Wigner’s elegant characterization of time reversal

There’s a nice post at The Eternal Universe illustrating discrete symmetries like time reversal. The father of this idea, Eugene Wigner, actually gave a very elegant characterization in his 1931 book:

The following four operations, carried out in succession on an arbitrary state, will result in the system returning to its original state. The first operation is time inversion, the second time displacement by t, the third again time inversion, and the last on again time displacement by t. (Wigner [1931] 1959, p.326.)

I’ve illustrated this assumption below, using the analogy:

  • Reversal of time = Flipping of toy car;
  • Evolution through time = Forward motion of car.

Wigner’s claim is then that whatever time reversal means, the initial state of the car below is the same as its final state.

The beauty is, if you write down this assumption in terms of quantum mechanical operators (and assume energy is positive), then you can quickly see an important property of time reversal — its antiunitarity (roughly, it involves conjugation). In terms of operators, Wigner assumed that:


This immediately implies that


But this is equivalent to the statement,


Now, T can’t be unitary, since then it would follow that Ti = iT, and we could cancel the i to get that THT^{-1} = -H. That’s false if energy is always positive. But Wigner’s theorem says that if a symmetry operator is not unitary, then it must be antiunitary — so T must be antiunitary.

Wigner’s argument seems to have been dropped from most modern textbooks. Perhaps the reason is that not all physical interactions are T-reversible, and Wigner’s assumption (illustrated here with cars) takes for granted that they are. On the other hand, all known physical interactions are CPT-reversible. So, a version of this argument still works if we reinterpret Wigner’s “time reversal” as “CPT reversal.”

6 thoughts on “Wigner’s elegant characterization of time reversal

  1. mattcat83

    Bryan, it looks like whatever text-editor you used to generate those equations just isn’t uploading properly. Was it in Latex? It says that “only textify.com can use mimetex on this server” and that you need to install mimtex.cgi … computers are still evil.

  2. BlackGriffen

    Correct me if I’m wrong, but isn’t CPT = I? So amending it with CPT reversal doesn’t help. I also wonder some at the positive energy requirement. Where does that come from? Is it possible to come up with a formalism where both positive and negative energy states decay toward zero energy? If so, could T then be made unitary?

  3. Bryan

    Hi BlackGriffen. No, CPT is not the identity — it’s the *laws* that are CPT invariant, not the states themselves. On the standard definition, CPT flips time, flips space, and reverses charge — which is not the identity operator. What “CPT-invariance” means is that if ψ(t) is a trajectory allowed by the dynamics of the theory, then so is CPTψ(-t).

    You’re right to question the positive energy requirement though — there are interactions with negative energy eigenstates. I think the only way to justify antiunitarity given this is through a symmetry argument of the kind I give here, which restricts attention to the *free* Hamiltonian. For that Hamilton, energy is certainly always positive, and so the assumption is justified.

  4. BlackGriffen

    The main reason I’m curious about the problem is that, again correct me if I’m wrong, but one of the two main problems with quantizing Einstein style gravity is that it has no lower bound to the allowable energies.

    Also, I’m a concrete kind of guy, so could you give me an example of a state that is not CPT invariant?


  5. Bryan

    BG — just think about a ball sliding down a hill. Say it has position and momentum x(t), p(t). The ball isn’t charged, so its state is invariant under C and P. But applying T (and hence CPT) takes each state x(t), p(t) to a new state x(-t), -p(-t). That is, T-reversal (and hence CPT-reversal) produces a ball sliding *up* a hill. This is a different state — it’s not the identity.

    The “invariance” is that if the trajectory of the first ball is possible, then the trajectory of the second ball is possible, too. So, if you watch a movie of a ball sliding down a hill, you can’t tell if you’re watching the original film in reverse. (Or, more generally, you can’t tell if you’re watching a CPT-reversed film.)