Comments on Soul Physics: Wigner's elegant characterization of time reversal...

BG -- just think about a ball sliding down a hill....

2010-03-26T07:25:44.981-04:00

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.)

The main reason I'm curious about the problem ...

2010-03-25T23:10:58.853-04:00

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?

Thanks.
BG

Hi BlackGriffen. No, CPT is not the identity -- it...

2010-03-25T20:55:25.165-04:00

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.

Correct me if I'm wrong, but isn't CPT = I...

2010-03-25T18:30:59.368-04:00

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?

Thanks Matt -- should be fixed now.

2010-03-25T10:23:54.742-04:00

Thanks Matt -- should be fixed now.

Bryan, it looks like whatever text-editor you used...

2010-03-25T10:11:13.170-04:00

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.