Wigner’s symmetry representation theorem

Introduction

I will assume familiarity with standard quantum mechanics, written using ‘Dirac notation’: \[\left\langle \psi|\varphi\right\rangle\]

This is an unnormalized transition amplitude between states: \(\varphi\rightarrow\psi\). The amplitude becomes a probability: \[P_{\varphi\rightarrow\psi}=\frac{\left|\left\langle \psi|\varphi\right\rangle \right|^{2}}{\left\langle \psi|\psi\right\rangle \left\langle \varphi|\varphi\right\rangle }\]

This represents the probability of measuring the system as being in the state \(\psi\) from a system prepared in the state \(\varphi\). There are linear/vector relations between states, and this is reflected in the Dirac ‘bra-kets’: \[\left\langle \psi|\alpha\varphi_{1}+\beta\varphi_{2}\right\rangle =\alpha\left\langle \psi|\varphi_{1}\right\rangle +\beta\left\langle \psi|\varphi_{2}\right\rangle\]

The final important relation is \(\left\langle \psi|\varphi\right\rangle =\left\langle \varphi|\psi\right\rangle ^{*}\), where the asterisk indicates complex conjugation. This has the implication that \(\left\langle \psi|\psi\right\rangle\) and \(\left\langle \varphi|\varphi\right\rangle\) are real. Also: \[P_{\varphi\rightarrow\psi}=\frac{\left\langle \psi|\varphi\right\rangle \left\langle \varphi|\psi\right\rangle }{\left\langle \psi|\psi\right\rangle \left\langle \varphi|\varphi\right\rangle }\]

An important observation is that for a non-zero complex number \(\alpha\), the state \(\alpha\varphi\) is the same as \(\varphi\). In fact if you work through the relations you find the probabilities are unchanged: \[P_{\alpha\varphi\rightarrow\beta\psi}=P_{\varphi\rightarrow\psi},\left(\alpha,\beta\neq0\right)\]

The states represented by the vectors \(\varphi,\psi\) can also be represented by \(\alpha\varphi,\beta\psi\) with \(\alpha,\beta\neq0\). The states correspond to the ‘rays’ \(\left[\varphi\right]=\left\{ \alpha\varphi:\alpha\in\mathbb{C}-\left\{ \varnothing\right\} \right\}\). The vector \(\varphi\) is a ‘representative’ of the ray \([\varphi]\).

We are interested in symmetry relations between states — in other words injective (one-to-one) mappings of states in the same or different system that maintain the probabilities between the transformed states. It will be found that these can all be represented by unitary (\(U\)) or antiunitary (\(A\)) operators (not both, in general) on the states vectors \(\varphi,\psi\). These operators, if they exist, satisfy \(\left\langle U\psi|U\varphi\right\rangle =\left\langle \psi|\varphi\right\rangle\) or \(\left\langle A\psi|A\varphi\right\rangle =\left\langle \varphi|\psi\right\rangle\). The operator \(U\) is linear: \(U\left(\alpha\varphi_{1}+\beta\varphi_{2}\right)=\alpha U\varphi_{1}+\beta U\varphi_{2}\). By contrast, \(A\) is antilinear: \(A\left(\alpha\varphi_{1}+\beta\varphi_{2}\right)=\alpha^{*}A\varphi_{1}+\beta^{*}A\varphi_{2}\). This is Wigner’s symmetry representation theorem. There are many complex proofs and extensions thereof. I will base my work on the two that I understand, given essentially by Bargmann [pdf] and Uhlhorn [only abstract and offline journal details]. The latter I have seen only in the form given by Chevalier. [My bra-ket definition is linear on the second place, and anti-linear on the first place, as usual in the physics literature. Chevalier uses the opposite convention, more common among mathematicians.]

Strictly the operators are only (anti)unitary if they are surjective (onto), allowing an inverse to be defined, giving bijectivity (one-to-one onto). Then the inverse operations can be related to the adjoints: \(U^{-1}=U^{\dagger}\) and \(A^{-1}=A^{\dagger}\). [The respective adjoints are defined by \(\left\langle U\psi|\varphi\right\rangle =\left\langle \psi|U^{\dagger}\varphi\right\rangle\) and \(\left\langle A\psi|\varphi\right\rangle =\left\langle A^{\dagger}\varphi|\psi\right\rangle\).]

We note that if the vector \(0\) is the only one that is mapped onto \(0\) by an additive mapping, then the mapping is injective since: \[A(\varphi)=A(\psi)\implies A(\varphi-\psi)=0\implies \varphi-\psi=0\implies \varphi=\psi.\] The symmetries we discuss are mappings of rays on rays, and the 0 vector represents no ray, therefore injectivity is almost automatic from the derivations.

We also include a bonus page, which derives from Chevalier’s comments on the Bargmann approach: Additive bounded mappings.