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?
> 
> 
> 



Can one derive Wigner’s theorem in the complex case from the real case?
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. 


