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that satisfy the equation are thought of as being on shell, though the classical theory does not allow negative values for the energy of a particle. This is because the propagator incorporates into one expression the cases in which the particle carries energy in one direction, and in which its
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1646:{\displaystyle \alpha ^{\mu }\partial _{\mu }{\mathcal {L}}={\frac {\partial {\mathcal {L}}}{\partial \phi }}\alpha ^{\mu }\partial _{\mu }\phi +{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\nu }\phi )}}\alpha ^{\mu }\partial _{\mu }\partial _{\nu }\phi }
2110:{\displaystyle \partial _{\mu }{\mathcal {L}}=\partial _{\nu }{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\nu }\phi )}}\partial _{\mu }\phi +{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\nu }\phi )}}\partial _{\mu }\partial _{\nu }\phi }
2126:
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1930:{\displaystyle \partial _{\mu }{\mathcal {L}}={\frac {\partial {\mathcal {L}}}{\partial \phi }}\partial _{\mu }\phi +{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\nu }\phi )}}\partial _{\mu }\partial _{\nu }\phi }
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1379:{\displaystyle \delta {\mathcal {L}}={\frac {\partial {\mathcal {L}}}{\partial \phi }}\delta \phi +{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi )}}\delta (\partial _{\mu }\phi )}
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1944:, since it is true for any fields configuration regardless of whether it respects the equations of motion (in this case, the EulerâLagrange equation given above). However, we can derive an
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2237:{\displaystyle \partial _{\nu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\nu }\phi )}}\partial _{\mu }\phi -\delta _{\mu }^{\nu }{\mathcal {L}}\right)=0}
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1107:{\displaystyle \partial _{\mu }{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi )}}={\frac {\partial {\mathcal {L}}}{\partial \phi }}}
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are in general allowed to be off shell, but the amplitude for the process will diminish depending on how far off shell they are. This is because the
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742:-dependence of the propagator is determined by the four-momenta of the incoming and outgoing particles. The propagator typically has
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of the virtual particle is the difference between the four-momenta of the incoming and outgoing particles.
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Arkani-Hamed, N. (Dec 21, 2012). "Scattering
Amplitudes and the Positive Grassmannian".
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2345:, which is only conserved on shell, that is, if the equations of motion are satisfied.
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of a particle. The equation for the mass shell is also often written in terms of the
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1247:{\displaystyle \delta {\mathcal {L}}=\alpha ^{\mu }\partial _{\mu }{\mathcal {L}}}
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Configurations of a system that do or do not satisfy classical equations of motion
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This is an instance of
Noether's theorem. Here, the conserved quantity is the
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Points on the hyperboloid surface (the "shell") are solutions to the equation.
1473:{\displaystyle \delta (\partial _{\mu }\phi )=\partial _{\mu }(\delta \phi )}
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1760:{\displaystyle \alpha ^{\mu }=(\epsilon ,0,...,0),(0,\epsilon ,...,0),...}
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1003:{\displaystyle S=\int d^{D}x{\mathcal {L}}(\phi ,\partial _{\mu }\phi )}
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varying the field and its derivative and setting the variation to zero
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carries energy in the other direction; negative and positive on-shell
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under the infinitesimal transformation. On the other hand, by
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equation by simply substituting the EulerâLagrange equation:
140:, configurations of a physical system that satisfy classical
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The EulerâLagrange equation for this action can be found by
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321:{\displaystyle E^{2}-|{\vec {p}}\,|^{2}c^{2}=m_{0}^{2}c^{4}}
1166:{\displaystyle x^{\mu }\rightarrow x^{\mu }+\alpha ^{\mu }}
926:{\displaystyle {\mathcal {L}}(\phi ,\partial _{\mu }\phi )}
191:
regarding differentiable symmetries of physical action and
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then simply represent opposing flows of positive energy.
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is a scalar, and so will infinitesimally transform as
517:{\displaystyle p^{\mu }p_{\mu }\equiv p^{2}=m_{0}^{2}}
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When speaking of the propagator, negative values for
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are termed off shell because they do not satisfy the
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Since this has to hold for independent translations
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2331:{\displaystyle \partial _{\nu }T^{\nu }{}_{\mu }=0}
585:The four-momentum of an exchanged virtual particle
60:. Unsourced material may be challenged and removed.
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1940:This is an example of an equation that holds
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707:Virtual particles corresponding to internal
524:. In the literature, one may also encounter
2389:Perimeter Institute for Theoretical Physics
836:. Unsourced material may be challenged and
582:if the metric signature used is (â,+,+,+).
575:{\displaystyle p^{\mu }p_{\mu }=-m_{0}^{2}}
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1117:Now, consider an infinitesimal spacetime
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2385:"A Deeper Dive: On-Shell and Off-Shell"
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179:formulation, extremal solutions to the
2603:Two-dimensional conformal field theory
152:); while those that do not are called
1409:{\displaystyle \delta {\mathcal {L}}}
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868:An example comes from considering a
834:adding citations to reliable sources
58:adding citations to reliable sources
2425:"Are virtual particles less real?"
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2276:{\displaystyle T^{\nu }{}_{\mu }}
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2383:Cachazo, Freddy (Dec 21, 2012).
806:
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670:{\displaystyle q^{2}=m_{X}^{2}}
334:massâenergy equivalence formula
45:needs additional citations for
2488:. Cambridge University Press,
2365:. Cambridge University Press,
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195:is another on-shell theorem.
187:give the on-shell equations.
211:Mass shell is a synonym for
3062:Quantum information science
3133:
378:{\displaystyle {\vec {p}}}
3087:
1173:. The Lagrangian density
356:in terms of the momentum
163:In quantum field theory,
697:{\displaystyle q_{\mu }}
625:{\displaystyle q_{\mu }}
185:EulerâLagrange equations
169:energyâmomentum relation
69:"On shell and off shell"
2758:2D free massless scalar
2651:Quantum electrodynamics
2578:QFT in curved spacetime
2486:Modern particle physics
2363:Modern particle physics
336:which gives the energy
3079:Quantum thermodynamics
3003:On shell and off shell
2998:Loop quantum cosmology
2840:N = 4 super YangâMills
2799:N = 1 super YangâMills
2666:Scalar electrodynamics
2656:Quantum chromodynamics
2558:Conformal field theory
2534:Quantum field theories
2423:Jaeger, Gregg (2019).
2332:
2277:
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2120:We can write this as:
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677:. The four-momentum
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3052:Quantum hydrodynamics
3047:Quantum hadrodynamics
2671:Scalar chromodynamics
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1767:, we may "divide" by
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1258:, we have in general
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735:{\displaystyle q^{2}}
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408:{\displaystyle m_{0}}
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183:are on shell and the
181:variational principle
175:for instance, in the
3117:Quantum field theory
3023:Quantum fluctuations
2993:Loop quantum gravity
2563:Lattice field theory
2484:Thomson, M. (2013).
2361:Thomson, M. (2013).
2343:stressâenergy tensor
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138:quantum field theory
54:improve this article
3057:Quantum information
2661:Quartic interaction
2444:2019Entrp..21..141J
2215:
746:on the mass shell.
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449:{\displaystyle c=1}
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173:classical mechanics
142:equations of motion
2943:NambuâJona-Lasinio
2871:Higher dimensional
2778:WessâZuminoâWitten
2568:Noncommutative QFT
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1389:Substituting for
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787:{\displaystyle E}
762:{\displaystyle E}
598:{\displaystyle X}
421:Einstein notation
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349:{\displaystyle E}
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193:conservation laws
189:Noether's theorem
165:virtual particles
146:on the mass shell
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16:(Redirected from
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3013:Quantum dynamics
2686:YangâMillsâHiggs
2641:Non-linear sigma
2631:EulerâHeisenberg
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2598:Topological QFT
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1500:
1490:
1485:
1484:
1448:
1429:
1418:
1417:
1391:
1390:
1360:
1335:
1328:
1316:
1295:
1283:
1263:
1262:
1227:
1217:
1199:
1198:
1175:
1174:
1153:
1140:
1127:
1122:
1121:
1094:
1082:
1058:
1051:
1039:
1027:
1022:
1021:
984:
955:
941:
940:
907:
886:
885:
878:Minkowski space
862:
851:
845:
842:
827:
811:
800:
776:
775:
751:
750:
722:
717:
716:
713:Feynman diagram
684:
679:
678:
639:
634:
633:
612:
607:
606:
587:
586:
541:
531:
526:
525:
486:
473:
463:
458:
457:
432:
431:
395:
390:
389:
358:
357:
338:
337:
308:
280:
268:
237:
232:
231:
201:
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
3130:
3128:
3120:
3119:
3109:
3108:
3102:
3101:
3088:
3085:
3084:
3082:
3081:
3076:
3071:
3070:
3069:
3059:
3054:
3049:
3044:
3043:
3042:
3032:
3031:
3030:
3020:
3015:
3010:
3005:
3000:
2995:
2990:
2985:
2980:
2978:Casimir effect
2974:
2972:
2968:
2967:
2964:
2963:
2961:
2960:
2955:
2953:Standard Model
2950:
2945:
2940:
2935:
2930:
2924:
2922:
2918:
2917:
2915:
2914:
2909:
2903:
2901:
2897:
2896:
2894:
2893:
2888:
2883:
2878:
2873:
2868:
2863:
2858:
2852:
2850:
2846:
2845:
2843:
2842:
2837:
2832:
2826:
2824:
2823:Superconformal
2820:
2819:
2817:
2816:
2811:
2806:
2804:SeibergâWitten
2801:
2796:
2790:
2788:
2787:Supersymmetric
2784:
2783:
2781:
2780:
2775:
2770:
2765:
2760:
2754:
2752:
2748:
2747:
2745:
2744:
2739:
2734:
2729:
2724:
2719:
2714:
2709:
2703:
2701:
2697:
2696:
2694:
2693:
2688:
2683:
2678:
2673:
2668:
2663:
2658:
2653:
2648:
2643:
2638:
2633:
2628:
2622:
2620:
2613:
2609:
2608:
2606:
2605:
2600:
2595:
2590:
2585:
2580:
2575:
2570:
2565:
2560:
2555:
2550:
2544:
2542:
2538:
2537:
2532:
2530:
2529:
2522:
2515:
2507:
2499:
2498:
2494:978-1107034266
2477:
2415:
2394:
2375:
2373:, pp. 117â119.
2371:978-1107034266
2353:
2352:
2350:
2347:
2339:
2338:
2327:
2324:
2319:
2310:
2306:
2300:
2296:
2270:
2261:
2257:
2245:
2244:
2233:
2230:
2226:
2220:
2213:
2208:
2204:
2200:
2197:
2192:
2188:
2181:
2178:
2173:
2169:
2165:
2162:
2155:
2150:
2143:
2137:
2133:
2118:
2117:
2106:
2101:
2097:
2091:
2087:
2080:
2077:
2072:
2068:
2064:
2061:
2054:
2049:
2043:
2040:
2035:
2031:
2024:
2021:
2016:
2012:
2008:
2005:
1998:
1993:
1985:
1981:
1977:
1972:
1965:
1961:
1938:
1937:
1926:
1921:
1917:
1911:
1907:
1900:
1897:
1892:
1888:
1884:
1881:
1874:
1869:
1863:
1860:
1855:
1851:
1844:
1841:
1834:
1829:
1823:
1818:
1811:
1807:
1781:
1777:
1756:
1753:
1750:
1747:
1744:
1741:
1738:
1735:
1732:
1729:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1684:
1681:
1678:
1675:
1670:
1666:
1654:
1653:
1642:
1637:
1633:
1627:
1623:
1617:
1613:
1606:
1603:
1598:
1594:
1590:
1587:
1580:
1575:
1569:
1566:
1561:
1557:
1551:
1547:
1540:
1537:
1530:
1525:
1519:
1514:
1507:
1503:
1497:
1493:
1469:
1466:
1463:
1460:
1455:
1451:
1447:
1444:
1441:
1436:
1432:
1428:
1425:
1403:
1398:
1387:
1386:
1375:
1372:
1367:
1363:
1359:
1356:
1350:
1347:
1342:
1338:
1334:
1331:
1324:
1319:
1313:
1310:
1307:
1301:
1298:
1291:
1286:
1280:
1275:
1270:
1241:
1234:
1230:
1224:
1220:
1216:
1211:
1206:
1184:
1160:
1156:
1152:
1147:
1143:
1139:
1134:
1130:
1115:
1114:
1100:
1097:
1090:
1085:
1079:
1073:
1070:
1065:
1061:
1057:
1054:
1047:
1042:
1034:
1030:
1011:
1010:
999:
996:
991:
987:
983:
980:
977:
972:
967:
962:
958:
954:
951:
948:
922:
919:
914:
910:
906:
903:
900:
895:
864:
863:
814:
812:
805:
799:
796:
783:
758:
729:
725:
691:
687:
664:
659:
655:
651:
646:
642:
619:
615:
594:
569:
564:
560:
556:
553:
548:
544:
538:
534:
511:
506:
502:
498:
493:
489:
485:
480:
476:
470:
466:
445:
442:
439:
429:speed of light
402:
398:
371:
368:
345:
330:
329:
315:
311:
305:
300:
296:
292:
287:
283:
277:
272:
263:
260:
253:
249:
244:
240:
215:, meaning the
200:
197:
128:
127:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3129:
3118:
3115:
3114:
3112:
3099:
3091:
3086:
3080:
3077:
3075:
3074:Quantum logic
3072:
3068:
3065:
3064:
3063:
3060:
3058:
3055:
3053:
3050:
3048:
3045:
3041:
3038:
3037:
3036:
3033:
3029:
3026:
3025:
3024:
3021:
3019:
3016:
3014:
3011:
3009:
3008:Quantum chaos
3006:
3004:
3001:
2999:
2996:
2994:
2991:
2989:
2986:
2984:
2983:Cosmic string
2981:
2979:
2976:
2975:
2973:
2969:
2959:
2956:
2954:
2951:
2949:
2946:
2944:
2941:
2939:
2936:
2934:
2931:
2929:
2926:
2925:
2923:
2919:
2913:
2910:
2908:
2905:
2904:
2902:
2898:
2892:
2889:
2887:
2884:
2882:
2879:
2877:
2874:
2872:
2869:
2867:
2864:
2862:
2859:
2857:
2856:Pure 4D N = 1
2854:
2853:
2851:
2847:
2841:
2838:
2836:
2833:
2831:
2828:
2827:
2825:
2821:
2815:
2812:
2810:
2807:
2805:
2802:
2800:
2797:
2795:
2792:
2791:
2789:
2785:
2779:
2776:
2774:
2771:
2769:
2766:
2764:
2761:
2759:
2756:
2755:
2753:
2749:
2743:
2740:
2738:
2737:ThirringâWess
2735:
2733:
2730:
2728:
2725:
2723:
2720:
2718:
2715:
2713:
2712:BulloughâDodd
2710:
2708:
2707:2D YangâMills
2705:
2704:
2702:
2698:
2692:
2689:
2687:
2684:
2682:
2679:
2677:
2674:
2672:
2669:
2667:
2664:
2662:
2659:
2657:
2654:
2652:
2649:
2647:
2644:
2642:
2639:
2637:
2634:
2632:
2629:
2627:
2624:
2623:
2621:
2617:
2614:
2610:
2604:
2601:
2599:
2596:
2594:
2591:
2589:
2586:
2584:
2583:String theory
2581:
2579:
2576:
2574:
2571:
2569:
2566:
2564:
2561:
2559:
2556:
2554:
2553:Axiomatic QFT
2551:
2549:
2548:Algebraic QFT
2546:
2545:
2543:
2539:
2535:
2528:
2523:
2521:
2516:
2514:
2509:
2508:
2505:
2495:
2491:
2487:
2481:
2478:
2473:
2469:
2464:
2459:
2454:
2449:
2445:
2441:
2437:
2433:
2426:
2419:
2416:
2410:
2405:
2398:
2395:
2390:
2386:
2379:
2376:
2372:
2368:
2364:
2358:
2355:
2348:
2346:
2344:
2325:
2322:
2317:
2308:
2304:
2298:
2286:
2285:
2284:
2268:
2259:
2255:
2231:
2228:
2224:
2211:
2206:
2202:
2198:
2195:
2190:
2176:
2171:
2141:
2135:
2123:
2122:
2121:
2104:
2099:
2089:
2075:
2070:
2041:
2038:
2033:
2019:
2014:
1983:
1975:
1963:
1951:
1950:
1949:
1947:
1943:
1924:
1919:
1909:
1895:
1890:
1861:
1858:
1853:
1842:
1821:
1809:
1797:
1796:
1795:
1779:
1775:
1754:
1751:
1748:
1745:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1709:
1703:
1700:
1697:
1694:
1691:
1688:
1685:
1682:
1679:
1673:
1668:
1664:
1640:
1635:
1625:
1615:
1611:
1601:
1596:
1567:
1564:
1559:
1549:
1545:
1538:
1517:
1505:
1495:
1491:
1483:
1482:
1481:
1464:
1461:
1453:
1445:
1439:
1434:
1423:
1396:
1370:
1365:
1354:
1345:
1340:
1311:
1308:
1305:
1299:
1278:
1268:
1261:
1260:
1259:
1257:
1232:
1222:
1218:
1214:
1204:
1158:
1154:
1150:
1145:
1141:
1132:
1128:
1120:
1098:
1077:
1068:
1063:
1032:
1020:
1019:
1018:
1016:
994:
989:
981:
978:
965:
960:
956:
952:
949:
946:
939:
938:
937:
936:
917:
912:
904:
901:
883:
880:. Consider a
879:
876:-dimensional
875:
871:
860:
857:
849:
846:December 2019
839:
835:
831:
825:
824:
820:
815:This section
813:
809:
804:
803:
797:
795:
781:
773:
756:
747:
745:
744:singularities
727:
723:
714:
710:
705:
689:
685:
662:
657:
653:
649:
644:
640:
617:
613:
592:
583:
567:
562:
558:
554:
551:
546:
542:
536:
532:
509:
504:
500:
496:
491:
487:
483:
478:
474:
468:
464:
443:
440:
437:
430:
426:
422:
418:
417:four-momentum
400:
396:
388:
366:
343:
335:
313:
309:
303:
298:
294:
290:
285:
281:
275:
258:
247:
242:
238:
230:
229:
228:
226:
222:
218:
214:
205:
198:
196:
194:
190:
186:
182:
178:
174:
170:
166:
161:
159:
155:
151:
147:
143:
139:
135:
124:
121:
113:
110:November 2014
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: â
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
3089:
3018:Quantum foam
3002:
2958:Stueckelberg
2912:ChernâSimons
2849:Supergravity
2588:Supergravity
2573:Gauge theory
2485:
2480:
2435:
2431:
2418:
2397:
2388:
2378:
2362:
2357:
2340:
2246:
2119:
1945:
1941:
1939:
1655:
1388:
1116:
1012:
873:
870:scalar field
867:
852:
843:
828:Please help
816:
798:Scalar field
772:antiparticle
748:
706:
632:, with mass
584:
331:
212:
210:
162:
157:
153:
149:
145:
131:
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
2900:Topological
2814:WessâZumino
2727:Sine-Gordon
2717:GrossâNeveu
2626:BornâInfeld
2593:Thermal QFT
2283:, we have:
1794:and write:
1119:translation
709:propagators
217:hyperboloid
144:are called
2681:YangâMills
2438:(2): 141.
2349:References
1017:, and is:
199:Mass shell
80:newspapers
3090:See also:
2809:Super QCD
2763:Liouville
2751:Conformal
2722:Schwinger
2409:1212.5605
2318:μ
2309:ν
2299:ν
2295:∂
2269:μ
2260:ν
2212:ν
2207:μ
2203:δ
2199:−
2196:ϕ
2191:μ
2187:∂
2177:ϕ
2172:ν
2168:∂
2161:∂
2149:∂
2136:ν
2132:∂
2105:ϕ
2100:ν
2096:∂
2090:μ
2086:∂
2076:ϕ
2071:ν
2067:∂
2060:∂
2048:∂
2039:ϕ
2034:μ
2030:∂
2020:ϕ
2015:ν
2011:∂
2004:∂
1992:∂
1984:ν
1980:∂
1964:μ
1960:∂
1942:off shell
1925:ϕ
1920:ν
1916:∂
1910:μ
1906:∂
1896:ϕ
1891:ν
1887:∂
1880:∂
1868:∂
1859:ϕ
1854:μ
1850:∂
1843:ϕ
1840:∂
1828:∂
1810:μ
1806:∂
1780:μ
1776:α
1722:ϵ
1680:ϵ
1669:μ
1665:α
1641:ϕ
1636:ν
1632:∂
1626:μ
1622:∂
1616:μ
1612:α
1602:ϕ
1597:ν
1593:∂
1586:∂
1574:∂
1565:ϕ
1560:μ
1556:∂
1550:μ
1546:α
1539:ϕ
1536:∂
1524:∂
1506:μ
1502:∂
1496:μ
1492:α
1465:ϕ
1462:δ
1454:μ
1450:∂
1440:ϕ
1435:μ
1431:∂
1424:δ
1397:δ
1371:ϕ
1366:μ
1362:∂
1355:δ
1346:ϕ
1341:μ
1337:∂
1330:∂
1318:∂
1309:ϕ
1306:δ
1300:ϕ
1297:∂
1285:∂
1269:δ
1233:μ
1229:∂
1223:μ
1219:α
1205:δ
1159:μ
1155:α
1146:μ
1138:→
1133:μ
1099:ϕ
1096:∂
1084:∂
1069:ϕ
1064:μ
1060:∂
1053:∂
1041:∂
1033:μ
1029:∂
995:ϕ
990:μ
986:∂
979:ϕ
953:∫
918:ϕ
913:μ
909:∂
902:ϕ
884:given by
817:does not
690:μ
618:μ
555:−
547:μ
537:μ
484:≡
479:μ
469:μ
387:rest mass
370:→
262:→
248:−
158:off shell
3111:Category
2886:Type IIB
2881:Type IIA
2866:4D N = 8
2861:4D N = 1
2830:6D (2,0)
2794:4D N = 1
2773:Polyakov
2732:Thirring
2541:Theories
2496:, p.119.
2472:33266857
1946:on shell
385:and the
225:momentum
150:on shell
18:On-shell
2988:History
2971:Related
2768:Minimal
2619:Regular
2463:7514619
2440:Bibcode
2432:Entropy
838:removed
823:sources
223:–
134:physics
94:scholar
2928:Chiral
2876:Type I
2691:Yukawa
2612:Models
2492:
2470:
2460:
2369:
935:action
933:. The
221:energy
177:action
96:
89:
82:
75:
67:
3067:links
3040:links
3028:links
2948:NMSSM
2933:Fermi
2676:Soler
2646:Proca
2428:(PDF)
2404:arXiv
711:in a
456:, as
423:with
419:; in
101:JSTOR
87:books
2938:MSSM
2835:ABJM
2742:Toda
2490:ISBN
2468:PMID
2367:ISBN
821:any
819:cite
332:the
73:news
2891:11D
2458:PMC
2448:doi
872:in
832:by
605:is
219:in
160:).
132:In
56:by
3113::
2907:BF
2466:.
2456:.
2446:.
2436:21
2434:.
2430:.
2387:.
2526:e
2519:t
2512:v
2474:.
2450::
2442::
2412:.
2406::
2391:.
2326:0
2323:=
2305:T
2256:T
2232:0
2229:=
2225:)
2219:L
2180:)
2164:(
2154:L
2142:(
2079:)
2063:(
2053:L
2042:+
2023:)
2007:(
1997:L
1976:=
1971:L
1899:)
1883:(
1873:L
1862:+
1833:L
1822:=
1817:L
1755:.
1752:.
1749:.
1746:,
1743:)
1740:0
1737:,
1734:.
1731:.
1728:.
1725:,
1719:,
1716:0
1713:(
1710:,
1707:)
1704:0
1701:,
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