1232:, it is difficult to experimentally examine this symmetry as it is not possible to transform an object under this symmetry. The indirect evidence of this symmetry is given by how accurately fundamental theories of physics that are invariant under this symmetry make predictions. Other indirect evidence is whether theories that are invariant under this symmetry lead to contradictions such as giving probabilities greater than 1. So far there has been no direct evidence that the fundamental constituents of the Universe are strings. The symmetry could also be a
25:
145:. They are less studied in physics because, unlike the rotations and translations of Poincaré symmetry, an object cannot be physically transformed by the inversion symmetry. Some physical theories are invariant under this symmetry, in these cases it is what is known as a 'hidden symmetry'. Other hidden symmetries of physics include
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Because the only invariants under this symmetry involve a minimum of 4 points, this symmetry cannot be a symmetry of point particle theory. Point particle theory relies on knowing the lengths of paths of particles through space-time (e.g., from
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The invariants for this symmetry in 4 dimensions is unknown however it is known that the invariant requires a minimum of 4 space-time points. In one dimension, the invariant is the well known
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These transformations are subgroups of general 1-1 conformal transformations on space-time. It is possible to extend these transformations to include all 1-1 conformal transformations on
231:) so that space-time is Euclidean and the equations are simpler. The Poincaré transformations are given by the coordinate transformation on space-time parametrized by the 4-vectors
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is an orthogonal matrix. This transformation is 1-1 meaning that each point is mapped to a unique point only if we theoretically include the points at infinity.
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gives a third transformation of the same form. The basic invariant under this transformation is the space-time length given by the distance between two
846:{\displaystyle V_{\mu }^{\prime }=\left(O_{\mu }^{\nu }V_{\nu }+P_{\tau }\right)\left(\delta _{\tau \mu }+Q_{\tau \mu }^{\nu }V_{\nu }\right)^{-1}.\,}
42:
1236:
meaning that although it is a symmetry of physics, the
Universe has 'frozen out' in one particular direction so this symmetry is no longer evident.
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Applying this transformation twice on a 4-vector gives a transformation of the same form. The new symmetry of 'inversion' is given by the 3-tensor
575:{\displaystyle V_{\mu }^{\prime }=\left(A_{\tau }^{\nu }V_{\nu }+B_{\tau }\right)\left(C_{\tau \mu }^{\nu }V_{\nu }+D_{\tau \mu }\right)^{-1}.}
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is a conformal function of the 4-dimensional invariant. A string field in endpoint-string theory is a function over the endpoints.
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We must also have an equivalent condition to the orthogonality condition of the
Poincaré transformations:
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began to publish on transformations of the plane generated by inversion in a circle of radius
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arithmetic. In the company of physicists employing the inversion transformation early on was
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Although it is natural to generalize the
Poincaré transformations in order to find hidden
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in which the strings are uniquely determined by their endpoints. The
304:{\displaystyle V_{\mu }^{\prime }=O_{\mu }^{\nu }V_{\nu }+P_{\mu }\,}
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in physics and thus narrow down the number of possible theories of
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Because one can divide the top and bottom of the transformation by
169:. His work initiated a large body of publications, now called
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Type of transformations applicable to coordinate space-time
185:, and the association with him leads it to be called the
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for this theory for a string starting at the endpoints
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is a 4-vector. Applying this transformation twice on a
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49:. Unsourced material may be challenged and removed.
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879:This symmetry becomes Poincaré symmetry if we set
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173:. The most prominently named mathematician became
1051:{\displaystyle {\frac {(x-X)(y-Y)}{(x-Y)(y-X)}}.}
1315:"THE POINCARE DISK MODEL OF HYPERBOLIC GEOMETRY"
197:In the following we shall use imaginary time (
177:once he reduced the planar transformations to
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109:Learn how and when to remove this message
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1102:). The symmetry can be a symmetry of a
646:{\displaystyle AA^{T}+BC=DD^{T}+CB\,}
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1251:Coordinate rotations and reflections
47:adding citations to reliable sources
931:the second condition requires that
699:to the unit matrix. We end up with
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679:we lose no generality by setting
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34:needs additional citations for
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141:transformations on coordinate
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1209:{\displaystyle \phi (x,X).\,}
193:Transformation on coordinates
1142:and ending at the endpoints
129:are a natural extension of
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416:{\displaystyle r=|x-y|.\,}
366:points given by 4-vectors
161:In 1831 the mathematician
58:"Inversion transformation"
127:inversion transformations
131:Poincaré transformations
175:August Ferdinand Möbius
1348:Functions and mappings
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965:Möbius transformations
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163:Ludwig Immanuel Magnus
1291:"Chapter 5 Inversion"
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1167:{\displaystyle (y,Y)}
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224:{\displaystyle t'=it}
1256:Spacetime symmetries
1246:Rotation group SO(3)
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898:{\displaystyle Q=0.}
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123:mathematical physics
43:improve this article
1230:high-energy physics
924:{\displaystyle Q=0}
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872:{\displaystyle Q.}
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672:{\displaystyle D,}
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171:inversive geometry
151:general covariance
1343:Conservation laws
1220:Physical evidence
1095:{\displaystyle y}
1075:{\displaystyle x}
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944:{\displaystyle O}
692:{\displaystyle D}
351:{\displaystyle P}
336:orthogonal matrix
327:{\displaystyle O}
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1296:. Archived from
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187:Kelvin transform
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1234:broken symmetry
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133:to include all
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1303:on 2021-07-16.
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179:complex number
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147:gauge symmetry
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1104:string theory
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99:November 2019
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88:
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70:
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63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
1309:
1298:the original
1285:
1271:superstrings
1261:CPT symmetry
1223:
1060:
958:
855:
655:
584:
425:
371:
367:
313:
232:
196:
166:
160:
126:
120:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
961:cross-ratio
183:Lord Kelvin
1332:Categories
1277:References
1226:symmetries
1108:propagator
955:Invariants
428:space-time
364:space-time
143:space-time
139:one-to-one
69:newspapers
1185:ϕ
1034:−
1019:−
1002:−
987:−
832:−
821:ν
811:ν
806:μ
803:τ
790:μ
787:τ
783:δ
766:τ
753:ν
743:ν
738:μ
720:′
715:μ
562:−
551:μ
548:τ
535:ν
525:ν
520:μ
517:τ
496:τ
483:ν
473:ν
468:τ
450:′
445:μ
399:−
370:and
296:μ
283:ν
273:ν
268:μ
255:′
250:μ
157:Early use
135:conformal
1338:Symmetry
1240:See also
360:4-vector
209:′
83:scholar
334:is an
314:where
85:
78:
71:
64:
56:
1318:(PDF)
1301:(PDF)
1294:(PDF)
963:from
905:When
90:JSTOR
76:books
338:and
149:and
62:news
1082:to
121:In
45:by
1334::
967::
893:0.
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189:.
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246:V
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219:t
216:i
213:=
206:t
167:R
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
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