1094:
regularizing the electron mass-energy) suffices to explain the system below a certain size. Similar regularization arguments work in other renormalization problems. For example, a theory may hold under one narrow set of conditions, but due to calculations involving infinities or singularities, it may breakdown under other conditions or scales. In the case of the electron, another way to avoid infinite mass-energy while retaining the point nature of the particle is to postulate tiny additional dimensions over which the particle could 'spread out' rather than restrict its motion solely over 3D space. This is precisely the motivation behind
1269:
physicists. They are inclined to think one master idea will be discovered that will solve all these problems together. I think it is asking too much to hope that anyone will be able to solve all these problems together. One should separate them one from another as much as possible and try to tackle them separately. And I believe the future development of physics will consist of solving them one at a time, and that after any one of them has been solved there will still be a great mystery about how to attack further ones."
1243:
survive in the future,…" He further observed that "One can distinguish between two main procedures for a theoretical physicist. One of them is to work from the experimental basis ... The other procedure is to work from the mathematical basis. One examines and criticizes the existing theory. One tries to pin-point the faults in it and then tries to remove them. The difficulty here is to remove the faults without destroying the very great successes of the existing theory."
145:. The regulator, also known as a "cutoff", models our lack of knowledge about physics at unobserved scales (e.g. scales of small size or large energy levels). It compensates for (and requires) the possibility of separation of scales that "new physics" may be discovered at those scales which the present theory is unable to model, while enabling the current theory to give accurate predictions as an "effective theory" within its intended scale of use.
1231:
that might reflect the underlying physics. The additional parameters of such a theory do not need to be removed (i.e. the theory needs no renormalization) and may provide some new information about the physics of quantum scattering, though they may turn out experimentally to be negligible. By contrast, any present regularization method introduces formal coefficients that must eventually be disposed of by renormalization.
1325:"Could it be that the real world consists of little X-ons which can be seen only at very tiny distances? And that in our measurements we are always observing on such a large scale that we can’t see these little X-ons, and that is why we get the differential equations? ... Are they also correct only as a smoothed-out imitation of a really much more complicated microscopic world?"
1207:, by introducing either unphysical particles with a negative metric or wrong statistics, or discrete space-time, or lowering the dimensionality of space-time, or some combination thereof). So the available regularization methods are understood as formalistic technical devices, devoid of any direct physical meaning. In addition, there are qualms about
736:
1220:
vertices in
Feynman series, and modify only the Feynman propagators to create a regularized Feynman series. This is the reasoning behind the formal Pauli–Villars covariant regularization by modification of Feynman propagators through auxiliary unphysical particles, cf. and representation of physical reality by
1364:
only but additional regularization - and hence new physics—is required uniquely for gravity. The regularizers model, and work around, the breakdown of QFT at small scales and thus show clearly the need for some other theory to come into play beyond QFT at these scales. A. Zee (Quantum Field Theory
1230:
conjectured there is a realistic regularization, which is implied by a theory that respects all the established principles of contemporary physics. So its propagators (i) do not need to be regularized, and (ii) can be regarded as such a regularization of the propagators used in quantum field theories
1202:
elements. These are independent of the particular regularization method used, and enable one to model perturbatively the measurable physical processes (cross sections, probability amplitudes, decay widths and lifetimes of excited states). However, so far no known regularized n-point Green's functions
1003:
1324:
noted about the use of differential equations: "... for neutron diffusion it is only an approximation that is good when the distance over which we are looking is large compared with the mean free path. If we looked more closely, we would see individual neutrons running around." And then he wondered,
1300:
reduction formula provides a perturbative S-matrix that: (i) is
Lorentz-invariant and unitary; (ii) involves only the QED particles; (iii) depends solely on QED parameters and those introduced by the modification of the Feynman propagators—for particular values of these parameters it is equal to the
1268:
Considering distinct theoretical problems, Dirac in 1963 suggested: "I believe separate ideas will be needed to solve these distinct problems and that they will be solved one at a time through successive stages in the future evolution of physics. At this point I find myself in disagreement with most
1249:
remarked in 1972, "Field-theoretic infinities first encountered in
Lorentz's computation of electron have persisted in classical electrodynamics for seventy and in quantum electrodynamics for some thirty-five years. These long years of frustration have left in the subject a curious affection for the
1295:
provides the most direct way from the
Lagrangian density to the corresponding Feynman series in its Lorentz-invariant form. The free-field part of the Lagrangian density determines the Feynman propagators, whereas the rest determines the vertices. As the QED vertices are considered to adequately
1242:
was persistently, extremely critical about procedures of renormalization. In 1963, he wrote, "… in the renormalization theory we have a theory that has defied all the attempts of the mathematician to make it sound. I am inclined to suspect that the renormalization theory is something that will not
1219:
As it seems that the vertices of non-regularized
Feynman series adequately describe interactions in quantum scattering, it is taken that their ultraviolet divergences are due to the asymptotic, high-energy behavior of the Feynman propagators. So it is a prudent, conservative approach to retain the
1093:
Regularization: Classical physics theory breaks down at small scales, e.g., the difference between an electron and a point particle shown above. Addressing this problem requires new kinds of additional physical constraints. For instance, in this case, assuming a finite electron radius (i.e.,
1250:
infinities and a passionate belief that they are an inevitable part of nature; so much so that even the suggestion of a hope that they may after all be circumvented - and finite values for the renormalization constants computed - is considered irrational."
1257:’s opinion, "History tells us that if we hit upon some obstacle, even if it looks like a pure formality or just a technical complication, it should be carefully scrutinized. Nature might be telling us something, and we should find out what it is."
1276:
is the domain of physics that we know most about, and presumably it will have to be put in order before we can hope to make any fundamental progress with other field theories, although these will continue to develop on the experimental basis."
540:
1365:
in a
Nutshell, 2003) considers this to be a benefit of the regularization framework—theories can work well in their intended domains but also contain information about their own limitations and point clearly to where new physics is needed.
861:
1340:. Feynman's preceding remark provides a possible physical reason for its existence; either that or it is just another way of saying the same thing (there is a fundamental unit of distance) but having no new information.
487:
191:
in space which is useful, in case the divergences arise from short-distance physical effects). The correct physical result is obtained in the limit in which the regulator goes away (in our example,
1203:
can be regarded as being based on a physically realistic theory of quantum-scattering since the derivation of each disregards some of the basic tenets of conventional physics (e.g., by not being
372:
1260:
The difficulty with a realistic regularization is that so far there is none, although nothing could be destroyed by its bottom-up approach; and there is no experimental basis for it.
1084:
1040:
492:
It is not always possible to define a regularization such that the limit of ε going to zero is independent of the regularization. In this case, one says that the theory contains an
428:
82:
792:
853:
1296:
describe interactions in QED scattering, it makes sense to modify only the free-field part of the
Lagrangian density so as to obtain such regularized Feynman series that the
731:{\displaystyle m_{\mathrm {em} }=\int {1 \over 2}E^{2}\,dV=\int _{r_{e}}^{\infty }{\frac {1}{2}}\left({q \over 4\pi r^{2}}\right)^{2}4\pi r^{2}\,dr={q^{2} \over 8\pi r_{e}},}
272:
215:
171:
Regularization procedures deal with infinite, divergent, and nonsensical expressions by introducing an auxiliary concept of a regulator (for example, the minimal distance
108:
308:
246:
1280:
Dirac’s two preceding remarks suggest that we should start searching for a realistic regularization in the case of quantum electrodynamics (QED) in the four-dimensional
189:
392:
313:
The existence of a limit as ε goes to zero and the independence of the final result from the regulator are nontrivial facts. The underlying reason for them lies in
822:
998:{\displaystyle r_{e}={e^{2} \over 4\pi \varepsilon _{0}m_{\mathrm {e} }c^{2}}=\alpha {\hbar \over m_{\mathrm {e} }c}\approx 2.8\times 10^{-15}\ \mathrm {m} .}
1102:. Rather than the existence of unknown new physics, assuming the existence of particle interactions with other surrounding particles in the environment,
489:
still give pretty accurate approximations. The physical reason why we can't take the limit of ε going to zero is the existence of new physics below Λ.
1332:
proposed that a quantum field theory can provide only an idealized, large-scale description of quantum dynamics, valid for distances larger than some
310:— are equal to the observed values. Such a constraint allows one to calculate a finite value for many other quantities that looked divergent.
101:
1301:
QED perturbative S-matrix; and (iv) exhibits the same symmetries as the QED perturbative S-matrix. Let us refer to such a regularization as
282:. Renormalization is based on the requirement that some physical quantities — expressed by seemingly divergent expressions such as
94:
433:
1337:
1313:
1546:
F. Villars (1960). "Regularization and Non-Singular
Interactions in Quantum Field Theory". In M. Fierz; V. F. Weiskopf (eds.).
1688:
Isham, C. J.; Salam, Abdus; Strathdee, J. (1971-04-15). "Infinity
Suppression in Gravity-Modified Quantum Electrodynamics".
163:
claims into the same equations. However, it is now well understood and has proven to yield useful, accurate predictions.
1297:
1731:
Isham, C. J.; Salam, Abdus; Strathdee, J. (1972-05-15). "Infinity
Suppression in Gravity-Modified Electrodynamics. II".
1221:
1672:
1357:
152:, another technique to control infinities without assuming new physics, by adjusting for self-interaction feedback.
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62:
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38:
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70:
1138:
1118:
765:
397:
74:
58:
42:
1305:, and start searching for the corresponding, modified free-field parts of the QED Lagrangian density.
831:
1798:
W. Heisenberg (1938). "Uber die in der Théorie der Elementarteilchen auftretende universelle Lange".
1292:
1285:
1171:
795:
513:
138:
251:
194:
1374:
1099:
274:. Regularization is the first step towards obtaining a completely finite and meaningful result; in
1211:
For a history and comments on this more than half-a-century old open conceptual problem, see e.g.
1273:
1179:
1143:
1043:
329:. Sometimes, taking the limit as ε goes to zero is not possible. This is the case when we have a
285:
223:
78:
1664:
1317:
1128:
496:. Anomalous theories have been studied in great detail and are often founded on the celebrated
174:
66:
377:
1786:
1320:
by using more detailed description than can be provided by differential field equations. And
155:
Regularization was for many decades controversial even amongst its inventors, as it combines
34:
1656:
1807:
1740:
1697:
1624:
1574:
1410:
1349:
1198:), and a suitable limiting procedure (a renormalization scheme) then leads to perturbative
1167:
1163:
524:
337:. However, even for these two examples, if the regulator only gives reasonable results for
275:
126:
523:
The mass of a charged particle should include the mass–energy in its electrostatic field (
8:
1281:
1191:
801:
517:
1811:
1787:
The Feynman Lectures on Physics. Vol. II, Section 12–7: The “underlying unity” of nature
1744:
1701:
1628:
1578:
1414:
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1254:
1087:
217:), but the virtue of the regulator is that for its finite value, the result is finite.
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1329:
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318:
1498:
1467:"The conceptual foundations and the philosophical aspects of renormalization theory"
141:
in order to make them finite by the introduction of a suitable parameter called the
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1748:
1705:
1632:
1582:
1478:
1426:
1418:
1183:
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1353:
1321:
1187:
1106:
offers an alternative strategy to resolve infinities in such classical problems.
1103:
279:
160:
149:
29:
1615:
P.A.M. Dirac (May 1963). "The Evolution of the Physicist's Picture of Nature".
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1833:
1819:
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1438:
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394:
is a superior energy cuttoff) and we are working with scales of the order of
1752:
1709:
220:
However, the result usually includes terms proportional to expressions like
1655:
P.A.M. Dirac (1990) . "Methods in theoretical physics". In A. Salam (ed.).
1360:. Infinities of the non-gravitational forces in QFT can be controlled via
1159:
322:
278:
it must be usually followed by a related, but independent technique called
1246:
330:
134:
1482:
1239:
1195:
1430:
1199:
527:). Assume that the particle is a charged spherical shell of radius
482:{\displaystyle \hbar c/\Lambda \ll \epsilon \ll \hbar c/\Lambda '}
755:, making it unable to be accelerated. Incidentally, the value of
752:
156:
122:
1563:"On the Invariant Regularization in Relativistic Quantum Theory"
1532:. Vol. 1. Cambridge University Press. Sec. 1.3 and Ch.9.
1190:
scheme. Regularization method results in regularized n-point
751:. This implies that the point particle would have infinite
1550:. New York: Interscience Publishers. pp. 78–106.
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1014:
864:
834:
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768:
543:
436:
400:
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343:
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226:
197:
177:
1396:"Regularization and renormalization of gauge fields"
1114:
Specific types of regularization procedures include
1730:
1687:
1453:
Finite Quantum Electrodynamics: The Causal Approach
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1078:
1034:
997:
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816:
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730:
481:
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386:
366:
302:
266:
240:
209:
183:
1831:
1308:
1797:
1777:(Cambridge University Press, Cambridge 1997).
1098:and other multi-dimensional models including
512:The problem of infinities first arose in the
500:or variations thereof (see, for example, the
367:{\displaystyle \epsilon \gg \hbar c/\Lambda }
333:and for nonrenormalizable couplings like the
102:
1654:
1614:
1560:
1548:Theoretical Physics in the Twentieth Century
1523:
1521:
507:
1541:
1539:
1527:
1464:
1316:, it would make physical sense to sidestep
1149:
1545:
1465:Cao, Tian Yu; Schweber, Silvan S. (1993).
109:
95:
1775:In Search of the Ultimate Building Blocks
1650:
1648:
1646:
1610:
1608:
1606:
1586:
1518:
1348:The need for regularization terms in any
794:equal to the electron mass is called the
683:
585:
1536:
1178:, a regularization method to circumvent
1079:{\displaystyle \hbar /m_{\mathrm {e} }c}
1035:{\displaystyle \alpha \approx 1/137.040}
248:which are not well-defined in the limit
1663:. Cambridge University Press. pp.
1343:
1832:
1643:
1603:
1561:Pauli, W.; Villars, F. (1949-07-01).
1214:
1154:
1303:the minimal realistic regularization
520:in the 19th and early 20th century.
1182:so as to obtain finite results for
16:Method used in mathematical physics
13:
1394:'t Hooft, G.; Veltman, M. (1972).
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988:
950:
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778:
775:
612:
553:
550:
534:. The mass–energy in the field is
472:
448:
413:
381:
361:
22:Renormalization and regularization
14:
1861:
1659:Unification of Fundamental Forces
1637:10.1038/scientificamerican0563-45
1515:(Springer-Verlag, New York 1993).
1358:physics beyond the standard model
1284:, starting with the original QED
1109:
787:{\displaystyle m_{\mathrm {em} }}
460:
423:{\displaystyle \hbar c/\Lambda '}
350:
1264:Minimal realistic regularization
1174:density, are computed using the
848:{\displaystyle \varepsilon _{0}}
1791:
1780:
1767:
1724:
1681:
1554:
1505:
1458:
1445:
1387:
1314:According to Bjorken and Drell
267:{\displaystyle \epsilon \to 0}
258:
210:{\displaystyle \epsilon \to 0}
201:
83:Point-splitting regularization
1:
1380:
1170:, implied by a corresponding
327:second order phase transition
1530:The Quantum Theory of Fields
1423:10.1016/0550-3213(72)90279-9
1309:Transport theoretic approach
1166:about quantum scattering of
1134:Zeta function regularization
1124:Pauli–Villars regularization
71:Zeta function regularization
63:Pauli–Villars regularization
7:
1368:
1298:Lehmann–Symanzik–Zimmermann
1234:
498:Atiyah–Singer index theorem
303:{\displaystyle 1/\epsilon }
241:{\displaystyle 1/\epsilon }
166:
10:
1866:
1356:is a major motivation for
1119:Dimensional regularization
741:which becomes infinite as
75:Causal perturbation theory
59:Dimensional regularization
43:Minimal subtraction scheme
1588:10.1103/revmodphys.21.434
1567:Reviews of Modern Physics
1338:Bjorken and Drell in 1965
1293:path-integral formulation
824:and restoring factors of
796:classical electron radius
514:classical electrodynamics
508:Classical physics example
184:{\displaystyle \epsilon }
133:is a method of modifying
1820:10.1002/andp.19384240105
1375:Zeldovich regularization
1186:containing loops, and a
1150:Realistic regularization
1100:multiple time dimensions
387:{\displaystyle \Lambda }
1753:10.1103/physrevd.5.2548
1710:10.1103/physrevd.3.1805
1318:ultraviolet divergences
1274:Quantum electrodynamics
1180:ultraviolet divergences
1144:Hadamard regularization
1044:fine-structure constant
325:and the existence of a
79:Hadamard regularization
1129:Lattice regularization
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67:Lattice regularization
1272:According to Dirac, "
1139:Causal regularization
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35:Renormalization group
1845:Quantum field theory
1528:S. Weinberg (1995).
1511:L. M.Brown, editor,
1350:quantum field theory
1344:Hints at new physics
1168:elementary particles
1164:quantum field theory
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525:electromagnetic mass
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276:quantum field theory
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148:It is distinct from
127:quantum field theory
1850:Summability methods
1840:Concepts in physics
1812:1938AnP...424...20H
1745:1972PhRvD...5.2548I
1702:1971PhRvD...3.1805I
1629:1963SciAm.208e..45D
1617:Scientific American
1579:1949RvMP...21..434P
1415:1972NuPhB..44..189T
1336:, expected also by
1282:Minkowski spacetime
817:{\displaystyle q=e}
616:
1800:Annalen der Physik
1483:10.1007/bf01255832
1334:fundamental length
1215:Pauli's conjecture
1155:Conceptual problem
1088:Compton wavelength
1076:
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995:
855:) turns out to be
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728:
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479:
430:, regulators with
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181:
1739:(10): 2548–2565.
1733:Physical Review D
1690:Physical Review D
1403:Nuclear Physics B
1328:Already in 1938,
1222:Feynman diagrams.
1205:Lorentz-invariant
1192:Green's functions
1090:of the electron.
986:
960:
930:
798:, which (setting
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335:Fermi interaction
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1696:(8): 1805–1817.
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1455:, Springer 1995.
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1209:renormalization.
1184:Feynman diagrams
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1513:Renormalization
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1362:renormalization
1354:quantum gravity
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1255:Gerard ’t Hooft
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1188:renormalization
1162:predictions by
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1104:renormalization
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39:On-shell scheme
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1723:
1680:
1673:
1642:
1602:
1573:(3): 434–444.
1553:
1535:
1517:
1504:
1457:
1444:
1409:(1): 189–213.
1385:
1384:
1382:
1379:
1378:
1377:
1370:
1367:
1345:
1342:
1310:
1307:
1265:
1262:
1236:
1233:
1216:
1213:
1156:
1153:
1151:
1148:
1147:
1146:
1141:
1136:
1131:
1126:
1121:
1111:
1110:Specific types
1108:
1075:
1069:
1064:
1059:
1055:
1031:
1027:
1023:
1020:
1017:
1006:
1005:
994:
990:
981:
978:
974:
970:
967:
964:
958:
952:
947:
942:
937:
934:
926:
922:
915:
910:
904:
900:
896:
893:
887:
883:
877:
872:
868:
842:
838:
813:
810:
807:
780:
777:
772:
758:
745:
739:
738:
727:
719:
715:
711:
708:
702:
698:
692:
689:
686:
680:
676:
672:
669:
664:
659:
651:
647:
643:
640:
636:
631:
624:
621:
614:
607:
603:
598:
594:
591:
588:
582:
578:
572:
569:
564:
561:
555:
552:
547:
530:
509:
506:
502:chiral anomaly
477:
474:
469:
465:
462:
459:
456:
453:
450:
446:
442:
439:
418:
415:
410:
406:
403:
383:
363:
359:
355:
352:
349:
346:
319:Kenneth Wilson
299:
295:
291:
263:
260:
257:
237:
233:
229:
206:
203:
200:
180:
168:
165:
131:regularization
117:
116:
114:
113:
106:
99:
91:
88:
87:
57:
54:Regularization
52:
51:
48:
47:
33:
28:
27:
24:
23:
15:
9:
6:
4:
3:
2:
1862:
1851:
1848:
1846:
1843:
1841:
1838:
1837:
1835:
1821:
1817:
1813:
1809:
1805:
1801:
1794:
1788:
1783:
1776:
1773:G. ’t Hooft,
1770:
1762:
1758:
1754:
1750:
1746:
1742:
1738:
1734:
1727:
1719:
1715:
1711:
1707:
1703:
1699:
1695:
1691:
1684:
1676:
1674:9780521371407
1670:
1666:
1661:
1660:
1651:
1649:
1647:
1638:
1634:
1630:
1626:
1622:
1618:
1611:
1609:
1607:
1598:
1594:
1589:
1584:
1580:
1576:
1572:
1568:
1564:
1557:
1549:
1542:
1540:
1531:
1524:
1522:
1514:
1508:
1500:
1496:
1492:
1488:
1484:
1480:
1477:(1): 33–108.
1476:
1472:
1468:
1461:
1454:
1448:
1440:
1436:
1432:
1428:
1424:
1420:
1416:
1412:
1408:
1404:
1397:
1390:
1386:
1376:
1373:
1372:
1366:
1363:
1359:
1355:
1351:
1341:
1339:
1335:
1331:
1326:
1323:
1319:
1315:
1306:
1304:
1299:
1294:
1289:
1287:
1283:
1278:
1275:
1270:
1261:
1258:
1256:
1251:
1248:
1244:
1241:
1232:
1229:
1224:
1223:
1212:
1210:
1206:
1201:
1197:
1193:
1189:
1185:
1181:
1177:
1176:Feynman rules
1173:
1169:
1165:
1161:
1145:
1142:
1140:
1137:
1135:
1132:
1130:
1127:
1125:
1122:
1120:
1117:
1116:
1115:
1107:
1105:
1101:
1097:
1096:string theory
1091:
1089:
1073:
1062:
1057:
1053:
1045:
1029:
1025:
1021:
1018:
1015:
992:
979:
976:
972:
968:
965:
962:
956:
945:
940:
935:
932:
924:
920:
908:
902:
898:
894:
891:
885:
881:
875:
870:
866:
858:
857:
856:
840:
836:
811:
808:
805:
797:
770:
754:
748:
725:
717:
713:
709:
706:
700:
696:
690:
687:
684:
678:
674:
670:
667:
662:
657:
649:
645:
641:
638:
634:
629:
622:
619:
605:
601:
596:
592:
589:
586:
580:
576:
570:
567:
562:
559:
545:
537:
536:
535:
526:
521:
519:
515:
505:
503:
499:
495:
490:
475:
467:
463:
457:
454:
451:
444:
440:
437:
416:
408:
404:
401:
357:
353:
347:
344:
336:
332:
328:
324:
320:
316:
311:
297:
293:
289:
281:
277:
261:
255:
235:
231:
227:
218:
204:
198:
178:
164:
162:
158:
153:
151:
146:
144:
140:
139:singularities
136:
132:
128:
125:, especially
124:
112:
107:
105:
100:
98:
93:
92:
90:
89:
84:
80:
76:
72:
68:
64:
60:
55:
50:
49:
44:
40:
36:
31:
26:
25:
21:
20:
1806:(1): 20–33.
1803:
1799:
1793:
1782:
1774:
1769:
1736:
1732:
1726:
1693:
1689:
1683:
1658:
1623:(5): 45–53.
1620:
1616:
1570:
1566:
1556:
1547:
1529:
1512:
1507:
1474:
1470:
1460:
1452:
1451:Scharf, G.:
1447:
1406:
1402:
1389:
1347:
1333:
1327:
1312:
1302:
1290:
1279:
1271:
1267:
1259:
1253:However, in
1252:
1245:
1238:
1225:
1218:
1160:Perturbative
1158:
1113:
1092:
1007:
743:
740:
522:
511:
491:
323:Leo Kadanoff
317:as shown by
315:universality
312:
219:
170:
154:
147:
142:
130:
120:
53:
1247:Abdus Salam
1196:propagators
762:that makes
331:Landau pole
137:which have
135:observables
1834:Categories
1381:References
1330:Heisenberg
1286:Lagrangian
1240:Paul Dirac
1172:Lagrangian
1761:0556-2821
1718:0556-2821
1597:0034-6861
1491:0039-7857
1439:0550-3213
1431:1874/4845
1288:density.
1054:ℏ
1019:≈
1016:α
977:−
969:×
963:≈
941:ℏ
936:α
899:ε
895:π
837:ε
710:π
671:π
642:π
613:∞
597:∫
563:∫
473:Λ
461:ℏ
458:≪
455:ϵ
452:≪
449:Λ
438:ℏ
414:Λ
402:ℏ
382:Λ
362:Λ
351:ℏ
348:≫
345:ϵ
298:ϵ
259:→
256:ϵ
236:ϵ
202:→
199:ϵ
179:ϵ
143:regulator
1499:46968305
1471:Synthese
1369:See also
1235:Opinions
1226:In 1949
1200:S-matrix
476:′
417:′
167:Overview
157:physical
1808:Bibcode
1741:Bibcode
1698:Bibcode
1625:Bibcode
1575:Bibcode
1411:Bibcode
1322:Feynman
1086:is the
1042:is the
1030:137.040
753:inertia
494:anomaly
374:(where
123:physics
1759:
1716:
1671:
1667:–143.
1595:
1497:
1489:
1437:
1046:, and
1008:where
985:
1495:S2CID
1399:(PDF)
1228:Pauli
1757:ISSN
1714:ISSN
1669:ISBN
1593:ISSN
1487:ISSN
1435:ISSN
1291:The
828:and
321:and
159:and
1816:doi
1749:doi
1706:doi
1665:125
1633:doi
1621:208
1583:doi
1479:doi
1427:hdl
1419:doi
1352:of
966:2.8
749:→ 0
516:of
504:).
121:In
1836::
1814:.
1804:32
1802:.
1755:.
1747:.
1735:.
1712:.
1704:.
1692:.
1645:^
1631:.
1619:.
1605:^
1591:.
1581:.
1571:21
1569:.
1565:.
1538:^
1520:^
1493:.
1485:.
1475:97
1473:.
1469:.
1433:.
1425:.
1417:.
1407:44
1405:.
1401:.
980:15
973:10
129:,
1822:.
1818::
1810::
1763:.
1751::
1743::
1737:5
1720:.
1708::
1700::
1694:3
1677:.
1639:.
1635::
1627::
1599:.
1585::
1577::
1501:.
1481::
1441:.
1429::
1421::
1413::
1194:(
1074:c
1068:e
1063:m
1058:/
1026:/
1022:1
993:.
989:m
957:c
951:e
946:m
933:=
925:2
921:c
914:e
909:m
903:0
892:4
886:2
882:e
876:=
871:e
867:r
841:0
826:c
812:e
809:=
806:q
779:m
776:e
771:m
759:e
757:r
746:e
744:r
726:,
718:e
714:r
707:8
701:2
697:q
691:=
688:r
685:d
679:2
675:r
668:4
663:2
658:)
650:2
646:r
639:4
635:q
630:(
623:2
620:1
606:e
602:r
593:=
590:V
587:d
581:2
577:E
571:2
568:1
560:=
554:m
551:e
546:m
531:e
529:r
468:/
464:c
445:/
441:c
409:/
405:c
358:/
354:c
294:/
290:1
262:0
232:/
228:1
205:0
110:e
103:t
96:v
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