1337:
498:
866:
There are many quantum field theories which, while not being exactly scale invariant, remain approximately scale invariant over a long range of distances. Such quantum field theories can be obtained by adding to free field theories interaction terms with small dimensionless
316:
1095:
also depends on the distance scale in such theories. In particular correlation functions of local operators are no longer simple powers but have a more complicated dependence on the distances, generally with logarithmic corrections.
1147:
In very special cases, it may happen when the couplings and the anomalous dimensions do not run at all, so that the theory is scale invariant at all distances and for any value of the coupling. For example, this occurs in the
871:. For example, in four spacetime dimensions one can add quartic scalar couplings, Yukawa couplings, or gauge couplings. Scaling dimensions of operators in such theories can be expressed schematically as
917:
663:
There are many scale invariant quantum field theories which are not free theories; these are called interacting. Scaling dimensions of operators in such theories may not be read off from a
514:
Free theories are the simplest scale-invariant quantum field theories. In free theories, one makes a distinction between the elementary operators, which are the fields appearing in the
268:
790:
646:
542:(in four spacetime dimensions, it is 1 for elementary bosonic fields including the vector potentials, 3/2 for elementary fermionic fields etc.). This scaling dimension is called the
1307:
816:
1268:
178:
311:
148:
76:
944:
750:
723:
606:
579:
493:{\displaystyle \langle O_{1}(\lambda x_{1})O_{2}(\lambda x_{2})\ldots \rangle =\lambda ^{-\Delta _{1}-\Delta _{2}-\ldots }\langle O_{1}(x_{1})O_{2}(x_{2})\ldots \rangle }
1093:
1026:
973:
836:
1229:
1130:
856:
689:
198:
667:; they are also not necessarily (half)integer. For example, in the scale (and conformally) invariant theory describing the critical points of the two-dimensional
1049:
997:
536:
218:
119:
1149:
1378:
999:. Such a separation of scaling dimensions into the classical and anomalous part is only meaningful when couplings are small, so that
1397:
1407:
752:
will generally give not a unique operator but infinitely many operators, and their dimension will not generally be equal to
1402:
518:, and the composite operators which are products of the elementary ones. The scaling dimension of an elementary operator
874:
1371:
695:
313:. More generally, correlation functions of several local operators must depend on the distances in such a way that
47:
664:
515:
226:
755:
611:
1196:
1133:
1063:
1273:
795:
1192:
503:
1234:
153:
86:, scaling dimensions of operators are fixed numbers, otherwise they are functions of the distance scale.
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273:
1141:
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539:
221:
98:
79:
43:
124:
52:
31:
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are also used). A composite operator obtained by taking a product of two operators of dimensions
1069:
1002:
949:
821:
1344:
1051:
868:
1201:
1102:
841:
674:
183:
1137:
95:
83:
17:
946:
is the dimension when all couplings are set to zero (i.e. the classical dimension), while
694:
Operator multiplication is subtle in interacting theories compared to free theories. The
651:
When interactions are turned on, the scaling dimension receives a correction called the
1348:
1034:
982:
521:
203:
104:
1391:
506:, which imposes further constraints on correlation functions of local operators.
668:
979:, and is expressed as a power series in the couplings collectively denoted as
1140:, and the anomalous dimensions stop running. Such a behavior is called an
792:. In the above two-dimensional Ising model example, the operator product
1336:
1099:
It may happen that the evolution of the couplings will lead to a value
46:
characterizes the rescaling properties of the operator under spacetime
27:
Number specifying how a quantum operator changes under dilations
1174:
Philippe Di
Francesco; Pierre Mathieu; David Sénéchal (1997).
1031:
Generally, due to quantum mechanical effects, the couplings
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1276:
1237:
1204:
1105:
1072:
1037:
1005:
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844:
824:
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731:
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614:
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319:
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229:
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186:
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127:
107:
55:
1136:
vanishes. Then at long distances the theory becomes
838:
whose dimension is 1 and not twice the dimension of
1301:
1262:
1223:
1124:
1087:
1043:
1020:
991:
967:
938:
911:
850:
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784:
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683:
640:
600:
573:
530:
492:
305:
262:
212:
192:
172:
142:
113:
70:
220:. This implies in particular that the two point
538:is determined by dimensional analysis from the
1319:Peskin, Michael E; Daniel V Schroeder (1995).
912:{\displaystyle \Delta =\Delta _{0}+\gamma (g)}
1372:
1062:) with the distance scale according to their
608:is a new operator whose dimension is the sum
8:
487:
429:
384:
320:
257:
230:
200:is a number called the scaling dimension of
1379:
1365:
1287:
1275:
1248:
1236:
1209:
1203:
1116:
1104:
1071:
1036:
1004:
984:
951:
930:
924:
888:
876:
843:
823:
797:
776:
763:
757:
736:
730:
709:
703:
676:
632:
619:
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185:
161:
155:
126:
106:
54:
862:Non scale-invariant quantum field theory
263:{\displaystyle \langle O(x)O(0)\rangle }
1321:An Introduction to quantum field theory
1161:
785:{\displaystyle \Delta _{1}+\Delta _{2}}
641:{\displaystyle \Delta _{1}+\Delta _{2}}
502:Most scale invariant theories are also
1302:{\displaystyle \epsilon =\phi _{1,3}}
1169:
1167:
1165:
811:{\displaystyle \sigma \times \sigma }
7:
1333:
1331:
1187:
1185:
1150:N=4 supersymmetric Yang–Mills theory
1066:. Therefore the anomalous dimension
90:Scale-invariant quantum field theory
1263:{\displaystyle \sigma =\phi _{1,2}}
173:{\displaystyle \lambda ^{-\Delta }}
927:
885:
878:
773:
760:
733:
706:
629:
616:
589:
562:
412:
399:
306:{\displaystyle (x^{2})^{-\Delta }}
298:
187:
165:
25:
1195:nomenclature, this theory is the
698:of two operators with dimensions
1335:
1052:do not remain constant, but vary
1082:
1076:
1015:
1009:
962:
956:
906:
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481:
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242:
236:
143:{\displaystyle x\to \lambda x}
131:
101:, by definition each operator
71:{\displaystyle x\to \lambda x}
59:
1:
1231:which contains the operators
1351:. You can help Knowledge by
1323:. Reading : Addison-Wesley.
939:{\displaystyle \Delta _{0}}
745:{\displaystyle \Delta _{2}}
718:{\displaystyle \Delta _{1}}
601:{\displaystyle \Delta _{2}}
574:{\displaystyle \Delta _{1}}
270:depends on the distance as
42:, of a local operator in a
18:Anomalous scaling dimension
1424:
1330:
1088:{\displaystyle \gamma (g)}
1021:{\displaystyle \gamma (g)}
968:{\displaystyle \gamma (g)}
696:operator product expansion
659:Interacting field theories
121:acquires under a dilation
831:{\displaystyle \epsilon }
691:whose dimension is 1/8.
1224:{\displaystyle M_{3,4}}
1125:{\displaystyle g=g_{*}}
1028:is a small correction.
851:{\displaystyle \sigma }
684:{\displaystyle \sigma }
193:{\displaystyle \Delta }
1398:Conformal field theory
1347:-related article is a
1303:
1264:
1225:
1193:conformal field theory
1176:Conformal field theory
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1408:Quantum physics stubs
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1178:. New York: Springer.
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671:there is an operator
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552:engineering dimension
533:
504:conformally invariant
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1403:Quantum field theory
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1142:infrared fixed point
1103:
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1056:quantum field theory
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222:correlation function
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99:quantum field theory
80:quantum field theory
53:
44:quantum field theory
977:anomalous dimension
653:anomalous dimension
548:canonical dimension
544:classical dimension
510:Free field theories
32:theoretical physics
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1054:(in the jargon of
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989:
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818:gives an operator
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1345:quantum mechanics
1044:{\displaystyle g}
992:{\displaystyle g}
531:{\displaystyle O}
213:{\displaystyle O}
114:{\displaystyle O}
36:scaling dimension
16:(Redirected from
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1190:
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1138:scale invariant
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96:scale invariant
92:
84:scale invariant
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28:
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15:
12:
11:
5:
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1111:
1108:
1084:
1081:
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1075:
1040:
1017:
1014:
1011:
1008:
988:
975:is called the
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905:
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26:
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3:
2:
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1338:
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1277:
1255:
1252:
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1245:
1241:
1238:
1216:
1213:
1210:
1206:
1198:
1197:minimal model
1194:
1188:
1186:
1182:
1177:
1170:
1168:
1166:
1162:
1155:
1153:
1151:
1145:
1143:
1139:
1135:
1134:beta-function
1117:
1113:
1109:
1106:
1097:
1079:
1073:
1065:
1064:beta-function
1061:
1057:
1053:
1038:
1029:
1012:
1006:
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978:
959:
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931:
903:
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802:
799:
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769:
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658:
656:
655:(see below).
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649:
633:
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593:
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549:
545:
541:
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97:
89:
87:
85:
81:
65:
62:
56:
49:
45:
41:
37:
33:
19:
1353:expanding it
1342:
1320:
1314:
1175:
1146:
1098:
1059:
1030:
976:
865:
693:
662:
652:
650:
551:
547:
543:
513:
501:
93:
39:
38:, or simply
35:
29:
669:Ising model
546:(the terms
1392:Categories
1156:References
1132:where the
665:Lagrangian
540:Lagrangian
516:Lagrangian
1285:ϕ
1278:ϵ
1246:ϕ
1239:σ
1118:∗
1074:γ
1007:γ
954:γ
928:Δ
898:γ
886:Δ
879:Δ
869:couplings
846:σ
826:ϵ
806:σ
803:×
800:σ
774:Δ
761:Δ
734:Δ
707:Δ
679:σ
630:Δ
617:Δ
590:Δ
563:Δ
488:⟩
485:…
430:⟨
425:…
422:−
413:Δ
409:−
400:Δ
396:−
392:λ
385:⟩
382:…
366:λ
337:λ
321:⟨
299:Δ
296:−
258:⟩
231:⟨
188:Δ
166:Δ
163:−
159:λ
150:a factor
135:λ
132:→
78:. If the
63:λ
60:→
48:dilations
40:dimension
919:, where
180:, where
1191:In the
34:, the
1343:This
94:In a
1349:stub
1270:and
725:and
581:and
550:and
1060:run
82:is
30:In
1394::
1184:^
1164:^
1152:.
1144:.
1058:,
858:.
648:.
1380:e
1373:t
1366:v
1355:.
1309:.
1295:3
1292:,
1289:1
1281:=
1256:2
1253:,
1250:1
1242:=
1217:4
1214:,
1211:3
1207:M
1114:g
1110:=
1107:g
1083:)
1080:g
1077:(
1039:g
1016:)
1013:g
1010:(
987:g
963:)
960:g
957:(
932:0
907:)
904:g
901:(
895:+
890:0
882:=
778:2
770:+
765:1
738:2
711:1
634:2
626:+
621:1
594:2
567:1
526:O
482:)
477:2
473:x
469:(
464:2
460:O
456:)
451:1
447:x
443:(
438:1
434:O
417:2
404:1
388:=
379:)
374:2
370:x
363:(
358:2
354:O
350:)
345:1
341:x
334:(
329:1
325:O
292:)
286:2
282:x
278:(
255:)
252:0
249:(
246:O
243:)
240:x
237:(
234:O
208:O
138:x
129:x
109:O
66:x
57:x
20:)
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