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nLab > Latest Changes: Wigner theorem

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 14th 2019

    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

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 14th 2019

    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

    • CommentRowNumber3.
    • CommentAuthorMiklós
    • CommentTimeNov 14th 2020
    • (edited Nov 14th 2020)

    .

    • CommentRowNumber4.
    • CommentAuthorMiklós
    • CommentTimeNov 28th 2021
    There is a minor inaccuracy in the statment, since anti-unitary operators are not linear (they are anti-linear).
    • CommentRowNumber5.
    • CommentAuthorMiklós
    • CommentTimeJul 13th 2022
    • (edited Jul 13th 2022)

    Is this theorem true at all? Let the function ff map (z 1,z 2,…)(z_1,z_2,\dots) to (z¯ 1,z 2,…)(\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.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 13th 2022

    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.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJul 13th 2022

    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

1 to 7 of 7

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