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I posted this on math StackExchange and got no replies, so I'm trying my luck here!

I'm a fourth year physics and math student who is writing up a report on some quantum mechanical symmetries and their consequences. The "audience" for my paper are either senior physics majors or first year graduate students in physics. Wigner's theorem's mathematical content is unfortunately beyond the scope of my report so I decided to "water it down". My question is this: is this an acceptably simplified version of Wigner's theorem?

Theorem (Wigner) Let $\Psi, \Phi$ be arbitrary state vectors in a Hilbert space $\mathscr{H}$. Suppose $\delta: \mathscr{H} \mapsto \mathscr{H}$ is bijective and Wigner symmetric (that is, under the mapping $\delta$, the transition probability $\langle \Psi | \Phi \rangle$ is unchanged). Then

  1. there exists a linear and unitary or antilinear and antiunitary operator $U$ such that $$\delta(\psi) = U \psi U^{-1},$$
  2. $U$ is unique up to an arbitrary phase factor.

Is there anything I'm misunderstanding with this formulation of Wigner's theorem?

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$\let\d=\delta \def\sH{\mathscr H}$ Some amendments are needed.

1) This is merely notational. Don't write $\d:\sH\mapsto\sH$ but rather $\d:\sH\to\sH$. "$\to$" is used between domain and codomain of a map. "$\mapsto$" between a point in domain and its image in codomain. Example: $$f:\Bbb C \to \Bbb R,\quad z \mapsto f(z) = |z|^2.$$

2) Transition probability is $|\langle\Psi|\Phi\rangle|^2$. This is the right definition ot transition probability in QM. The difference is also fundamental for the meaning of theorem, as conservation of $\langle\Psi|\Phi\rangle$ is much stronger and is essentially the theorem's thesis.

3) Unitary and antiunitary imply linear and antilinear respectively. So those clauses could and should be suppressed.

4) Not $\d(\psi) = U \psi U^{-1}$, but $\d(\Psi) = U \Psi$ (note capitals).

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  • $\begingroup$ In addition to Elio's remarks, I stress that the hypotheses of Wigner theorem demand that $\delta$ tranforms rays into rays bijectively, not vectors into vectors. $\endgroup$ – Valter Moretti Dec 11 '18 at 19:54
  • $\begingroup$ @Valter Moretti Hi Valter! You're right. In my excuse I tried to adhere to OP's request: is this an acceptably simplified version of Wigner's theorem? And decided to leave rays apart. Of course it remains to be seen if this is an "acceptably simplified version". $\endgroup$ – Elio Fabri Dec 11 '18 at 20:29
  • $\begingroup$ Hi Elio. Barring the issue of the rays, it is acceptable, I think. One other minor remark it is that the nature unitary or antiunitary of $U$ is decided by $\delta$ unless $\dim H =1$ where both cases are admitted for the same $\delta$. $\endgroup$ – Valter Moretti Dec 11 '18 at 20:52

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