:
4918:
4913:
4908:
4904:
4897:
4892:
4887:
4882:
4879:
4875:
4871:
4868:
4865:
4863:
4858:
4852:
4848:
4845:
4839:
4836:
4830:
4827:
4824:
4818:
4815:
4809:
4806:
4803:
4800:
4796:
4790:
4786:
4779:
4776:
4772:
4768:
4765:
4761:
4755:
4752:
4748:
4741:
4738:
4734:
4730:
4727:
4724:
4721:
4717:
4708:
4704:
4697:
4692:
4687:
4684:
4678:
4675:
4668:
4664:
4661:
4655:
4652:
4642:
4638:
4635:
4632:
4625:
4621:
4618:
4615:
4612:
4610:
4608:
4605:
4602:
4596:
4593:
4587:
4584:
4581:
4578:
4575:
4574:
4540:Ludvig Faddeev
4532:
4531:
4520:
4513:
4509:
4506:
4503:
4498:
4495:
4491:
4484:
4481:
4478:
4473:
4468:
4464:
4457:
4452:
4447:
4442:
4436:
4433:
4429:
4423:
4420:
4416:
4407:
4404:
4400:
4395:
4391:
4388:
4382:
4377:
4372:
4367:
4361:
4358:
4352:
4348:
4344:
4341:
4335:
4332:
4328:
4324:
4321:
4318:
4315:
4312:
4309:
4306:
4285:path integrals
4280:
4277:
4255:renormalizable
4242:
4235:
4229:
4225:
4222:
4219:
4215:
4210:
4206:
4202:
4198:
4193:
4189:
4185:
4157:
4151:
4145:
4141:
4138:
4135:
4128:
4123:
4119:
4115:
4111:
4108:
4105:
4083:
4082:
4071:
4063:
4058:
4054:
4050:
4047:
4042:
4037:
4034:
4030:
4021:
4018:
4014:
4005:
4002:
3999:
3995:
3988:
3985:
3980:
3975:
3972:
3968:
3962:
3958:
3926:
3921:
3917:
3902:
3901:
3890:
3884:
3881:
3876:
3873:
3866:
3863:
3854:
3850:
3823:
3815:
3812:
3808:
3802:
3799:
3796:
3793:
3789:
3782:
3779:
3773:
3768:
3765:
3758:
3755:
3724:
3716:
3711:
3708:
3704:
3695:
3691:
3687:
3683:
3677:
3672:
3669:
3665:
3661:
3656:
3652:
3647:
3632:
3631:
3617:
3614:
3610:
3605:
3599:
3595:
3591:
3586:
3582:
3577:
3573:
3568:
3564:
3559:
3555:
3551:
3546:
3540:
3536:
3532:
3527:
3523:
3518:
3514:
3509:
3505:
3500:
3496:
3492:
3487:
3481:
3477:
3473:
3468:
3464:
3459:
3455:
3450:
3446:
3441:
3420:
3419:
3405:
3402:
3397:
3392:
3386:
3383:
3379:
3370:
3366:
3361:
3356:
3351:
3346:
3340:
3337:
3333:
3324:
3320:
3315:
3310:
3305:
3300:
3294:
3291:
3287:
3278:
3274:
3269:
3246:
3245:
3234:
3228:
3225:
3220:
3215:
3209:
3206:
3202:
3196:
3192:
3187:
3159:
3151:
3146:
3143:
3139:
3133:
3129:
3125:
3120:
3117:
3113:
3098:
3097:
3086:
3080:
3077:
3072:
3067:
3064:
3060:
3051:
3048:
3044:
3035:
3032:
3029:
3025:
3018:
3015:
3010:
3005:
3002:
2998:
2992:
2988:
2959:
2953:
2950:
2947:
2944:
2941:
2938:
2933:
2930:
2927:
2924:
2919:
2914:
2911:
2907:
2878:
2875:
2872:
2868:
2864:
2859:
2856:
2853:
2849:
2836:indices (e.g.
2822:
2821:
2810:
2802:
2797:
2794:
2790:
2781:
2777:
2770:
2764:
2761:
2758:
2754:
2748:
2744:
2740:
2735:
2731:
2726:
2705:
2704:
2688:
2683:
2679:
2670:
2665:
2661:
2652:
2649:
2646:
2642:
2635:
2632:
2627:
2622:
2618:
2612:
2608:
2604:
2599:
2594:
2590:
2584:
2580:
2576:
2571:
2566:
2563:
2559:
2530:
2524:
2521:
2516:
2512:
2508:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2475:group one has
2435:
2430:
2426:
2404:
2403:
2392:
2384:
2379:
2375:
2366:
2362:
2355:
2349:
2346:
2341:
2337:
2330:
2327:
2322:
2318:
2302:is defined as
2282:
2279:
2275:
2248:
2243:
2239:
2230:
2226:
2222:
2219:
2216:
2194:
2193:
2182:
2174:
2170:
2161:
2158:
2155:
2151:
2144:
2141:
2137:
2131:
2127:
2120:
2115:
2111:
2106:
2101:
2093:
2090:
2086:
2079:
2076:
2070:
2066:
2060:
2056:
2047:
2043:
2038:
2034:
2031:
1984:
1980:
1965:
1964:
1948:
1943:
1940:
1936:
1930:
1927:
1924:
1920:
1913:
1910:
1904:
1901:
1898:
1893:
1889:
1885:
1882:
1879:
1873:
1870:
1864:
1861:
1855:
1852:
1845:
1757:
1739:
1716:BPST instanton
1710:
1699:
1698:
1689:
1688:
1680:
1679:
1678:
1669:
1668:
1660:
1659:
1658:
1657:
1656:
1648:
1645:
1600:Wolfgang Pauli
1595:
1592:
1550:Standard Model
1534:
1526:
1514:
1506:
1494:
1486:
1444:Yoichiro Nambu
1416:
1390:abelian groups
1364:
1361:
1357:Chen Ning Yang
1344:
1341:
1338:
1335:
1309:
1306:
1302:
1281:
1278:
1275:
1272:
1234:
1231:
1229:
1226:
1222:Standard Model
1164:Chen Ning Yang
1155:
1154:
1152:
1151:
1144:
1137:
1129:
1126:
1125:
1118:
1117:
1112:
1107:
1102:
1097:
1092:
1087:
1082:
1077:
1072:
1067:
1062:
1057:
1052:
1047:
1042:
1037:
1032:
1027:
1022:
1017:
1012:
1007:
1002:
997:
992:
987:
982:
977:
972:
967:
962:
957:
952:
947:
942:
937:
932:
927:
922:
917:
912:
907:
902:
897:
892:
887:
882:
877:
872:
867:
862:
857:
852:
847:
842:
837:
832:
827:
822:
817:
812:
807:
802:
797:
792:
787:
782:
777:
772:
767:
762:
757:
752:
747:
742:
737:
732:
727:
722:
717:
712:
707:
702:
697:
692:
687:
682:
677:
672:
667:
662:
657:
652:
647:
642:
637:
632:
627:
622:
617:
612:
607:
602:
597:
592:
587:
582:
577:
572:
567:
562:
557:
552:
547:
542:
537:
532:
527:
522:
517:
512:
507:
502:
497:
492:
487:
482:
477:
472:
467:
462:
457:
452:
447:
442:
437:
432:
426:
425:
422:
421:
418:
417:
414:
413:
408:
403:
398:
393:
387:
384:
383:
380:
379:
376:
375:
370:
365:
360:
354:
351:Standard Model
349:
348:
345:
344:
341:
340:
335:
330:
325:
320:
318:Dirac equation
314:
311:
310:
307:
306:
303:
302:
300:Wick's theorem
297:
292:
287:
285:Regularization
282:
277:
272:
267:
262:
257:
252:
247:
242:
237:
232:
227:
222:
217:
211:
208:
207:
204:
203:
200:
199:
194:
192:Noether charge
189:
184:
179:
177:Gauge symmetry
174:
169:
164:
159:
154:
149:
143:
138:
137:
134:
133:
130:
129:
124:
119:
114:
109:
104:
99:
94:
89:
83:
80:
79:
76:
75:
69:
68:
63:
55:
54:
41:
38:
32:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8498:
8487:
8484:
8482:
8479:
8478:
8476:
8463:
8455:
8450:
8444:
8441:
8439:
8438:Quantum logic
8436:
8432:
8429:
8428:
8427:
8424:
8422:
8419:
8417:
8414:
8412:
8409:
8405:
8402:
8401:
8400:
8397:
8393:
8390:
8389:
8388:
8385:
8383:
8380:
8378:
8375:
8373:
8372:Quantum chaos
8370:
8368:
8365:
8363:
8360:
8358:
8355:
8353:
8350:
8348:
8347:Cosmic string
8345:
8343:
8340:
8339:
8337:
8333:
8323:
8320:
8318:
8315:
8313:
8310:
8308:
8305:
8303:
8300:
8298:
8295:
8293:
8290:
8289:
8287:
8283:
8277:
8274:
8272:
8269:
8268:
8266:
8262:
8256:
8253:
8251:
8248:
8246:
8243:
8241:
8238:
8236:
8233:
8231:
8228:
8226:
8223:
8221:
8220:Pure 4D N = 1
8218:
8217:
8215:
8211:
8205:
8202:
8200:
8197:
8195:
8192:
8191:
8189:
8185:
8179:
8176:
8174:
8171:
8169:
8166:
8164:
8161:
8159:
8156:
8155:
8153:
8149:
8143:
8140:
8138:
8135:
8133:
8130:
8128:
8125:
8123:
8120:
8119:
8117:
8113:
8107:
8104:
8102:
8101:Thirring–Wess
8099:
8097:
8094:
8092:
8089:
8087:
8084:
8082:
8079:
8077:
8076:Bullough–Dodd
8074:
8072:
8071:2D Yang–Mills
8069:
8068:
8066:
8062:
8056:
8053:
8051:
8048:
8046:
8043:
8041:
8038:
8036:
8033:
8031:
8028:
8026:
8023:
8021:
8018:
8016:
8013:
8011:
8008:
8006:
8003:
8001:
7998:
7996:
7993:
7991:
7988:
7987:
7985:
7981:
7978:
7974:
7968:
7965:
7963:
7960:
7958:
7955:
7953:
7950:
7948:
7947:String theory
7945:
7943:
7940:
7938:
7935:
7933:
7930:
7928:
7925:
7923:
7920:
7918:
7917:Axiomatic QFT
7915:
7913:
7912:Algebraic QFT
7910:
7909:
7907:
7903:
7899:
7892:
7887:
7885:
7880:
7878:
7873:
7872:
7869:
7855:on 2009-01-16
7854:
7850:
7846:
7842:
7832:on 2021-06-03
7831:
7827:
7823:
7819:
7815:
7811:
7810:
7805:
7801:
7800:
7796:
7783:
7779:
7775:
7770:
7765:
7760:
7759:
7755:
7754:
7749:
7747:981-238-934-2
7743:
7739:
7736:. Singapore:
7735:
7731:
7727:
7723:
7721:0-19-851961-3
7717:
7713:
7709:
7704:
7700:
7694:
7690:
7686:
7682:
7678:
7677:
7673:
7672:
7667:
7657:
7652:
7648:
7644:
7640:
7636:
7632:
7628:
7624:
7620:
7616:
7610:
7607:
7602:
7598:
7594:
7590:
7586:
7582:
7577:
7572:
7569:(7): 074034.
7568:
7564:
7563:
7548:
7545:
7540:
7536:
7532:
7528:
7524:
7520:
7516:
7512:
7507:
7502:
7498:
7494:
7493:
7485:
7482:
7477:
7471:
7467:
7460:
7457:
7452:
7448:
7441:
7438:
7433:
7429:
7425:
7421:
7415:
7413:
7409:
7403:
7402:gr-qc/0012054
7398:
7391:
7389:
7387:
7383:
7377:
7372:
7368:
7364:
7360:
7356:
7355:
7350:
7346:
7342:
7336:
7333:
7328:
7326:9781461301950
7322:
7318:
7317:
7314:
7306:
7303:
7298:
7292:
7288:
7281:
7279:
7277:
7275:
7273:
7269:
7264:
7260:
7256:
7252:
7248:
7244:
7240:
7233:
7231:
7227:
7216:
7212:
7206:
7203:
7196:
7191:
7188:
7186:
7183:
7181:
7178:
7176:
7173:
7171:
7168:
7166:
7163:
7161:
7158:
7156:
7153:
7151:
7148:
7146:
7140:
7138:
7135:
7133:
7130:
7128:
7125:
7123:
7120:
7118:
7115:
7113:
7110:
7108:
7105:
7103:
7102:Coulomb gauge
7100:
7098:
7095:
7094:
7089:
7087:
7085:
7080:
7078:
7073:
7069:
7065:
7061:
7056:
7054:
7050:
7046:
7039:Open problems
7038:
7036:
7030:
7026:
7022:
7018:
7014:
7010:
7005:
7003:
6999:
6992:
6988:
6983:
6963:
6960:
6957:
6953:
6929:
6921:
6917:
6908:
6905:
6901:
6895:
6882:
6879:
6876:
6872:
6863:
6853:
6837:
6835:
6831:
6823:
6815:
6788:
6781:
6773:
6768:
6764:
6754:
6751:
6748:
6740:
6737:
6734:
6731:
6727:
6717:
6709:
6704:
6700:
6693:
6688:
6675:
6670:
6660:
6654:
6651:
6644:
6640:
6637:
6633:
6626:
6618:
6614:
6604:
6601:
6598:
6590:
6587:
6583:
6573:
6565:
6555:
6545:
6540:
6527:
6522:
6512:
6509:
6505:
6501:
6498:
6495:
6489:
6486:
6477:
6471:
6468:
6460:
6456:
6448:
6447:
6446:
6422:
6419:
6410:
6404:
6401:
6393:
6389:
6385:
6380:
6371:
6361:
6353:
6350:
6346:
6339:
6334:
6328:
6316:
6308:
6305:
6301:
6294:
6289:
6286:
6268:
6260:
6255:
6251:
6244:
6239:
6233:
6218:
6210:
6205:
6201:
6194:
6189:
6183:
6175:
6172:
6169:
6165:
6156:
6153:
6150:
6146:
6139:
6134:
6124:
6119:
6114:
6110:
6101:
6098:
6094:
6090:
6087:
6084:
6080:
6071:
6061:
6053:
6050:
6046:
6039:
6034:
6028:
6013:
6005:
6000:
5996:
5989:
5984:
5978:
5966:
5958:
5953:
5949:
5942:
5937:
5931:
5923:
5913:
5910:
5907:
5903:
5896:
5891:
5881:
5878:
5872:
5869:
5865:
5861:
5858:
5855:
5852:
5843:
5833:
5825:
5821:
5814:
5809:
5803:
5788:
5780:
5775:
5771:
5764:
5759:
5753:
5742:
5732:
5729:
5726:
5722:
5709:
5701:
5691:
5681:
5675:
5671:
5663:
5658:
5648:
5645:
5639:
5636:
5632:
5628:
5625:
5622:
5620:
5612:
5609:
5600:
5594:
5591:
5585:
5574:
5573:
5572:
5566:
5562:
5552:
5529:
5523:
5519:
5510:
5507:
5503:
5494:
5481:
5478:
5475:
5471:
5462:
5452:
5442:
5439:
5434:
5430:
5424:
5414:
5401:
5391:
5383:
5376:
5371:
5361:
5358:
5355:
5350:
5346:
5335:
5334:
5333:
5311:
5303:
5300:
5288:
5283:
5273:
5267:
5264:
5260:
5255:
5252:
5247:
5244:
5240:
5232:
5231:
5230:
5209:
5203:
5200:
5196:
5187:
5184:
5180:
5175:
5171:
5168:
5162:
5157:
5147:
5141:
5138:
5132:
5129:
5124:
5120:
5112:
5111:
5110:
5088:
5083:
5076:
5068:
5064:
5054:
5046:
5036:
5029:
5023:
5015:
5011:
5001:
4993:
4983:
4975:
4968:
4963:
4953:
4950:
4947:
4941:
4933:
4930:
4926:
4919:
4911:
4906:
4902:
4895:
4890:
4880:
4877:
4873:
4869:
4866:
4850:
4846:
4843:
4834:
4828:
4825:
4822:
4813:
4804:
4801:
4794:
4788:
4784:
4777:
4774:
4770:
4766:
4759:
4753:
4750:
4746:
4739:
4736:
4732:
4728:
4725:
4722:
4715:
4706:
4702:
4695:
4685:
4682:
4673:
4650:
4630:
4616:
4613:
4611:
4603:
4600:
4591:
4585:
4582:
4576:
4565:
4564:
4563:
4561:
4557:
4553:
4549:
4545:
4541:
4537:
4536:gauge freedom
4518:
4511:
4504:
4496:
4493:
4489:
4479:
4471:
4466:
4462:
4455:
4450:
4440:
4434:
4431:
4427:
4421:
4418:
4414:
4405:
4402:
4398:
4393:
4389:
4386:
4380:
4375:
4365:
4359:
4356:
4350:
4346:
4342:
4339:
4330:
4319:
4316:
4310:
4304:
4294:
4293:
4292:
4286:
4278:
4276:
4274:
4270:
4262:
4256:
4240:
4233:
4227:
4223:
4220:
4217:
4213:
4208:
4204:
4200:
4196:
4191:
4187:
4183:
4155:
4149:
4143:
4139:
4136:
4133:
4126:
4121:
4117:
4113:
4109:
4106:
4103:
4086:
4069:
4061:
4056:
4052:
4048:
4045:
4040:
4035:
4032:
4028:
4019:
4016:
4012:
4003:
4000:
3997:
3993:
3986:
3983:
3978:
3973:
3970:
3966:
3960:
3945:
3944:
3943:
3924:
3919:
3915:
3888:
3882:
3879:
3874:
3871:
3861:
3852:
3848:
3837:
3836:
3835:
3821:
3813:
3810:
3806:
3800:
3797:
3794:
3791:
3787:
3780:
3777:
3771:
3766:
3763:
3753:
3738:
3722:
3714:
3709:
3706:
3702:
3693:
3689:
3685:
3681:
3675:
3670:
3667:
3663:
3659:
3654:
3650:
3645:
3615:
3612:
3608:
3603:
3597:
3593:
3589:
3584:
3580:
3575:
3571:
3566:
3562:
3557:
3553:
3549:
3544:
3538:
3534:
3530:
3525:
3521:
3516:
3512:
3507:
3503:
3498:
3494:
3490:
3485:
3479:
3475:
3471:
3466:
3462:
3457:
3453:
3448:
3444:
3439:
3428:
3427:
3426:
3425:
3403:
3400:
3395:
3390:
3384:
3381:
3377:
3368:
3364:
3359:
3354:
3349:
3344:
3338:
3335:
3331:
3322:
3318:
3313:
3308:
3303:
3298:
3292:
3289:
3285:
3276:
3272:
3267:
3255:
3254:
3253:
3251:
3232:
3226:
3223:
3218:
3213:
3207:
3204:
3200:
3194:
3190:
3185:
3173:
3172:
3171:
3157:
3149:
3144:
3141:
3137:
3131:
3127:
3123:
3118:
3115:
3111:
3084:
3078:
3075:
3070:
3065:
3062:
3058:
3049:
3046:
3042:
3033:
3030:
3027:
3023:
3016:
3013:
3008:
3003:
3000:
2996:
2990:
2975:
2974:
2973:
2970:
2957:
2948:
2945:
2942:
2939:
2917:
2912:
2909:
2905:
2876:
2873:
2870:
2866:
2862:
2857:
2854:
2851:
2847:
2835:
2830:
2828:
2808:
2800:
2795:
2792:
2788:
2779:
2775:
2768:
2762:
2759:
2756:
2752:
2746:
2742:
2738:
2733:
2729:
2724:
2713:
2712:
2711:
2710:
2686:
2681:
2677:
2668:
2663:
2659:
2650:
2647:
2644:
2640:
2633:
2630:
2625:
2620:
2616:
2610:
2602:
2597:
2592:
2588:
2582:
2574:
2569:
2564:
2561:
2557:
2546:
2545:
2544:
2543:The relation
2541:
2528:
2522:
2519:
2514:
2510:
2506:
2503:
2500:
2497:
2494:
2491:
2488:
2485:
2472:
2462:
2454:
2433:
2428:
2424:
2412:
2390:
2382:
2377:
2373:
2364:
2360:
2353:
2347:
2344:
2339:
2328:
2325:
2320:
2316:
2305:
2304:
2303:
2301:
2280:
2277:
2273:
2241:
2237:
2228:
2224:
2217:
2214:
2203:
2180:
2172:
2168:
2159:
2156:
2153:
2149:
2142:
2139:
2135:
2129:
2125:
2118:
2113:
2109:
2104:
2099:
2091:
2088:
2084:
2077:
2074:
2068:
2064:
2058:
2054:
2045:
2041:
2036:
2032:
2029:
2019:
2018:
2017:
2015:
2004:, indexed by
2003:
1982:
1978:
1946:
1941:
1938:
1934:
1928:
1925:
1922:
1918:
1911:
1908:
1902:
1899:
1891:
1887:
1880:
1877:
1871:
1868:
1862:
1859:
1830:
1829:
1828:
1827:
1813:
1809:
1800:
1796:
1786:
1778:
1774:
1770:
1756:
1752:
1746:
1743:is the third
1738:
1727:
1723:
1717:
1709:
1705:
1693:
1684:
1673:
1664:
1654:
1646:
1644:
1641:
1637:
1633:
1628:
1625:
1621:
1617:
1613:
1612:Kaluza, Klein
1609:
1605:
1601:
1593:
1591:
1589:
1586:problem is a
1585:
1581:
1577:
1576:Phenomenology
1573:
1571:
1563:
1559:
1555:
1552:combines the
1551:
1543:
1479:
1475:
1466:
1457:
1453:
1449:
1445:
1441:
1437:
1432:
1428:
1414:
1404:
1400:
1395:
1391:
1385:
1383:
1377:
1375:
1371:
1362:
1360:
1358:
1339:
1333:
1325:
1307:
1304:
1300:
1276:
1270:
1262:
1258:
1256:
1252:
1248:
1244:
1240:
1232:
1227:
1225:
1223:
1215:
1211:
1208:) as well as
1203:
1199:
1195:
1192:
1188:
1182:
1177:
1173:
1169:
1165:
1161:
1150:
1145:
1143:
1138:
1136:
1131:
1130:
1128:
1127:
1121:
1116:
1113:
1111:
1108:
1106:
1103:
1101:
1098:
1096:
1093:
1091:
1090:Zamolodchikov
1088:
1086:
1085:Zamolodchikov
1083:
1081:
1078:
1076:
1073:
1071:
1068:
1066:
1063:
1061:
1058:
1056:
1053:
1051:
1048:
1046:
1043:
1041:
1038:
1036:
1033:
1031:
1028:
1026:
1023:
1021:
1018:
1016:
1013:
1011:
1008:
1006:
1003:
1001:
998:
996:
993:
991:
988:
986:
983:
981:
978:
976:
973:
971:
968:
966:
963:
961:
958:
956:
953:
951:
948:
946:
943:
941:
938:
936:
933:
931:
928:
926:
923:
921:
918:
916:
913:
911:
908:
906:
903:
901:
898:
896:
893:
891:
888:
886:
883:
881:
878:
876:
873:
871:
868:
866:
863:
861:
858:
856:
853:
851:
848:
846:
843:
841:
838:
836:
833:
831:
828:
826:
823:
821:
818:
816:
813:
811:
808:
806:
803:
801:
798:
796:
793:
791:
788:
786:
783:
781:
778:
776:
773:
771:
768:
766:
763:
761:
758:
756:
753:
751:
748:
746:
743:
741:
738:
736:
733:
731:
728:
726:
723:
721:
718:
716:
713:
711:
708:
706:
703:
701:
698:
696:
693:
691:
688:
686:
683:
681:
678:
676:
673:
671:
668:
666:
663:
661:
658:
656:
653:
651:
648:
646:
643:
641:
638:
636:
633:
631:
628:
626:
623:
621:
618:
616:
613:
611:
608:
606:
603:
601:
598:
596:
593:
591:
588:
586:
583:
581:
578:
576:
573:
571:
568:
566:
563:
561:
558:
556:
553:
551:
548:
546:
543:
541:
538:
536:
533:
531:
528:
526:
523:
521:
518:
516:
513:
511:
508:
506:
503:
501:
498:
496:
493:
491:
488:
486:
483:
481:
478:
476:
473:
471:
468:
466:
463:
461:
458:
456:
453:
451:
448:
446:
443:
441:
438:
436:
433:
431:
428:
427:
420:
419:
412:
409:
407:
404:
402:
399:
397:
396:Supersymmetry
394:
392:
391:String theory
389:
388:
382:
381:
374:
371:
369:
366:
364:
361:
359:
356:
355:
352:
347:
346:
339:
336:
334:
331:
329:
326:
324:
321:
319:
316:
315:
309:
308:
301:
298:
296:
293:
291:
288:
286:
283:
281:
278:
276:
273:
271:
268:
266:
263:
261:
258:
256:
253:
251:
248:
246:
243:
241:
238:
236:
233:
231:
228:
226:
223:
221:
218:
216:
213:
212:
206:
205:
198:
195:
193:
190:
188:
185:
183:
180:
178:
175:
173:
170:
168:
165:
163:
160:
158:
155:
153:
150:
148:
145:
144:
141:
136:
135:
128:
125:
123:
120:
118:
115:
113:
110:
108:
105:
103:
100:
98:
95:
93:
90:
88:
85:
84:
78:
77:
74:
71:
70:
66:
61:
57:
56:
53:
49:
44:
30:
19:
8453:
8382:Quantum foam
8322:Stueckelberg
8276:Chern–Simons
8213:Supergravity
8044:
7952:Supergravity
7937:Gauge theory
7857:. Retrieved
7853:the original
7834:. Retrieved
7830:the original
7825:
7807:
7785:. Retrieved
7733:
7707:
7684:
7681:Frampton, P.
7630:
7626:
7609:
7566:
7560:
7547:
7496:
7490:
7484:
7465:
7459:
7446:
7440:
7431:
7430:(memorial).
7427:
7358:
7352:
7335:
7316:
7312:
7305:
7286:
7246:
7242:
7218:. Retrieved
7214:
7205:
7137:Lorenz gauge
7081:
7066:'s list of "
7057:
7042:
7009:a posteriori
7006:
6984:
6838:
6807:
6444:
5570:
5548:
5331:
5228:
5108:
4547:
4544:Victor Popov
4533:
4282:
4279:Quantization
4260:
4087:
4084:
3903:
3633:
3421:
3247:
3099:
2971:
2833:
2831:
2823:
2706:
2542:
2470:
2405:
2195:
1966:
1823:
1798:
1784:
1776:
1772:
1768:
1754:
1750:
1745:Pauli matrix
1736:
1725:
1721:
1707:
1703:
1629:
1597:
1574:
1470:SU(2) × U(1)
1460:SU(2) × U(1)
1433:
1430:
1406:
1387:
1379:
1374:Robert Mills
1366:
1259:
1247:group theory
1243:Emmy Noether
1239:Hermann Weyl
1236:
1214:strong force
1206:U(1) × SU(2)
1180:
1172:gauge theory
1168:Robert Mills
1159:
1158:
1119:
965:Stueckelberg
705:Jona-Lasinio
295:Vacuum state
280:Quantization
126:
122:Gauge theory
102:Strong force
87:Field theory
29:
8264:Topological
8178:Wess–Zumino
8091:Sine-Gordon
8081:Gross–Neveu
7990:Born–Infeld
7957:Thermal QFT
7619:Veltman, M.
7249:(1): 1–23.
7077:confinement
6834:decay rates
4548:ghost field
3735:Define the
2002:Lie algebra
1640:Abdus Salam
1624:gauge field
1580:confinement
1476:to produce
1202:weak forces
1191:non-abelian
1174:based on a
1105:Zinn-Justin
955:Sommerfield
880:Pomeranchuk
850:Osterwalder
845:Oppenheimer
775:Łopuszański
600:Fredenhagen
401:Technicolor
8475:Categories
8045:Yang–Mills
7859:2008-11-24
7836:2018-08-30
7787:2015-05-05
7434:: 137–146.
7420:Atiyah, M.
7341:Yang, C.N.
7220:2024-04-09
7197:References
7175:Weyl gauge
7150:Propagator
2709:commutator
2196:Here, the
1826:Lagrangian
1730:-slice of
1651:See also:
1620:Lagrangian
1472:mix after
1216:(based on
1194:Lie groups
1100:Zimmermann
995:Vainshtein
740:Kontsevich
685:Iliopoulos
660:Heisenberg
485:Bogoliubov
423:Scientists
275:Propagator
162:T-symmetry
157:P-symmetry
152:C-symmetry
140:Symmetries
97:Weak force
81:Background
18:Yang–Mills
8454:See also:
8173:Super QCD
8127:Liouville
8115:Conformal
8086:Schwinger
7814:EMS Press
7778:Gross, D.
7689:Wiley-VCH
7656:1874/4845
7601:119434312
7345:Mills, R.
7263:0034-6861
6909:μ
6896:μ
6892:∂
6857:¯
6769:ν
6752:−
6741:ν
6738:μ
6705:μ
6661:∫
6641:
6615:ε
6602:−
6559:¯
6556:ε
6513:∫
6510:−
6502:
6490:ε
6481:¯
6478:ε
6423:ε
6414:¯
6411:ε
6386:×
6354:ν
6340:δ
6335:δ
6309:μ
6295:δ
6290:δ
6256:ν
6245:δ
6240:δ
6206:μ
6195:δ
6190:δ
6125:∫
6099:−
6091:
6085:×
6054:ν
6040:δ
6035:δ
6001:μ
5990:δ
5985:δ
5954:ν
5943:δ
5938:δ
5924:μ
5920:∂
5882:∫
5870:−
5862:
5856:×
5822:ε
5815:δ
5810:δ
5776:μ
5765:δ
5760:δ
5743:μ
5739:∂
5695:¯
5692:ε
5682:δ
5672:δ
5649:∫
5637:−
5629:
5613:ε
5604:¯
5601:ε
5511:μ
5495:μ
5491:∂
5456:¯
5425:μ
5421:∂
5415:μ
5411:∂
5395:¯
5362:∫
5359:−
5301:⋅
5298:∂
5274:∫
5268:ξ
5256:−
5204:ν
5201:μ
5188:ν
5185:μ
5172:
5148:∫
5133:−
5040:¯
5037:ε
5012:ε
4987:¯
4954:∫
4934:μ
4907:μ
4881:∫
4870:
4838:¯
4817:¯
4808:∂
4799:∂
4764:∂
4720:∂
4686:
4654:¯
4617:∫
4604:ε
4595:¯
4592:ε
4497:μ
4467:μ
4441:∫
4422:ν
4419:μ
4406:ν
4403:μ
4390:
4366:∫
4351:−
4343:
4320:∫
4221:−
4137:−
4057:ν
4049:−
4036:ν
4033:μ
4020:μ
3974:ν
3971:μ
3961:μ
3957:∂
3920:μ
3904:A source
3875:ν
3872:μ
3865:~
3853:μ
3814:σ
3811:ρ
3801:σ
3798:ρ
3795:ν
3792:μ
3788:ε
3767:ν
3764:μ
3757:~
3710:κ
3707:ν
3694:μ
3671:κ
3668:ν
3655:μ
3598:μ
3585:κ
3567:ν
3539:ν
3526:μ
3508:κ
3480:κ
3467:ν
3449:μ
3385:μ
3382:κ
3369:ν
3339:ν
3336:μ
3323:κ
3293:κ
3290:ν
3277:μ
3208:ν
3205:μ
3195:μ
3145:ν
3142:μ
3119:ν
3116:μ
3066:ν
3063:μ
3047:μ
3004:ν
3001:μ
2991:μ
2987:∂
2949:−
2946:−
2943:−
2913:ν
2910:μ
2906:η
2796:ν
2793:μ
2760:−
2747:ν
2734:μ
2682:ν
2664:μ
2621:μ
2611:ν
2607:∂
2603:−
2593:ν
2583:μ
2579:∂
2565:ν
2562:μ
2520:−
2507:…
2429:μ
2378:μ
2345:−
2340:μ
2336:∂
2321:μ
2274:δ
2218:
2085:δ
2033:
2014:curvature
1942:ν
1939:μ
1929:ν
1926:μ
1903:−
1881:
1863:−
1808:instanton
1308:θ
1035:Wetterich
1020:Weisskopf
970:Sudarshan
920:Schwinger
835:Nishijima
800:Maldacena
765:Leutwyler
730:Kinoshita
630:Goldstone
620:Gell-Mann
535:Doplicher
312:Equations
8486:Symmetry
8250:Type IIB
8245:Type IIA
8230:4D N = 8
8225:4D N = 1
8194:6D (2,0)
8158:4D N = 1
8137:Polyakov
8096:Thirring
7905:Theories
7780:(1992).
7756:Articles
7683:(2008).
7621:(1972).
7539:42504317
7531:16711979
7422:(2017).
7347:(1954).
7090:See also
7070:". Here
7025:glueball
3100:Putting
2198:f
1050:Wightman
1015:Weinberg
1005:Virasoro
985:Tomonaga
980:Thirring
975:Symanzik
935:Semenoff
910:Schrader
875:Polyakov
795:Majorana
735:Klebanov
690:Ivanenko
680:'t Hooft
650:Guralnik
595:Fröhlich
590:Fritzsch
585:Frampton
500:Buchholz
445:Bargmann
435:Anderson
235:Crossing
8352:History
8335:Related
8132:Minimal
7983:Regular
7816:, 2001
7635:Bibcode
7581:Bibcode
7511:Bibcode
7363:Bibcode
7029:hybrids
7017:hadrons
7011:in the
2459:is the
2451:is the
2409:is the
2298:), the
2000:of the
1793:on the
1718:on the
1392:, e.g.
1370:isospin
1060:Wilczek
1025:Wentzel
1000:Veltman
945:Shirkov
940:Shifman
930:Seiberg
915:Schwarz
895:Rubakov
820:Naimark
770:Lipatov
760:Lehmann
725:Kendall
615:Gelfand
610:Glashow
570:Feynman
550:Faddeev
545:Englert
515:Coleman
505:Cachazo
490:Brodsky
475:Bjorken
465:Berezin
455:Belavin
215:Anomaly
73:History
8292:Chiral
8240:Type I
8055:Yukawa
7976:Models
7847:. The
7744:
7718:
7695:
7599:
7537:
7529:
7472:
7323:
7293:
7261:
6969:
6950:
6927:
6914:
6888:
6869:
6847:
6793:
6761:
6724:
6697:
6679:
6611:
6580:
6549:
6531:
6343:
6332:
6298:
6284:
6278:
6248:
6237:
6228:
6198:
6187:
6162:
6143:
6105:
6043:
6032:
6023:
5993:
5982:
5946:
5935:
5900:
5876:
5818:
5807:
5798:
5768:
5757:
5748:
5719:
5685:
5679:
5667:
5643:
5534:
5516:
5500:
5487:
5468:
5446:
5407:
5380:
5343:
5317:
5292:
5214:
5193:
5166:
5109:being
5061:
5008:
4972:
4899:
4781:
4743:
4712:
4699:
4680:
4671:
4645:
4628:
4516:
4486:
4459:
4438:
4411:
4384:
4337:
4302:
4238:
4180:
4160:
4100:
4067:
4025:
4009:
3990:
3953:
3930:
3912:
3886:
3845:
3819:
3747:
3720:
3699:
3642:
3634:since
3619:
3436:
3407:
3374:
3328:
3282:
3263:
3252:holds
3230:
3181:
3155:
3108:
3082:
3055:
3039:
3020:
2983:
2955:
2902:
2882:
2844:
2806:
2785:
2772:
2766:
2721:
2692:
2674:
2656:
2637:
2554:
2526:
2483:
2453:vector
2439:
2421:
2388:
2370:
2357:
2351:
2332:
2313:
2286:
2270:
2250:
2234:
2212:
2178:
2165:
2146:
2122:
2097:
2051:
2027:
1988:
1975:
1952:
1838:
1734:where
1542:photon
1520:, and
1446:, and
1204:(i.e.
1115:Zumino
1080:Yukawa
1070:Witten
1065:Wilson
1055:Wigner
990:Tyutin
950:Skyrme
900:Ruelle
870:Plefka
865:Peskin
855:Parisi
815:Møller
805:Migdal
790:Maiani
785:Lüders
750:Landau
745:Kuraev
720:Källén
710:Jordan
695:Jackiw
635:Gribov
525:DeWitt
520:Dashen
510:Callan
480:Bleuer
450:Becchi
440:Anselm
8431:links
8404:links
8392:links
8312:NMSSM
8297:Fermi
8040:Soler
8010:Proca
7764:arXiv
7674:Books
7597:S2CID
7571:arXiv
7535:S2CID
7501:arXiv
7397:arXiv
6445:with
4550:(see
1765:with
1622:of a
1465:SU(3)
1399:SU(2)
1218:SU(3)
1110:Zuber
960:Stora
925:Segal
905:Salam
890:Proca
885:Popov
860:Pauli
840:Oehme
830:Neveu
825:Nambu
810:Mills
700:Jaffe
675:Hagen
670:Higgs
645:Gupta
640:Gross
625:Glimm
605:Furry
575:Fierz
565:Fermi
560:Fayet
555:Fadin
540:Dyson
530:Dirac
495:Brout
470:Bethe
430:Adler
209:Tools
8302:MSSM
8199:ABJM
8106:Toda
7742:ISBN
7716:ISBN
7693:ISBN
7527:PMID
7470:ISBN
7432:2017
7321:ISBN
7291:ISBN
7259:ISSN
7027:and
6832:and
4542:and
3737:dual
2200:are
1775:=1,
1771:=2,
1700:The
1616:Fock
1560:and
1558:weak
1546:U(1)
1454:and
1200:and
1166:and
1075:Yang
1045:Wick
1040:Weyl
1030:Wess
1010:Ward
715:Jost
665:Hepp
655:Haag
580:Fock
460:Bell
8255:11D
7651:hdl
7643:doi
7589:doi
7519:doi
7371:doi
7251:doi
6638:exp
6499:exp
6088:exp
5859:exp
5626:exp
4867:exp
4683:exp
4340:exp
4263:= 4
2469:SU(
1814:on
1802:of
1606:of
1179:SU(
1095:Zee
780:Low
755:Lee
8477::
8271:BF
7824:.
7812:,
7806:,
7740:.
7714:.
7710:.
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7631:44
7629:.
7625:.
7617:;
7595:.
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7565:.
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7411:^
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7213:.
7086:.
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5169:tr
4387:tr
3248:A
2215:tr
2030:tr
1878:tr
1787:=0
1779:=0
1614:,
1590:.
1500:,
1442:,
1257:.
7890:e
7883:t
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7033:σ
6994:g
6964:c
6961:b
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6930:.
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6918:c
6906:b
6902:A
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6880:b
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6782:y
6779:(
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6758:)
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6684:d
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6666:d
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6645:(
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6571:(
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6314:(
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6219:x
6216:(
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6130:d
6120:4
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6095:(
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5866:(
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5831:(
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5676:i
5664:x
5659:4
5654:d
5646:g
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5610:,
5595:,
5592:j
5589:[
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5559:ξ
5555:c
5530:)
5524:c
5520:c
5508:b
5504:A
5482:c
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5476:a
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5463:a
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5443:g
5440:+
5435:a
5431:c
5402:a
5392:c
5384:(
5377:x
5372:4
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5356:=
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5312:2
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5295:(
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5253:=
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5210:)
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5176:(
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5130:=
5125:F
5121:S
5089:}
5084:]
5080:)
5077:x
5074:(
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5052:(
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5021:(
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4851:]
4847:A
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4805:,
4802:c
4795:[
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4775:+
4771:]
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4754:f
4751:g
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4733:]
4729:A
4726:,
4723:A
4716:[
4707:F
4703:S
4696:i
4691:{
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4674:c
4667:d
4663:[
4660:]
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4641:d
4637:[
4634:]
4631:A
4624:d
4620:[
4614:=
4607:]
4601:,
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4583:j
4580:[
4577:Z
4519:,
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4505:x
4502:(
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4477:(
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4327:d
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4314:]
4311:j
4308:[
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4261:D
4241:.
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4224:4
4218:D
4214:(
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4205:[
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4197:]
4192:2
4188:g
4184:[
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4118:[
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4110:]
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4104:[
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4046:=
4041:c
4029:F
4017:b
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3807:F
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3604:]
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3124:=
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2937:(
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2918:=
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2863:=
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2725:[
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2575:=
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2511:n
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2498:c
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2326:=
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2221:(
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2110:T
2105:[
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2006:a
1983:a
1979:T
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1860:=
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1816:ℝ
1804:ℝ
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