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For every nonzero vector $\phi$ in a Hilbert space $\mathcal{H}$, let us denote $[\phi] := \mathbb{C}\phi$ the ray associated to $\phi$.

Let $S$ be the set of rays. For all non-zero $\phi,\psi$, $<[\phi],[\psi]> := \vert \langle \phi,\psi\rangle\vert$ is well-defined.

Wigner's theorem states that every bijection $f$ of $S$ preserving $<\cdot, \cdot>$ comes from a unitary or antiunitary operator. Is it still true if we replace the hypothesis "preserve $<\cdot ,\cdot>$" by "for all rays $r_1$, $r_2$, if $<r_1,r_2> = 0$, then $<f(r_1),f(r_2)> = 0$"?

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Yes, it is true: U. Uhlhorn, Representation of symmetry transformations in quantum mechanics, Ark. Fys. 23 (1963), 307–340.

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  • $\begingroup$ Any proof of Uhlhorn's result in a book? "Ark. Phys." has no internet archive where one can find the article as PDF. Thank you! $\endgroup$
    – DanielC
    Jan 29, 2023 at 17:48
  • $\begingroup$ Here you can find a generalisation of the result and this paper is included in the Euclid Project: Orthogonality preserving transformations on the set of n -dimensional subspaces of a Hilbert space Peter Šemrl Illinois J. Math. 48(2): 567-573 (Summer 2004). DOI: 10.1215/ijm/1258138399 $\endgroup$ Jan 29, 2023 at 18:17
  • $\begingroup$ Thank you very much! I wonder why Wigner didn’t think of using the fundamental theorem of projective geometry (I suppose he didn’t since the result follows easily from it). $\endgroup$
    – Plop
    Jan 30, 2023 at 21:46

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