Symmetry transformations on a quantum system; Definitions

Question

We define a symmetry transformation of a system to be any transformation that, when performed, does not change the outcome of a measurement. Wigner's symmetry theorem says that any symmetry of a quantum system is represented by a linear and unitary operator which acts on the Hilbert space of physical states $\mathscr{H}$. So for any symmetry $\mathscr{E}$ there corresponds a unitary transformation $\mathcal{U}(\mathscr{E})$ acting on $\mathscr{H}$.

Suppose $\hat{A}$ is the Hermitian operator corresponding to some observable $A$ which has eigenvalue $\lambda\in\mathbb{R}$ with eigenvector $|\Phi\rangle$. Then, if the system is in a state $|\Psi\rangle$, the probability of measuring the value $\lambda$ of the observable $A$ is given by the Born rule; \begin{align} \text{Prob}(\lambda,\hat{A},\Psi)=\frac{|\langle\Phi|\Psi\rangle|^2}{\langle\Psi|\Psi\rangle\langle\Phi|\Phi\rangle}~. \end{align} We define a symmetry of a quantum system to be one that preserves the above probabilities. However, any unitary operator $\tilde{\mathcal{U}}$ acting on $\mathscr{H}$ will preserve these probabilities; \begin{align} \text{Prob}(\lambda,\hat{A'},\Psi')=\frac{|\langle\tilde{\mathcal{U}}\Phi|\tilde{\mathcal{U}}\Psi\rangle|^2}{\langle\tilde{\mathcal{U}}\Psi|\tilde{\mathcal{U}}\Psi\rangle\langle\tilde{\mathcal{U}}\Phi|\tilde{\mathcal{U}}\Phi\rangle}= \frac{|\langle\Phi|\tilde{\mathcal{U}}^{\dagger}\tilde{\mathcal{U}}\Psi\rangle|^2}{\langle\Psi|\tilde{\mathcal{U}}^{\dagger}\tilde{\mathcal{U}}\Psi\rangle\langle\Phi|\tilde{\mathcal{U}}^{\dagger}\tilde{\mathcal{U}}\Phi\rangle}=\text{Prob}(\lambda,\hat{A},\Psi). \end{align} This would imply that any unitary transformation acting on $\mathscr{H}$ corresponds to a symmetry transformation of the system. I doubt this is true, so where did I go wrong in my definitions and how do I rectify it?

Answer 1

There are many definitions of symmetry in Quantum Theory. However your idea is not correct: symmetries, acting on states, do change outcomes of measurements, at least for one observable. Otherwise we are not speaking of a symmetry but of a gauge transformation.

In the Hilbert space formulation, and I will stick to this case here, and in the absence of superselection rules, a symmetry is a bijective operation that preserves some structure of the space of the states or of the space of observables. As far as I know there are four notions and they are those listed below.

1. Wigner symmetry: a bijective map from the space of rays (unit vectors up to phases) to the space of rays that preserves the probability transitions.
2. Kadison symmetry: a bijective map from the space of states (positive trace-class operators with unit trace) that preserves the convex linear combinations of states, i.e. it preserves the weights of the mixtures of pure states.
3. Kadison symmetry in dual formulation: a bijective map from the lattice of elementary observables (orthogonal projectors in the Hilbert space) into itself preserving the orthocomplemented $\sigma$-complete lattice structure. I.e. it preserves the logical structure of quantum theory.
4. Segal symmetry: a bijective map from the set of bounded everywhere defined self-adjoint operators into the same set that preserves the structure of Jordan algebra of that set, i.e. it preserves the structure of the set of observables.

Each of these definitions can be physically motivated. All definitions lead to the same characterization theorem (for 3 and 4 the Hilbert space must be separable with dimension $\neq 2$ to take advantage of Gleason's theorem).

Theorem [Wigner-Kadison-Segal]. A symmetry of type 1-4 is described by a unitary or antiunitary (depending on the symmetry if the Hilbert space has dimension $>1$ otherwise both possibilities are allowed). Every unitary or anti unitary operator defines a symmetry of type 1-4 simultaneously.

A fundamental symmetry depending on time and continuous with respect to the natural topology (depending on the type 1-4), is the temporal evolution.

A symmetry of type 1-4 (also depending on time) is said to be a dynamical symmetry if it commutes with the temporal evolution.

(At this point, as more or less everybody knows, a theorem similar to Noether's one arises as a consequence of the Stone's theorem provided, as I supposed, the Hilbert space is complex).

When superselection rules enter the game (namely, the center of the von Neumann algebra of observables is non-trivial), the picture becomes more delicate and, for instance, Wigner's notion ceases to be a good notion because the same notion of transition probability becomes ambiguous (pure states and rays are not one-to-one). The same problem arises when the von Neumann algebra of observables admits a non Abelian commutant, as in chromodynamics (even if the center is trivial).