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Wigner's theorem states that every symmetry of complex Hilbert space is either unitary or anti-unitary, up to multiplication by a unit scalar function. Here $f:\mathcal{H} \rightarrow \mathcal{H}$ is called a symmetry if it is a bijection and $\left| \langle f(\mathbf{v}),f(\mathbf{w}) \rangle \right| = \left| \langle \mathbf{v},\mathbf{w} \rangle \right| $ for all pairs of vectors.

In the real case, the assertion is that every symmetry must be unitary (again up to multiplication by a unit scalar function). I follow more easily the a proof of the real case, so I wonder.

Is there a reasonable way to derive the complex version from the real version of Wigner’s theorem?

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  • $\begingroup$ The statement of Wigner's theorem as you write it does not make much sense: what is a symmetry of a complex Hilbert space? Rather, I would phrase the theorem as: any symmetry of a quantum state space is representable by either a unitary or antiunitary operator of the underlying complex Hilbert space. $\endgroup$
    – johndoe
    Sep 12, 2012 at 17:46
  • $\begingroup$ I opted for the classical physics language over the mathematically precise statement. For the original form of Wigner's theorem this seems best to me, but I may have been hanging with too many physicists lately to be trusted here. For generaliations one is better off discussing projective Hilbert space, as you suggest. I have in mind the elegant paper by Freed and Moore " twisted equivariant matter." arXiv:1208.5055 $\endgroup$
    – Terry Loring
    Sep 12, 2012 at 18:30
  • $\begingroup$ This sort of does what you want, but I don't vouch for its correctness: arxiv.org/abs/0802.3624 It has similarities to this argument but uses a step from affine geometry: academia.edu/1865537/… $\endgroup$
    – Dan Piponi
    Feb 5, 2013 at 0:19

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I think the answer to the question is "no", but from the follow-up comment I gather the underlying question is "what proof of Wigner's theorem can one give to physics students". Here's what I would do, inspired by http://arxiv.org/abs/0808.0779

I would focus on the case of an $N=2$ dimensional Hilbert space of a spin-1/2 particle. Then every state is associated with a unit vector $\hat{n}$ on the Bloch sphere, pointing in the direction of the spin. The transition probability between two states $\hat{n}_1$ and $\hat{n}_2$ is given by the angle between the two vectors, $\frac{1}{2}(1+\hat{n}_1\cdot\hat{n}_2)$, so that orthogonal states correspond to opposite points on the sphere. We seek the most general transformation that preserves the transition probability, so it should preserve the angle between any two vectors.

I would then invoke a theorem which I hope is familiar to the students, that any angle-preserving transformation of the sphere onto itself is either a rotation or a rotation followed by a reflection. Wigner's theorem for $N=2$ then follows from the fact that a rotation in the Bloch sphere corresponds to a unitary operation on the state and a reflection corresponds to complex conjugation of the state.

arxiv:0808.0779 shows how the case $N>2$ can be obtained by induction starting from $N=2$, but I think the $N=2$ proof is instructive enough for physics students.

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  • $\begingroup$ This is very helpful. I will leave the question open a while to see if there is a positive answer. There is not likely to be a way to prove that the answer is no. Continuing this train of thought: in the real case, I can consider the unit ball in three-space and the quotient space is $RP^2.$ I assume if I ponder a little longer I see how in a special case with real scalars I only see the unitary option. I really like to see how changing scalar field changes results. $\endgroup$
    – Terry Loring
    Sep 12, 2012 at 19:56
  • $\begingroup$ I suppose real projective space and complex projective space are enough different that it is unlikely one can derive either version of Wigner's theorem from the other. I think I have my answer. $\endgroup$
    – Terry Loring
    Sep 13, 2012 at 2:10
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Not an answer to the question, but there is Freed's short proof of the Wigner theorem.

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  • $\begingroup$ Thanks for the link. However, I am hoping for a proof physics students can follow easily. Of course, there are physics students and there are physics students. $\endgroup$
    – Terry Loring
    Sep 12, 2012 at 16:55

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