20:
28:
1030:
923:β . An addition operation corresponds to a translation in the first embedding. The translations to the second embedding are special conformal transformations, forming translations at infinity. Addition by these transformations reciprocates the terms before addition, then returns the result by another reciprocation. This operation is called the
898:
306:
736:
455:
567:
713:
96:
893:{\displaystyle {\begin{pmatrix}0&1\\1&0\end{pmatrix}}{\begin{pmatrix}1&0\\t&1\end{pmatrix}}{\begin{pmatrix}0&1\\1&0\end{pmatrix}}={\begin{pmatrix}1&t\\0&1\end{pmatrix}}.}
470:
1033:
365:
617:
911:
The translations form a subgroup of the linear fractional group acting on a projective line. There are two embeddings into the projective line of
977:
301:{\displaystyle x'^{\mu }={\frac {x^{\mu }-b^{\mu }x^{2}}{1-2b\cdot x+b^{2}x^{2}}}={\frac {x^{2}}{|x-bx^{2}|^{2}}}(x^{\mu }-b^{\mu }x^{2})\,.}
939:
using infinity but excluding zero. The translations at infinity thus form another subgroup of the homography group on the projective line.
44:
1140:
577:
75:
19:
1130:
1087:"On the advancements of conformal transformations and their associated symmetries in geometry and theoretical physics *"
947:
The term special conformal transformation ("speziellen konformen
Transformationen" in German) was first used in 1962 by
27:
1135:
993:
604:
1048:"Zur physikalischen Deutung und darstellungstheoretischen Analyse der konformen Transformationen von Raum und Zeit"
461:
912:
60:
328:
52:
67:
936:
932:
924:
36:
1086:
312:
1047:
1106:
1067:
992:
Galeriu, CΗlin (2019) "Electric charge in hyperbolic motion: the special conformal solution",
973:
1015:(1911) "On the application of quaternions to some recent developments of electrical theory",
1098:
1059:
1029:
1012:
997:
573:
56:
562:{\displaystyle K_{\mu }=-i(2x_{\mu }x^{\nu }\partial _{\nu }-x^{2}\partial _{\mu })\,.}
1124:
928:
71:
589:
63:, which is the generator of linear fractional transformations that is not affine.
948:
450:{\displaystyle {\frac {x'^{\mu }}{x'^{2}}}={\frac {x^{\mu }}{x^{2}}}-b^{\mu }\,.}
1102:
572:
Special conformal transformations have been used to study the force field of an
1001:
708:{\displaystyle U(q:1){\begin{pmatrix}1&0\\t&1\end{pmatrix}}=U(q+t:1).}
1110:
1071:
1063:
903:
The matrix describes the action of a special conformal transformation.
722:) includes of translations at infinity with respect to the embedding
972:. Graduate texts in contemporary physics. Springer. pp. 97β98.
588:
The inversion can also be taken to be multiplicative inversion of
23:
A coordinate grid prior to a special conformal transformation
856:
817:
781:
745:
644:
59:
of a special conformal transformation involves use of
31:
The same grid after a special conformal transformation
739:
620:
473:
368:
99:
892:
707:
561:
449:
300:
90:A special conformal transformation can be written
16:Special class of linear fractional transformations
8:
1034:Associative Composition Algebra/Homographies
851:
812:
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968:Di Francesco; Mathieu, SΓ©nΓ©chal (1997).
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18:
960:
391:
376:
105:
1017:Proceedings of the Royal Irish Academy
7:
543:
520:
14:
80:special conformal transformations
1028:
45:linear fractional transformation
41:special conformal transformation
931:the parallel operator forms an
699:
681:
636:
624:
552:
493:
291:
255:
242:
217:
76:spherical wave transformations
1:
1085:Kastrup, H.A. (2008-09-18).
1103:10.1002/andp.200852009-1005
1019:29:1β9, particularly page 9
994:European Journal of Physics
605:projective line over a ring
1157:
347:), and another inversion (
311:It is a composition of an
1064:10.1002/andp.19624640706
1002:10.1088/1361-6404/ab3df6
611:) include translations:
61:multiplicative inversion
1046:Kastrup, H. A. (1962).
913:homogeneous coordinates
718:The homography group G(
584:Projective presentation
462:infinitesimal generator
1141:Conformal field theory
970:Conformal field theory
894:
709:
595:. The complex algebra
563:
451:
302:
32:
24:
927:. In the case of the
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710:
599:can be extended to P(
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303:
53:affine transformation
30:
22:
737:
618:
607:. Homographies on P(
471:
366:
97:
68:mathematical physics
1131:Projective geometry
86:Vector presentation
37:projective geometry
1136:Conformal mappings
1091:Annalen der Physik
1052:Annalen der Physik
935:in an alternative
933:addition operation
925:parallel operation
890:
881:
842:
806:
770:
705:
669:
559:
447:
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33:
25:
1097:(9β10): 631β690.
979:978-0-387-94785-3
578:hyperbolic motion
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1058:(7β8): 388β428.
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907:Group property
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603:) through the
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72:conformal maps
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10:
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1025:
1022:
1018:
1014:
1013:Arthur Conway
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989:
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975:
971:
964:
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929:complex plane
926:
922:
918:
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904:
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606:
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598:
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591:
590:biquaternions
583:
581:
579:
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556:
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510:
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500:
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372:
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360:
358:
354:
351: β
350:
346:
342:
339: β
338:
335: β
334:
330:
326:
322:
319: β
318:
314:
295:
286:
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272:
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50:
46:
42:
38:
29:
21:
1094:
1090:
1080:
1055:
1051:
1041:
1036:at Wikibooks
1024:
1016:
1008:
988:
969:
963:
949:Hans Kastrup
946:
920:
916:
910:
902:
727:
723:
719:
717:
608:
600:
596:
592:
587:
571:
459:
356:
352:
348:
344:
340:
336:
332:
324:
320:
316:
310:
89:
79:
65:
48:
40:
34:
329:translation
55:. Thus the
1125:Categories
955:References
70:, certain
57:generation
1111:0003-3804
1072:0003-3804
548:μ
544:∂
530:−
525:ν
521:∂
515:ν
505:μ
488:−
480:μ
439:μ
431:−
414:μ
381:μ
313:inversion
277:μ
269:−
264:μ
225:−
169:⋅
160:−
140:μ
132:−
127:μ
110:μ
74:known as
392:′
377:′
106:′
47:that is
943:History
919:β and
1109:
1070:
996:40(6)
976:
937:field
730::1);
355:/z =
327:), a
323:/x =
43:is a
1107:ISSN
1068:ISSN
974:ISBN
726:β U(
460:Its
78:are
39:, a
1099:doi
1095:520
1060:doi
1056:464
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576:in
464:is
359:β²)
66:In
51:an
49:not
35:In
1127::
1105:.
1093:.
1089:.
1066:.
1054:.
1050:.
951:.
915::
580:.
343:=
82:.
1113:.
1101::
1074:.
1062::
1000::
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877:1
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802:1
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790:0
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724:q
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700:)
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691:t
688:+
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628:q
625:(
622:U
609:B
601:B
597:B
593:B
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538:2
534:x
511:x
501:x
497:2
494:(
491:i
485:=
476:K
445:.
435:b
424:2
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410:x
404:=
396:2
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325:y
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207:x
201:=
193:2
189:x
183:2
179:b
175:+
172:x
166:b
163:2
157:1
150:2
146:x
136:b
123:x
116:=
102:x
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