Beyond the CPT theorem

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Virtually all known laws of physics are invariant under the CPT transformation — that is, the combined operations of Charge conjugation (C), Parity or “mirror flipping” (P), and Time reversal (T). What that means is the following. Start with a trajectory \psi(t) through state space, which represents some possible way for a system to change over time according to the known laws of physics. Now transform that trajectory, by reversing the order of all the states, and then applying C, P and T to each of them:

    \[ \psi(t) \mapsto CPT\psi(-t). \]

CPT-invariance means that the resulting trajectory CPT\psi(-t) will also be a possible according to the laws. One can check that this is equivalent to the statement that CPT commutes with the Hamiltonian, [CPT,H]=0.

Why is CPT so often symmetry? There is a theorem that explains it: if we characterize quantum field theory in a very plausible and general way (such as by the Wightman axioms or Haag axioms), and in particular assume that it admits a unitary representation of the Poincaré group, then CPT-invariance is guaranteed. This result is called the CPT theorem. See Borchers & Yngvason for a very readable proof in the Haag framework.

Ok, that’s the background for today. Now:

Question: as we move forward, and begin to adopt theories that go beyond the standard model of particle physics, will we continue to have a CPT theorem or something like it?

It is widely believe that the CPT may fail in generic extensions of quantum theory. In particular, the requirement of a unitary representation of the Poincaré group is pretty strong, and may not hold in the kind of general context of interest in quantum gravity. Just search for CPT-violation on the arxiv to see what I mean.

But there is a sense in which something “like” a CPT theorem probably will hold in physics beyond the standard model. That sense is this: every unitary dynamics admits infinitely many “time reversing transformations” (i.e., time reversal plus some other linear symmetries) under which the dynamics is invariant. Here’s a more careful statement of this fact.

Fact. Let \mathcal{H} be a separable Hilbert space with a unitary group U_t = e^{-itH} describing the quantum dynamics, and let T be the (antiunitary) time reversal operator. Then there exists a unitary operator \Theta such that the dynamics is \Theta T-invariant, in that [\Theta T,H]=0.

Think of \Theta as some kind of generalized symmetry transformation, similar to CP, but something else entirely. It is in this sense that this fact expresses something like the CPT-theorem, although unlike the CPT-theorem the mathematics is completely trivial.

There are two steps to seeing why this “Fact” is true. The first is to observe that, for every self-adjoint operator H, there is something called a conjugation operator K_H such that [K_H,H]=0. Here’s how it’s defined. The self-adjoint operator H comes with its own basis set for the Hilbert space, {v_1,v_2,v_3,\dots}. That’s because of the spectral theorem. So for every vector \psi in the Hilbert space there are complex constants c_i that allow you to write that vector,

    \[\psi = c_1v_1 + c_2v_2 + c_3v_3 + \cdots.\]

The conjugation operator K_H is just the operator defined by conjugating all the complex constants of a vector written in the H-basis,

    \[K_H\psi = c_1^*v_1 + c_2^*v_2 + c_3^*v_3 + \cdots.\]

So that’s pretty easy. And it’s easy to check that K_H satisfies some special properties: it is antilinear K_H(a\psi+b\phi)=a^*K_H\psi+b^*K_H\phi, antiunitary \langle K_H\psi,K_H\phi\rangle = \langle \psi,\phi \rangle^*, and it commutes with the H that we used to define it [K_H,H]=0. We will use all of these properties in the next step.

The second step to seeing why our “Fact” is true is to recognize that if K_1 and K_2 are any two antiunitary operators, then they are related by a unitary operator, K_2 = UK_1. It’s a nice exercise to check for yourself that this is true, but if you get stuck, try here.

Since time reversal T and the conjugation operator K_H for the Hamiltonian are both anitunitary, this means that K_H is related to T by some unitary operator \Theta:

    \[K_H = \Theta T.\]

So, there is always a unitary operator \Theta such that \Theta T commutes with the Hamiltonian H.

Above, I said there were actually infinitely many such operators. Puzzle: Can you work out why?

If you’re impatient, here’s the reason. Let f:\mathbb{R}\rightarrow\mathbb{C} be a function (a Borel function if we’re being pedantic), and let f(H) be the corresponding Hilbert space operator as a function of the Hamiltonian H. (For example, if f(x)=x^2, then f(H)=H^2=H\circ H.) Every such function f(H) commutes with H. And we already know that K_H does as well. So their composition commutes with H as well:

    \[0 = [f(H)K_H,H] = [f(H)\Theta T,H] = [\Theta^\prime T, H],\]

where \Theta^\prime = f(H)\Theta. Since there are infinitely many such functions, this means that there are infinitely many such operators \Theta^\prime.

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