Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 14th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to * Valentine Bargman, _Note on Wigner's theorem on symmetry transformations_, Journal of Mathematical Physics 5.7 (1964): 862-868 ([doi:10.1063/1.1704188](https://doi.org/10.1063/1.1704188)) <a href="https://ncatlab.org/nlab/revision/diff/Wigner+theorem/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/Wigner+theorem/8">v8</a>, <a href="https://ncatlab.org/nlab/show/Wigner+theorem">current</a>

   added pointer to

   • Valentine Bargman, Note on Wigner’s theorem on symmetry transformations, Journal of Mathematical Physics 5.7 (1964): 862-868 (doi:10.1063/1.1704188)

   diff, v8, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 14th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexand added this one (thanks to David R.): * C. S. Sharma and D. F. Almeida, _Additive isometries on a quaternionic Hilbert space_, Journal of Mathematical Physics 31, 1035 (1990) ([doi:10.1063/1.528779](https://doi.org/10.1063/1.528779)) <a href="https://ncatlab.org/nlab/revision/diff/Wigner+theorem/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/Wigner+theorem/8">v8</a>, <a href="https://ncatlab.org/nlab/show/Wigner+theorem">current</a>

   and added this one (thanks to David R.):

   • C. S. Sharma and D. F. Almeida, Additive isometries on a quaternionic Hilbert space, Journal of Mathematical Physics 31, 1035 (1990) (doi:10.1063/1.528779)

   diff, v8, current

3. • CommentRowNumber3.
   • CommentAuthorMiklós
   • CommentTimeNov 14th 2020
   • (edited Nov 14th 2020)
   • PermaLink
   Author: Miklós
   Format: MarkdownItex.

   .

4. • CommentRowNumber4.
   • CommentAuthorMiklós
   • CommentTimeNov 28th 2021
   • PermaLink
   Author: Miklós
   Format: TextThere is a minor inaccuracy in the statment, since anti-unitary operators are not linear (they are anti-linear).

   There is a minor inaccuracy in the statment, since anti-unitary operators are not linear (they are anti-linear).
5. • CommentRowNumber5.
   • CommentAuthorMiklós
   • CommentTimeJul 13th 2022
   • (edited Jul 13th 2022)
   • PermaLink
   Author: Miklós
   Format: MarkdownItexIs this theorem true at all? Let the function $f$ map $(z_1,z_2,\dots)$ to $(\overline z_1, z_2,\dots)$ where the coordinates refer to a Hilbert basis. This is a surjective norm-preserving transformation but isn't unitary or anti-unitary even up to phase.

   Is this theorem true at all? Let the function $f$ map $(z_1,z_2,\dots)$ to $(\overline z_1, z_2,\dots)$ where the coordinates refer to a Hilbert basis. This is a surjective norm-preserving transformation but isn’t unitary or anti-unitary even up to phase.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJul 13th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexThe statement in the entry was missing the condition that the map sends lines to lines, i.e. that it is a map of projective spaces. I have made a quick edit, but no time for more for the moment.

   The statement in the entry was missing the condition that the map sends lines to lines, i.e. that it is a map of projective spaces. I have made a quick edit, but no time for more for the moment.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJul 13th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have now expanded a fair bit, written out the actual statement (starting in a new section "Preliminaries" [here](https://ncatlab.org/nlab/show/Wigner+theorem#Preliminaries)) and also adding an "Idea"-section ([here](https://ncatlab.org/nlab/show/Wigner+theorem#Idea)). <a href="https://ncatlab.org/nlab/revision/diff/Wigner+theorem/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/Wigner+theorem/13">v13</a>, <a href="https://ncatlab.org/nlab/show/Wigner+theorem">current</a>

   I have now expanded a fair bit, written out the actual statement (starting in a new section “Preliminaries” here) and also adding an “Idea”-section (here).

   diff, v13, current