Linear or conjugate operators and automorphisms on the lattice of subspaces

[Fredrik]

I'm reading about symmetries in QM in "Geometry of quantum theory" by Varadarajan. In one of the proofs, he refers to theorem 2.1, which is stated without proof. He says that the theorem is proved in "Linear algebra and projective geometry" by Baer. That isn't very helpful, since he doesn't even mention what part of the book I should be looking at. Also, since I have no interest in vector spaces over fields (or division rings) other than ℂ, I'm not particularly eager to search for the relevant stuff in Baer's book. So I'm wondering if someone recognizes this theorem, and can give me better reference.

I think I will only need a simplified version of the theorem: Let V be a finite-dimensional inner product space over ℂ. Let L be the lattice of subspaces of V, partially ordered by inclusion. For each T:V→V, we define $\alpha_T:L\to L$ by $\alpha_T(M)=T(M)$ for all M in L.

(a) If T:V→V is bijective, and either linear or conjugate linear, then $\alpha_T$ is an automorphism of L.
(b) If $\alpha:L\to L$ is an automorphism of L, then there's either a linear bijection T:V→V such that $\alpha=\alpha_T$, or a conjugate linear bijection T:V→V such that $\alpha=\alpha_T$. This T is unique up to multiplication by a complex number.

Part (a) is probably easy to prove. I just haven't tried it yet. I expect (b) to be a bit tricky though.

Edit: The thread title should say "linear or conjugate linear operators", not "unitary/antiunitary operators". I was half-way through this post before I realized that the theorem doesn't even mention an inner product.

 

[micromass]

I found the relevant theorem in Baer's text. It is essentially section III.1 that is important.
I'll type the proof out for you in the complex case, but it may take a while because it's fairly long.

However, I'm not sure that the theorem in your OP even holds true. Both Varadarajan and Baer say something different. They say that for each automorphism $\alpha:L\rightarrow L$, there exists an automorphism $\theta:\mathbb{C}\rightarrow \mathbb{C}$ and a bijective map $T:V\rightarrow V$ such that

$$T(\alpha v+\beta w)=\theta(\alpha) T(v)+\theta(\beta)T(w)$$

This map T is linear if $\theta$ is the identity and is conjugate-lineari if $\theta$ is the complex conjugation. But there are more automorphisms of $\mathbb{C}$ than just those two.

PS I edited the thread title for you.

 

[Fredrik]

Thanks. Varadarajan mentioned after the theorem that there are infinitely many automorphisms of ℂ, and added that "the identity and complex conjugation are the only analytically well-behaved ones (e.g. measurable, bounded, etc)". But you're right, the theorem doesn't say that there's a well-behaved automorphism, it just says that there's an automorphism...

I will take a look at the theorem in Baer tomorrow. Thanks for finding it for me. You may want to wait until I've had a chance to look at it before you start typing up a version of it just for me.

By the way, the theorem I'm actually interested in is 4.29. Its proof refers to theorem 4.27, and the proof of 4.27 refers back to theorem 2.1. I still haven't read enough to know what exactly I need from theorem 2.1 or Baer's book.

I'm also reading an article from 1964 by Bargmann, "Note on Wigner's theorem on symmetry operations", that's supposed to prove a version of theorem 4.29 in an "elementary" way. It looks like it doesn't require anything fancier than the projection theorem, but it's written in a way that makes it hard to follow anyway.