1221:, which is an excellent approximation when it comes to describing the behavior of microscopic particles in weak gravitational fields like those found on Earth. In order to describe situations in which gravity is strong enough to influence (quantum) matter, yet not strong enough to require quantization itself, physicists have formulated quantum field theories in curved spacetime. These theories rely on general relativity to describe a curved background spacetime, and define a generalized quantum field theory to describe the behavior of quantum matter within that spacetime.
33:
3652:
3045:
1380:. QFT in curved spacetime is expected to be a viable approximation to the theory of quantum gravity when spacetime curvature is not significant on the Planck scale. However, the fact that the true theory of quantum gravity remains unknown means that the precise criteria for when QFT on curved spacetime is a good approximation are also unknown.
2440:
One expects it to be a good approximation to full quantum gravity provided the typical frequencies of the gravitational background are very much less than the Planck frequency and provided, with a suitable measure for energy, the energy of created particles is very much less than the energy of the
2411:
In particular, due to the weakness of gravitational forces, the back reaction of the spacetime metric to the energy momentum tensor of the quantum fields may be neglected, in a first approximation, and one is left with the problem of quantum field theory on
Lorentzian manifolds. Surprisingly, this
1315:
procedure to the theory of quantum fields developed on curved backgrounds. Several rigorous results concerning QFT in the presence of a black hole have been obtained. In particular the algebraic approach allows one to deal with the problems mentioned above arising from the absence of a preferred
1182:. This theory uses a semi-classical approach; it treats spacetime as a fixed, classical background, while giving a quantum-mechanical description of the matter and energy propagating through that spacetime. A general prediction of this theory is that particles can be created by time-dependent
1448:
Kay, B. S. (2023). "Quantum Field Theory in Curved
Spacetime (2nd edition) (article prepared for the second edition of the Encyclopaedia of Mathematical Physics, edited by M. Bojowald and R. J. Szabo, to be published by Elsevier)".
1348:
where an accelerating observer observes black body radiation. Other prediction of quantum fields in curved spaces include, for example, the radiation emitted by a particle moving along a geodesic and the interaction of
2503:
Quantum field theory in curved spacetime is a semiclassical approximation to quantum gravity theory, where the curved background spacetime is treated classically, while the matter fields in the curved spacetime are
1246:. Even then, as in flat spacetime, the asymptotic particle interpretation depends on the observer (i.e., different observers may measure different numbers of asymptotic particles on a given spacetime).
1311:
has been implemented in order to include an algebraic version of quantum field theory in curved spacetime. Indeed, the viewpoint of local quantum physics is suitable to generalize the
2412:
seemingly modest approach leads to far-reaching conceptual and mathematical problems and to spectacular predictions, the most famous one being the
Hawking radiation of black holes.
1292:
correspond to negative frequencies. This is why a state which looks like a vacuum to one observer cannot look like a vacuum state to another observer; it could even appear as a
2379:
Quantum field theory on curved spacetime, which might be considered as an intermediate step towards quantum gravity, already has no distinguished particle interpretation.
1316:
reference vacuum state, the absence of a natural notion of particle and the appearance of unitarily inequivalent representations of the algebra of observables.
282:
3082:
1503:"Lectures on quantum field theory in curved spacetime (Lecture Note 39/2008 Max Planck Institute for Mathematics in the Natural Sciences (2008))"
1149:
3505:
3315:
2137:"Do static sources respond to massive scalar particles from the Hawking radiation as uniformly accelerated ones do in the inertial vacuum?"
3397:
3356:
202:
1265:. This is because a mode decomposition of a field into positive and negative frequency modes is not invariant under diffeomorphisms. If
2633:
242:
1325:
1193:), or by time-independent gravitational fields that contain horizons. The most famous example of the latter is the phenomenon of
3160:
2739:
3495:
3264:
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1650:
1558:
1889:"Scalar radiation from a radially infalling source into a Schwarzschild black hole in the framework of quantum field theory"
3387:
3075:
1007:
3125:
1519:
Khavkine, Igor; Moretti, Valter (2015), Brunetti, Romeo; Dappiaggi, Claudio; Fredenhagen, Klaus; Yngvason, Jakob (eds.),
1289:
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in a fixed classical spacetime and among other thing predicts the creation of particles by time-varying spacetimes and
1142:
3655:
3048:
2548:
2529:
2404:
1749:
1485:
2272:; Parikh, Maulik K.; van der Schaar, Jan Pieter (28 April 2006). "Universal correction to the inflationary vacuum".
1832:"Corrigendum: Scalar radiation emitted from a source rotating around a black hole (2000 Class. Quantum Grav. 17 19)"
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A two-page chart outline of the basic principles governing the behavior of quantum fields in general relativity.
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2204:"Interaction of Hawking radiation with static sources in de Sitter and Schwarzschild--de Sitter spacetimes"
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The theory of quantum field theory in curved spacetime may be considered as an intermediate step towards
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252:
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1239:), can the notion of incoming and outgoing particle be recovered, thus enabling one to define an
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in QFT, so merely formulating QFT in curved spacetime is not a true theory of quantum gravity.
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1967:
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1527:, Mathematical Physics Studies, Cham: Springer International Publishing, pp. 191â251,
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24:
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8:
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2003:"Interaction of Hawking radiation with static sources outside a Schwarzschild black hole"
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2455:"Simulating quantum field theory in curved spacetime with quantum many-body systems"
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2331:"Quantum Gravity from the Point of View of Locally Covariant Quantum Field Theory"
1521:"Algebraic QFT in Curved Spacetime and Quasifree Hadamard States: An Introduction"
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Quantum Field Theory on Curved
Spacetimes: Concepts and Mathematical Foundations
1979:
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1232:. Only in certain situations, such as in asymptotically flat spacetimes (zero
1228:, on curved spacetimes quantum fields lose their interpretation as asymptotic
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or ground state canonically. The concept of a vacuum is not invariant under
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3203:
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2706:
2701:
2696:
2671:
2269:
2068:
Crispino, LuĂs C. B.; Higuchi, Atsushi; Matsas, George E. A. (1998-09-22).
1887:
Oliveira, Leandro A.; Crispino, LuĂs C. B.; Higuchi, Atsushi (2018-02-16).
1759:
1671:
Crispino, LuĂs C. B.; Higuchi, Atsushi; Matsas, George E. A. (2008-07-01).
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1944:
Brito, JoĂŁo P. B.; Bernar, Rafael P.; Crispino, LuĂs C. B. (2020-06-11).
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2246:
2179:
2112:
2045:
2001:
Higuchi, Atsushi; Matsas, George E. A.; Sudarsky, Daniel (1998-10-22).
1715:
1478:
Quantum field theory in curved space-time and black hole thermodynamics
1198:
747:
657:
522:
462:
134:
69:
2453:
Yang, Run-Qiu; Liu, Hui; Zhu, Shining; Luo, Le; Cai, Rong-Gen (2020).
2434:
2220:
2202:
Castiñeiras, J.; Costa e Silva, I. P.; Matsas, G. E. A. (2003-10-31).
2153:
2135:
Castiñeiras, J.; Costa e Silva, I. P.; Matsas, G. E. A. (2003-03-27).
2086:
2019:
1946:"Synchrotron geodesic radiation in Schwarzschild--de Sitter spacetime"
1791:
3366:
2979:
2847:
2759:
2731:
1775:"Scalar radiation emitted from a source rotating around a black hole"
1328:
in quantum field theory in curved spacetime geometry is known as the
1293:
2424:
Kay, Bernard S. (2006). "Quantum field theory in curved spacetime".
1830:
Crispino, L. C. B.; Higuchi, A.; Matsas, G. E. A. (September 2016).
3464:
2887:
2686:
2471:
1962:
1773:
Crispino, L. C. B.; Higuchi, A.; Matsas, G. E. A. (November 1999).
1455:
1240:
1187:
707:
2611:
2606:
Summary Chart of Intro Steps to
Quantum Fields in Curved Spacetime
2347:
1689:
1533:
2070:"Interaction of Hawking radiation and a static electric charge"
1258:
2201:
2134:
1356:
This formalism is also used to predict the primordial density
1744:. P. C. W. Davies. Cambridge : Cambridge University Press.
2324:
2268:
1344:. The latter can be understood as a manifestation of the
2441:
background gravitational field or of its matter sources.
1886:
2067:
2000:
1829:
1772:
1670:
16:
Extension of quantum field theory to curved spacetime
2541:
Aspects of quantum field theory in curved space-time
1943:
1371:
1249:Another observation is that unless the background
1280:of exp will contain negative frequencies even if
3666:
2557:
1168:quantum field theory in curved spacetime (QFTCS)
2428:. Academic Press (Elsevier). pp. 202â214.
1518:
2390:
3076:
2627:
2519:
2452:
1364:. These predictions are calculated using the
1143:
2262:
1336:. This approach studies the interaction of
3090:
3083:
3069:
2634:
2620:
2576:
2560:Introduction to Quantum Effects in Gravity
1589:"Particle Creation in Expanding Universes"
1525:Advances in Algebraic Quantum Field Theory
1288:correspond to positive frequencies, while
1150:
1136:
31:
2520:Birrell, N. D.; Davies, P. C. W. (1982).
2488:
2470:
2433:
2364:
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2111:
2085:
2044:
2018:
1961:
1912:
1855:
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1714:
1688:
1532:
1454:
2582:Quantum Field Theory in Curved Spacetime
1360:spectrum arising in different models of
2538:
1737:
1673:"The Unruh effect and its applications"
1625:
1500:
1276:) is a diffeomorphism, in general, the
3667:
3161:Two-dimensional conformal field theory
2335:Communications in Mathematical Physics
1637:, World Scientific, pp. 167â188,
1586:
1468:
1466:
3064:
2615:
2426:Encyclopedia of Mathematical Physics
1472:
1353:with particles outside black holes.
2641:
2558:Mukhanov, V.; Winitzki, S. (2007).
2423:
1463:
1447:
13:
2513:
1631:"Particle Creation by Black Holes"
14:
3691:
3656:Template:Quantum mechanics topics
3049:Template:Quantum mechanics topics
2599:
3651:
3650:
3044:
3043:
2765:Topological quantum field theory
2490:10.1103/PhysRevResearch.2.023107
1418:Topological quantum field theory
1372:Approximation to quantum gravity
1299:Since the end of the 1980s, the
2750:Jackiw–Teitelboim gravity
2446:
2417:
2384:
2318:
2195:
2128:
2061:
1994:
1937:
1893:The European Physical Journal C
1880:
1823:
1403:History of quantum field theory
1319:
2955:Causal dynamical triangulation
2823:Black hole information paradox
2584:. Cambridge University Press.
2522:Quantum fields in curved space
2274:Journal of High Energy Physics
1914:10.1140/epjc/s10052-018-5604-8
1848:10.1088/0264-9381/33/20/209502
1766:
1741:Quantum fields in curved space
1731:
1664:
1619:
1580:
1512:
1494:
1441:
1257:, there is no way to define a
331:Renormalization group equation
1:
2921:Wheeler–DeWitt equation
2329:; Rejzner, Katarzyna (2016).
2304:10.1088/1126-6708/2006/04/057
1836:Classical and Quantum Gravity
1779:Classical and Quantum Gravity
1434:
160:Spontaneous symmetry breaking
120:Symmetry in quantum mechanics
2843:Cosmic censorship hypothesis
2662:BatalinâVilkovisky formalism
7:
3620:Quantum information science
1980:10.1103/PhysRevD.101.124019
1551:10.1007/978-3-319-21353-8_5
1390:
1296:under suitable hypotheses.
1204:
10:
3696:
2939:Hartle–Hawking state
2838:Bousso's holographic bound
2818:Black hole complementarity
2238:10.1103/PhysRevD.68.084022
2171:10.1103/PhysRevD.67.067502
2104:10.1103/PhysRevD.58.084027
2037:10.1103/PhysRevD.58.104021
1801:10.1088/0264-9381/17/1/303
1643:10.1142/9789814539395_0011
1613:10.1103/PhysRevLett.21.562
1408:Local quantum field theory
1368:or modifications thereto.
1301:local quantum field theory
1213:, which form the basis of
155:Explicit symmetry breaking
3645:
3528:
3478:
3457:
3406:
3380:
3344:
3308:
3257:
3176:
3169:
3098:
3035:
3002:
2947:
2931:Euclidean quantum gravity
2929:
2908:Canonical quantum gravity
2906:
2873:
2866:
2858:Gravitational singularity
2828:Black-hole thermodynamics
2808:
2773:
2730:
2712:RyuâTakayanagi conjecture
2649:
2366:10.1007/s00220-016-2676-x
1707:10.1103/RevModPhys.80.787
1677:Reviews of Modern Physics
1635:Euclidean Quantum Gravity
1587:Parker, L. (1968-08-19).
1366:Bunch–Davies vacuum
321:BargmannâWigner equations
248:Path Integral Formulation
2985:Superfluid vacuum theory
2740:2+1D topological gravity
2459:Physical Review Research
1413:Statistical field theory
326:Schwinger-Dyson equation
3316:2D free massless scalar
3209:Quantum electrodynamics
3136:QFT in curved spacetime
2975:Noncommutative geometry
2722:WeinbergâWitten theorem
2717:Trans-Planckian problem
2539:Fulling, S. A. (1989).
1738:Birrell, N. D. (1982).
1593:Physical Review Letters
1501:Fewster, C. J. (2008).
356:Electroweak interaction
351:Quantum electrodynamics
316:WheelerâDeWitt equation
193:Background field method
3637:Quantum thermodynamics
3561:On shell and off shell
3556:Loop quantum cosmology
3398:N = 4 super YangâMills
3357:N = 1 super YangâMills
3224:Scalar electrodynamics
3214:Quantum chromodynamics
3116:Conformal field theory
3092:Quantum field theories
2657:AdS/CFT correspondence
1290:annihilation operators
1253:has a global timelike
1226:cosmological constants
1217:, are defined in flat
1211:quantum field theories
361:Quantum chromodynamics
218:Effective field theory
3610:Quantum hydrodynamics
3605:Quantum hadrodynamics
3229:Scalar chromodynamics
2883:Bosonic string theory
2795:Semiclassical gravity
2692:Holographic principle
2682:Gravitational anomaly
306:KleinâGordon equation
238:LSZ reduction formula
3675:Quantum field theory
3581:Quantum fluctuations
3551:Loop quantum gravity
3121:Lattice field theory
2916:Loop quantum gravity
2775:Quantum field theory
1184:gravitational fields
1172:quantum field theory
399:Theory of everything
233:Lattice field theory
203:Correlation function
25:Quantum field theory
3615:Quantum information
3219:Quartic interaction
2785:BunchâDavies vacuum
2777:in curved spacetime
2677:FaddeevâPopov ghost
2580:; Toms, D. (2009).
2481:2020PhRvR...2b3107Y
2395:(2009). "Preface".
2357:2016CMaPh.345..741B
2296:2006JHEP...04..057G
2230:2003PhRvD..68h4022C
2163:2003PhRvD..67f7502C
2096:1998PhRvD..58h4027C
2029:1998PhRvD..58j4021H
1972:2020PhRvD.101l4019B
1905:2018EPJC...78..133O
1699:2008RvMP...80..787C
1605:1968PhRvL..21..562P
1543:2014arXiv1412.5945K
1326:perturbation theory
1176:Minkowski spacetime
1170:is an extension of
1164:theoretical physics
378:Incomplete theories
3501:NambuâJona-Lasinio
3429:Higher dimensional
3336:WessâZuminoâWitten
3126:Noncommutative QFT
2970:Group field theory
2898:Superstring theory
2853:Firewall (physics)
2393:Fredenhagen, Klaus
2327:Fredenhagen, Klaus
1398:General relativity
1286:Creation operators
243:Partition function
170:Topological charge
90:General relativity
85:Special relativity
3662:
3661:
3524:
3523:
3058:
3057:
3015:Eternal inflation
3010:Quantum cosmology
2998:
2997:
2790:Hawking radiation
2755:Liouville gravity
2591:978-0-521-87787-9
2569:978-0-521-86834-1
2325:Brunetti, Romeo;
2208:Physical Review D
2141:Physical Review D
2074:Physical Review D
2007:Physical Review D
1950:Physical Review D
1652:978-981-02-0515-7
1560:978-3-319-21352-1
1428:Quantum spacetime
1351:Hawking radiation
1342:Hawking radiation
1278:Fourier transform
1195:Hawking radiation
1160:
1159:
223:Expectation value
198:BRST quantization
145:Poincaré symmetry
100:YangâMills theory
80:Quantum mechanics
3687:
3654:
3653:
3571:Quantum dynamics
3244:YangâMillsâHiggs
3199:Non-linear sigma
3189:EulerâHeisenberg
3174:
3173:
3085:
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2833:Bekenstein bound
2650:Central concepts
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2506:
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2415:
2414:
2391:BĂ€r, Christian;
2388:
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2381:
2368:
2350:
2322:
2316:
2315:
2289:
2270:Greene, Brian R.
2266:
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1659:
1623:
1617:
1616:
1584:
1578:
1577:
1576:
1575:
1536:
1516:
1510:
1509:
1507:
1498:
1492:
1491:
1470:
1461:
1460:
1458:
1445:
1423:Quantum geometry
1362:cosmic inflation
1303:approach due to
1270:
1180:curved spacetime
1152:
1145:
1138:
213:Effective action
140:Lorentz symmetry
65:Electromagnetism
35:
21:
20:
3695:
3694:
3690:
3689:
3688:
3686:
3685:
3684:
3680:Quantum gravity
3665:
3664:
3663:
3658:
3641:
3593:Quantum gravity
3520:
3479:Particle theory
3474:
3453:
3402:
3376:
3340:
3304:
3258:Low dimensional
3253:
3194:GinzburgâLandau
3165:
3156:Topological QFT
3094:
3089:
3059:
3054:
3053:
3031:
3020:FRW/CFT duality
2994:
2943:
2925:
2902:
2862:
2804:
2776:
2769:
2726:
2645:
2643:Quantum gravity
2640:
2602:
2592:
2570:
2551:
2532:
2516:
2514:Further reading
2511:
2510:
2451:
2447:
2422:
2418:
2407:
2389:
2385:
2323:
2319:
2267:
2263:
2200:
2196:
2133:
2129:
2066:
2062:
1999:
1995:
1942:
1938:
1885:
1881:
1828:
1824:
1771:
1767:
1752:
1736:
1732:
1669:
1665:
1657:
1655:
1653:
1624:
1620:
1585:
1581:
1573:
1571:
1561:
1517:
1513:
1505:
1499:
1495:
1488:
1471:
1464:
1446:
1442:
1437:
1432:
1393:
1383:Gravity is not
1378:quantum gravity
1374:
1334:quantum gravity
1322:
1313:renormalization
1268:
1263:diffeomorphisms
1219:Minkowski space
1207:
1191:pair production
1156:
1127:
1126:
1125:
1123:
417:
409:
408:
404:Quantum gravity
379:
371:
370:
366:Higgs mechanism
346:
336:
335:
311:Proca equations
296:
288:
287:
283:Wightman Axioms
268:Renormalization
228:Feynman diagram
183:
175:
174:
115:
105:
104:
55:
40:
38:Feynman diagram
17:
12:
11:
5:
3693:
3683:
3682:
3677:
3660:
3659:
3646:
3643:
3642:
3640:
3639:
3634:
3629:
3628:
3627:
3617:
3612:
3607:
3602:
3601:
3600:
3590:
3589:
3588:
3578:
3573:
3568:
3563:
3558:
3553:
3548:
3543:
3538:
3536:Casimir effect
3532:
3530:
3526:
3525:
3522:
3521:
3519:
3518:
3513:
3511:Standard Model
3508:
3503:
3498:
3493:
3488:
3482:
3480:
3476:
3475:
3473:
3472:
3467:
3461:
3459:
3455:
3454:
3452:
3451:
3446:
3441:
3436:
3431:
3426:
3421:
3416:
3410:
3408:
3404:
3403:
3401:
3400:
3395:
3390:
3384:
3382:
3381:Superconformal
3378:
3377:
3375:
3374:
3369:
3364:
3362:SeibergâWitten
3359:
3354:
3348:
3346:
3345:Supersymmetric
3342:
3341:
3339:
3338:
3333:
3328:
3323:
3318:
3312:
3310:
3306:
3305:
3303:
3302:
3297:
3292:
3287:
3282:
3277:
3272:
3267:
3261:
3259:
3255:
3254:
3252:
3251:
3246:
3241:
3236:
3231:
3226:
3221:
3216:
3211:
3206:
3201:
3196:
3191:
3186:
3180:
3178:
3171:
3167:
3166:
3164:
3163:
3158:
3153:
3148:
3143:
3138:
3133:
3128:
3123:
3118:
3113:
3108:
3102:
3100:
3096:
3095:
3088:
3087:
3080:
3073:
3065:
3056:
3055:
3052:
3051:
3037:
3036:
3033:
3032:
3030:
3029:
3028:
3027:
3022:
3017:
3006:
3004:
3000:
2999:
2996:
2995:
2993:
2992:
2990:Twistor theory
2987:
2982:
2977:
2972:
2967:
2962:
2957:
2951:
2949:
2945:
2944:
2942:
2941:
2935:
2933:
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2926:
2924:
2923:
2918:
2912:
2910:
2904:
2903:
2901:
2900:
2895:
2890:
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2879:
2877:
2868:
2864:
2863:
2861:
2860:
2855:
2850:
2845:
2840:
2835:
2830:
2825:
2820:
2814:
2812:
2806:
2805:
2803:
2802:
2797:
2792:
2787:
2781:
2779:
2771:
2770:
2768:
2767:
2762:
2757:
2752:
2747:
2742:
2736:
2734:
2728:
2727:
2725:
2724:
2719:
2714:
2709:
2704:
2699:
2694:
2689:
2684:
2679:
2674:
2669:
2664:
2659:
2653:
2651:
2647:
2646:
2639:
2638:
2631:
2624:
2616:
2610:
2609:
2601:
2600:External links
2598:
2597:
2596:
2590:
2574:
2568:
2555:
2549:
2536:
2530:
2515:
2512:
2509:
2508:
2445:
2416:
2405:
2383:
2341:(3): 741â779.
2317:
2287:hep-th/0512243
2261:
2194:
2127:
2060:
2013:(10): 104021.
1993:
1956:(12): 124019.
1936:
1879:
1842:(20): 209502.
1822:
1765:
1750:
1730:
1683:(3): 787â838.
1663:
1651:
1629:(1993-05-01),
1627:Hawking, S. W.
1618:
1599:(8): 562â564.
1579:
1559:
1511:
1493:
1486:
1462:
1439:
1438:
1436:
1433:
1431:
1430:
1425:
1420:
1415:
1410:
1405:
1400:
1394:
1392:
1389:
1385:renormalizable
1373:
1370:
1338:quantum fields
1321:
1318:
1309:Daniel Kastler
1255:Killing vector
1215:standard model
1206:
1203:
1158:
1157:
1155:
1154:
1147:
1140:
1132:
1129:
1128:
1121:
1120:
1115:
1110:
1105:
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1095:
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905:
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880:
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865:
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855:
850:
845:
840:
835:
830:
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820:
815:
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800:
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401:
396:
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372:
369:
368:
363:
358:
353:
347:
344:Standard Model
342:
341:
338:
337:
334:
333:
328:
323:
318:
313:
308:
303:
301:Dirac equation
297:
294:
293:
290:
289:
286:
285:
280:
278:Wick's theorem
275:
270:
265:
263:Regularization
260:
255:
250:
245:
240:
235:
230:
225:
220:
215:
210:
205:
200:
195:
190:
184:
181:
180:
177:
176:
173:
172:
167:
165:Noether charge
162:
157:
152:
150:Gauge symmetry
147:
142:
137:
132:
127:
122:
116:
111:
110:
107:
106:
103:
102:
97:
92:
87:
82:
77:
72:
67:
62:
56:
53:
52:
49:
48:
42:
41:
36:
28:
27:
15:
9:
6:
4:
3:
2:
3692:
3681:
3678:
3676:
3673:
3672:
3670:
3657:
3649:
3644:
3638:
3635:
3633:
3632:Quantum logic
3630:
3626:
3623:
3622:
3621:
3618:
3616:
3613:
3611:
3608:
3606:
3603:
3599:
3596:
3595:
3594:
3591:
3587:
3584:
3583:
3582:
3579:
3577:
3574:
3572:
3569:
3567:
3566:Quantum chaos
3564:
3562:
3559:
3557:
3554:
3552:
3549:
3547:
3544:
3542:
3541:Cosmic string
3539:
3537:
3534:
3533:
3531:
3527:
3517:
3514:
3512:
3509:
3507:
3504:
3502:
3499:
3497:
3494:
3492:
3489:
3487:
3484:
3483:
3481:
3477:
3471:
3468:
3466:
3463:
3462:
3460:
3456:
3450:
3447:
3445:
3442:
3440:
3437:
3435:
3432:
3430:
3427:
3425:
3422:
3420:
3417:
3415:
3414:Pure 4D N = 1
3412:
3411:
3409:
3405:
3399:
3396:
3394:
3391:
3389:
3386:
3385:
3383:
3379:
3373:
3370:
3368:
3365:
3363:
3360:
3358:
3355:
3353:
3350:
3349:
3347:
3343:
3337:
3334:
3332:
3329:
3327:
3324:
3322:
3319:
3317:
3314:
3313:
3311:
3307:
3301:
3298:
3296:
3295:ThirringâWess
3293:
3291:
3288:
3286:
3283:
3281:
3278:
3276:
3273:
3271:
3270:BulloughâDodd
3268:
3266:
3265:2D YangâMills
3263:
3262:
3260:
3256:
3250:
3247:
3245:
3242:
3240:
3237:
3235:
3232:
3230:
3227:
3225:
3222:
3220:
3217:
3215:
3212:
3210:
3207:
3205:
3202:
3200:
3197:
3195:
3192:
3190:
3187:
3185:
3182:
3181:
3179:
3175:
3172:
3168:
3162:
3159:
3157:
3154:
3152:
3149:
3147:
3144:
3142:
3141:String theory
3139:
3137:
3134:
3132:
3129:
3127:
3124:
3122:
3119:
3117:
3114:
3112:
3111:Axiomatic QFT
3109:
3107:
3106:Algebraic QFT
3104:
3103:
3101:
3097:
3093:
3086:
3081:
3079:
3074:
3072:
3067:
3066:
3063:
3050:
3042:
3039:
3038:
3034:
3026:
3023:
3021:
3018:
3016:
3013:
3012:
3011:
3008:
3007:
3005:
3001:
2991:
2988:
2986:
2983:
2981:
2978:
2976:
2973:
2971:
2968:
2966:
2965:Dual graviton
2963:
2961:
2958:
2956:
2953:
2952:
2950:
2946:
2940:
2937:
2936:
2934:
2932:
2928:
2922:
2919:
2917:
2914:
2913:
2911:
2909:
2905:
2899:
2896:
2894:
2891:
2889:
2886:
2884:
2881:
2880:
2878:
2876:
2875:String theory
2872:
2869:
2865:
2859:
2856:
2854:
2851:
2849:
2846:
2844:
2841:
2839:
2836:
2834:
2831:
2829:
2826:
2824:
2821:
2819:
2816:
2815:
2813:
2811:
2807:
2801:
2798:
2796:
2793:
2791:
2788:
2786:
2783:
2782:
2780:
2778:
2772:
2766:
2763:
2761:
2758:
2756:
2753:
2751:
2748:
2746:
2743:
2741:
2738:
2737:
2735:
2733:
2729:
2723:
2720:
2718:
2715:
2713:
2710:
2708:
2705:
2703:
2700:
2698:
2695:
2693:
2690:
2688:
2685:
2683:
2680:
2678:
2675:
2673:
2670:
2668:
2665:
2663:
2660:
2658:
2655:
2654:
2652:
2648:
2644:
2637:
2632:
2630:
2625:
2623:
2618:
2617:
2614:
2607:
2604:
2603:
2593:
2587:
2583:
2579:
2575:
2571:
2565:
2561:
2556:
2552:
2550:0-521-34400-X
2546:
2542:
2537:
2533:
2531:0-521-23385-2
2527:
2523:
2518:
2517:
2505:
2500:
2496:
2491:
2486:
2482:
2478:
2473:
2468:
2465:(2): 023107.
2464:
2460:
2456:
2449:
2442:
2436:
2435:gr-qc/0601008
2431:
2427:
2420:
2413:
2408:
2406:9783642027802
2402:
2398:
2394:
2387:
2380:
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2367:
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2279:
2275:
2271:
2265:
2257:
2253:
2248:
2243:
2239:
2235:
2231:
2227:
2222:
2221:gr-qc/0308015
2217:
2214:(8): 084022.
2213:
2209:
2205:
2198:
2190:
2186:
2181:
2176:
2172:
2168:
2164:
2160:
2155:
2154:gr-qc/0211053
2150:
2147:(6): 067502.
2146:
2142:
2138:
2131:
2123:
2119:
2114:
2109:
2105:
2101:
2097:
2093:
2088:
2087:gr-qc/9804066
2083:
2080:(8): 084027.
2079:
2075:
2071:
2064:
2056:
2052:
2047:
2042:
2038:
2034:
2030:
2026:
2021:
2020:gr-qc/9806093
2016:
2012:
2008:
2004:
1997:
1989:
1985:
1981:
1977:
1973:
1969:
1964:
1959:
1955:
1951:
1947:
1940:
1932:
1928:
1924:
1920:
1915:
1910:
1906:
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1898:
1894:
1890:
1883:
1875:
1871:
1867:
1863:
1858:
1853:
1849:
1845:
1841:
1837:
1833:
1826:
1818:
1814:
1810:
1806:
1802:
1798:
1793:
1792:gr-qc/9901006
1788:
1784:
1780:
1776:
1769:
1761:
1757:
1753:
1751:0-521-23385-2
1747:
1743:
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1734:
1726:
1722:
1717:
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1696:
1691:
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1678:
1674:
1667:
1654:
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1644:
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1632:
1628:
1622:
1614:
1610:
1606:
1602:
1598:
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1583:
1570:
1566:
1562:
1556:
1552:
1548:
1544:
1540:
1535:
1530:
1526:
1522:
1515:
1504:
1497:
1489:
1487:0-226-87025-1
1483:
1480:. Chicago U.
1479:
1475:
1469:
1467:
1457:
1452:
1444:
1440:
1429:
1426:
1424:
1421:
1419:
1416:
1414:
1411:
1409:
1406:
1404:
1401:
1399:
1396:
1395:
1388:
1386:
1381:
1379:
1369:
1367:
1363:
1359:
1354:
1352:
1347:
1343:
1339:
1335:
1331:
1330:semiclassical
1327:
1317:
1314:
1310:
1306:
1302:
1297:
1295:
1291:
1287:
1283:
1279:
1275:
1271:
1264:
1260:
1256:
1252:
1251:metric tensor
1247:
1245:
1243:
1238:
1235:
1231:
1227:
1224:For non-zero
1222:
1220:
1216:
1212:
1202:
1200:
1196:
1192:
1189:
1185:
1181:
1178:to a general
1177:
1173:
1169:
1165:
1153:
1148:
1146:
1141:
1139:
1134:
1133:
1131:
1130:
1124:
1119:
1116:
1114:
1111:
1109:
1106:
1104:
1101:
1099:
1096:
1094:
1093:Zamolodchikov
1091:
1089:
1088:Zamolodchikov
1086:
1084:
1081:
1079:
1076:
1074:
1071:
1069:
1066:
1064:
1061:
1059:
1056:
1054:
1051:
1049:
1046:
1044:
1041:
1039:
1036:
1034:
1031:
1029:
1026:
1024:
1021:
1019:
1016:
1014:
1011:
1009:
1006:
1004:
1001:
999:
996:
994:
991:
989:
986:
984:
981:
979:
976:
974:
971:
969:
966:
964:
961:
959:
956:
954:
951:
949:
946:
944:
941:
939:
936:
934:
931:
929:
926:
924:
921:
919:
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389:Supersymmetry
387:
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384:String theory
382:
381:
375:
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367:
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34:
30:
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26:
23:
22:
19:
3647:
3576:Quantum foam
3516:Stueckelberg
3470:ChernâSimons
3407:Supergravity
3146:Supergravity
3135:
3131:Gauge theory
3040:
3003:Applications
2893:Supergravity
2800:Unruh effect
2774:
2707:Quantum foam
2702:Planck units
2697:IR/UV mixing
2672:Causal patch
2581:
2559:
2540:
2521:
2502:
2462:
2458:
2448:
2439:
2425:
2419:
2410:
2399:. Springer.
2396:
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2207:
2197:
2144:
2140:
2130:
2077:
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2063:
2010:
2006:
1996:
1953:
1949:
1939:
1896:
1892:
1882:
1857:11449/162073
1839:
1835:
1825:
1785:(1): 19â32.
1782:
1778:
1768:
1740:
1733:
1680:
1676:
1666:
1656:, retrieved
1634:
1621:
1596:
1592:
1582:
1572:, retrieved
1524:
1514:
1496:
1477:
1443:
1382:
1375:
1358:perturbation
1355:
1346:Unruh effect
1332:approach to
1323:
1320:Applications
1298:
1281:
1273:
1266:
1248:
1241:
1234:cosmological
1223:
1208:
1167:
1161:
1122:
968:Stueckelberg
698:Jona-Lasinio
273:Vacuum state
258:Quantization
95:Gauge theory
75:Strong force
60:Field theory
18:
3458:Topological
3372:WessâZumino
3285:Sine-Gordon
3275:GrossâNeveu
3184:BornâInfeld
3151:Thermal QFT
2960:Causal sets
2810:Black holes
2247:11449/23527
2180:11449/23239
2113:11449/65534
2046:11449/65552
1716:11449/24446
1474:Wald, R. M.
1305:Rudolf Haag
1199:black holes
1197:emitted by
1108:Zinn-Justin
958:Sommerfield
883:Pomeranchuk
848:Osterwalder
843:Oppenheimer
773:ĆopuszaĆski
593:Fredenhagen
394:Technicolor
3669:Categories
3239:YangâMills
3025:Multiverse
2867:Approaches
2745:CGHS model
2732:Toy models
2667:CA-duality
2578:Parker, L.
2504:quantized.
2472:1906.01927
2280:(4): 057.
1963:2006.08887
1899:(2): 133.
1658:2021-08-15
1574:2022-01-14
1456:2308.14517
1435:References
1103:Zimmermann
998:Vainshtein
873:Polchinski
733:Kontsevich
678:Iliopoulos
653:Heisenberg
478:Bogoliubov
416:Scientists
253:Propagator
135:T-symmetry
130:P-symmetry
125:C-symmetry
113:Symmetries
70:Weak force
54:Background
3648:See also:
3367:Super QCD
3321:Liouville
3309:Conformal
3280:Schwinger
3041:See also:
2980:Spin foam
2760:RST model
2499:218502756
2348:1306.1058
1988:219708236
1923:1434-6052
1874:126192949
1866:0264-9381
1809:0264-9381
1725:119223632
1690:0710.5373
1569:119179440
1534:1412.5945
1294:heat bath
1237:curvature
1230:particles
1209:Ordinary
1038:Wetterich
1023:Weisskopf
973:Sudarshan
923:Schwinger
833:Nishijima
798:Maldacena
763:Leutwyler
723:Kinoshita
623:Goldstone
613:Gell-Mann
528:Doplicher
295:Equations
3444:Type IIB
3439:Type IIA
3424:4D N = 8
3419:4D N = 1
3388:6D (2,0)
3352:4D N = 1
3331:Polyakov
3290:Thirring
3099:Theories
2888:M-theory
2848:ER = EPR
2687:Graviton
2375:55608399
2312:16290999
2256:41250020
2189:33007353
2122:15522105
2055:14575175
1931:55070002
1817:14018854
1476:(1995).
1391:See also
1284:> 0.
1205:Overview
1188:graviton
1053:Wightman
1018:Weinberg
1008:Virasoro
988:Tomonaga
983:Thirring
978:Symanzik
938:Semenoff
913:Schrader
878:Polyakov
793:Majorana
728:Klebanov
683:Ivanenko
673:'t Hooft
643:Guralnik
588:Fröhlich
583:Fritzsch
578:Frampton
493:Buchholz
438:Bargmann
428:Anderson
208:Crossing
3546:History
3529:Related
3326:Minimal
3177:Regular
2562:. CUP.
2543:. CUP.
2524:. CUP.
2477:Bibcode
2353:Bibcode
2292:Bibcode
2226:Bibcode
2159:Bibcode
2092:Bibcode
2025:Bibcode
1968:Bibcode
1901:Bibcode
1760:7462032
1695:Bibcode
1601:Bibcode
1539:Bibcode
1244:-matrix
1063:Wilczek
1028:Wentzel
1003:Veltman
948:Shirkov
943:Shifman
933:Seiberg
918:Schwarz
898:Rubakov
818:Naimark
768:Lipatov
758:Lehmann
738:Kreimer
718:Kendall
608:Gelfand
603:Glashow
563:Feynman
543:Faddeev
538:Englert
508:Coleman
498:Cachazo
483:Brodsky
468:Bjorken
458:Berezin
448:Belavin
188:Anomaly
46:History
3486:Chiral
3434:Type I
3249:Yukawa
3170:Models
2948:Others
2588:
2566:
2547:
2528:
2497:
2403:
2373:
2310:
2254:
2187:
2120:
2053:
1986:
1929:
1921:
1872:
1864:
1815:
1807:
1758:
1748:
1723:
1649:
1567:
1557:
1484:
1324:Using
1259:vacuum
1186:(multi
1118:Zumino
1083:Yukawa
1073:Witten
1068:Wilson
1058:Wigner
993:Tyutin
953:Skyrme
903:Ruelle
868:Plefka
863:Peskin
853:Parisi
813:MĂžller
803:Migdal
788:Maiani
783:LĂŒders
748:Landau
743:Kuraev
713:KÀllén
703:Jordan
688:Jackiw
628:Gribov
518:DeWitt
513:Dashen
503:Callan
473:Bleuer
443:Becchi
433:Anselm
3625:links
3598:links
3586:links
3506:NMSSM
3491:Fermi
3234:Soler
3204:Proca
2495:S2CID
2467:arXiv
2430:arXiv
2371:S2CID
2343:arXiv
2308:S2CID
2282:arXiv
2252:S2CID
2216:arXiv
2185:S2CID
2149:arXiv
2118:S2CID
2082:arXiv
2051:S2CID
2015:arXiv
1984:S2CID
1958:arXiv
1927:S2CID
1870:S2CID
1813:S2CID
1787:arXiv
1721:S2CID
1685:arXiv
1565:S2CID
1529:arXiv
1506:(PDF)
1451:arXiv
1174:from
1113:Zuber
963:Stora
928:Segal
908:Salam
893:Proca
888:Popov
858:Pauli
838:Oehme
828:Neveu
823:Nambu
808:Mills
693:Jaffe
668:Hagen
663:Higgs
638:Gupta
633:Gross
618:Glimm
598:Furry
568:Fierz
558:Fermi
553:Fayet
548:Fadin
533:Dyson
523:Dirac
488:Brout
463:Bethe
423:Adler
182:Tools
3496:MSSM
3393:ABJM
3300:Toda
2586:ISBN
2564:ISBN
2545:ISBN
2526:ISBN
2401:ISBN
2278:2006
1919:ISSN
1862:ISSN
1805:ISSN
1756:OCLC
1746:ISBN
1647:ISBN
1555:ISBN
1482:ISBN
1307:and
1078:Yang
1048:Wick
1043:Weyl
1033:Wess
1013:Ward
708:Jost
658:Hepp
648:Haag
573:Fock
453:Bell
3449:11D
2485:doi
2361:doi
2339:345
2300:doi
2242:hdl
2234:doi
2175:hdl
2167:doi
2108:hdl
2100:doi
2041:hdl
2033:doi
1976:doi
1954:101
1909:doi
1852:hdl
1844:doi
1797:doi
1711:hdl
1703:doi
1639:doi
1609:doi
1547:doi
1162:In
1098:Zee
778:Low
753:Lee
3671::
3465:BF
2501:.
2493:.
2483:.
2475:.
2461:.
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2005:.
1982:.
1974:.
1966:.
1952:.
1948:.
1925:.
1917:.
1907:.
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