What is CPT, really?

Question

The naive statement for the "CPT theorem" one usually finds in the literature is "relativistic theories should be CPT invariant". It is clear that this statement is not true as written, e.g. topological theories are typically not invariant under CPT. A much more precise statement of CPT is found e.g. in Freed's "Five lectures on SUSY", namely (paraphrased)

In a local QFT the CPT theorem states that representations of the connected component of the Poincaré group can be lifted to representations of the whole group (i.e., containing reflections and time-inversions).

This is much better, because it explicitly excludes topological theories (inasmuch as these have no propagating degrees of freedom, i.e., the Hilbert space does not contain irreps of Poincaré). It also deals with the Hilbert space directly, and so it applies to e.g. non-lagrangian theories.

That being said, I am still unsure what the "theorem" is really doing for us. Is it really a theorem, or rather an axiom? Are we to impose it when constructing theories, or should it follow automatically?

The main reason I am confused can be illustrated by considering the standard construction of supermultiplets. For example, if we take a massless multiplet whose highest weight has helicity 0, and act on the latter with the SUSY generators, we also find states of helicity 1/2 and 1. At this point, every book says that, by CPT, the correct multiplet must contain the CPT conjugate, i.e., states of helicity -1/2 and -1. One thus obtains the standard vector multiplet. This application of CPT exactly follows Freed's statement: the first half 0,1/2,1 is a good irrep of the connected component of (super)Poincaré, but does not lift by itself; we are to enlarge it by its conjugate so that the result does lift.

It seems that here we are imposing CPT invariance, rather than observing that it holds. In other words, what if I refused to include the CPT conjugate in the multiplet? Then CPT would be violated, and so the theorem is not really a theorem, for I can construct theories where it does not hold. Instead, it seems that, in constructing theories, I should impose CPT, i.e., it is an axiom. Is this understanding correct? Or perhaps it turns out that if I tried to construct a theory with the half multiplet only, i.e., helicity 0,1/2,1 (and no conjugate), the result ends up being pathological for some reason?

^{A similar situation is found when constructing non-supersymmetric states. Here a state of helicity +1 is typically packaged together with its CPT conjugate -1, but this is done for phenomenological reasons: as Weinberg explains (page 73), electromagnetic phenomena is observed to be invariant under parity, and so the existence of a state of helicity +1 requires the existence of one with helicity -1. But if we are interested in QFT for purely theoretical reasons, then it is perfectly sensible to try and construct theories of particles of helicity +1 that violate parity symmetry -- this is specially so for SUSY, where no phenomenological data exists!}

Answer 1

As you have mentioned, when we Wick rotate to Euclidean signature, the four-component Lorentz group $O(d,1)$ becomes the two-component $O(d+1)$. Let's suppose we have infinitesimal Lorentz symmetry. Then our Euclidean signature correlation functions enjoy the full symmetry of $SO(d+1)$, which is the connected component of the identity. Not all of these symmetries will descend to operators on the Hilbert space. However, transformations that fix a spatial slice will define such operators for us. An example of such an operator is a $\pi$ rotation in a plane containing one direction of space and one direction of time. This will give us our CRT symmetry (involving just a single reflection of space---we can get CPT for odd $d$ by combining with some space rotations).

You can decide for yourself whether or not to regard this as a proof. However, I do not know any counterexamples. You mention that some TQFTs don't have CPT symmetry. I suppose you are talking about chiral theories, but note that while CPT (or CRT) is anti-unitary, it also reverses the orientation of space, so something like a Chern-Simons term is actually invariant. Maybe you mean something else though?

By the way C, R, and T are all meaningless on their own (without extra assumptions on their existence). The way I like to think about the theorem is that it says: reflect your system somehow. Now reverse the direction of time. The theorem says there is guaranteed to be some internal transformation "C" which if we now apply C we have performed a symmetry. For free complex fermions for instance C happens to be charge conjugation if R and T are the usual ones. For real free fermions C is the identity.

Also on that note what we call CRT is possibly ambiguous up to internal symmetries (as well as up to rotations in space). In a recent paper of ours, we needed to actually pin down a "canonical" CRT, which is basically the one which comes "directly" from the argument I made above, for some anomaly matching reasons. We were able to do this by breaking all internal symmetries by hand, but you can also think about the analytic continuation more carefully to write an expression for the CRT transformation in terms of the boost matrix $M$ in the $x^0, x^1$ plane where we do our rotating. It is

$$O(x^0,x^1,...) \mapsto (i^F e^{i \pi M}O(-x^0,-x^1,...)e^{-i\pi M})^\dagger,$$

where $F$ is the fermion parity of $O$.

Answer 2

What makes CPT special

As an analogy, consider Noether's theorem. The justification for calling a conserved quantity "energy" doesn't come from considering any single theory by itself. It comes from the idea that time-translation symmetry combined with the action principle always gives a conserved quantity, along with a recipe for constructing the conserved quantity in terms of those general ingredients. In other words, the justification for calling it "energy" comes from looking at a whole family of theories.

Similarly, the justification for calling a symmetry CPT (instead of just PT-like) comes from the idea that a certain list of conditions (Lorentz symmetry, microcausality, ...) always imply such a symmetry. In other words, it comes from considering a whole family of theories, some of which might have more than one PT-like symmetry (modulo the full Poincaré group). The theorem is what picks out one of those PT-like symmetries as special, and that's the one we call CPT.

...but not as special as sometimes advertized

What is the most general formulation of the CPT theorem? I don't think that dust has settled yet,$^{[1]}$ but one condition that seems to be essential is Lorentz symmetry. CPT symmetry is not required for a QFT to be consistent, just like Lorentz symmetry is not required for consistency.$^{[2]}$ CPT symmetry, like Lorentz symmetry, is presumably something that we should only expect to hold as an approximation in a sufficiently small region of spacetime, in a QFT with a Lorentzian background metric.

^{${[1]}$ I can't rule out the possibility that somebody will discover some natural generalization of the CPT theorem that doesn't rely on any concept of spacetime symmetry but that implies the usual CPT theorem in the case of Lorentz-symmetric theories.}

^{${[2]}$ We can always start with a lattice QFT, and the difficult question of the existence of a nontrivial continuum limit is beside the point here.}

Theorem or axiom?

A CPT theorem singles out one PT-like symmetry (modulo Poincaré transformations) as being special, even in theories that have more than one. As an axiom, we might as well just call it a PT axiom, because the axiom doesn't care if a theory overachieves by having more than one PT-like symmetry, as long as it has at least one.

Consider Freed's description of the CPT theorem as stating that representations of the connected component of the Poincaré group can be lifted to representations of the whole group (at least the subgroup generated by an even number of reflections). Is the lift always unique? If not, then we have the same issue as before: we might as well just call it a PT-axiom, because the axiom doesn't care if a theory overachieves by admitting more than one such lift.

Supermultiplets

If membership in a (super)multiplet is supposed to be governed by spacetime symmetries, then we need to decide — as a matter of convention — whether or not that government should include PT-like symmetries. That's only an option if the theory actually has any PT-like symmetries, which we could enforce either by directly imposing the existence of such a symmetry as an axiom, or by imposing the conditions of a CPT theorem as axioms.

what if I refused to include the CPT conjugate in the multiplet? Then CPT would be violated...

If a theory doesn't have any states that otherwise could be used to "complete" the multiplet (C)PT-style, then it must violate one of the conditions of a (C)PT theorem — but not Lorentz symmetry, because that's implied by SUSY. So it must violate some other condition, like microcausality, and that would normally be called "pathological." I haven't tried to contrive any examples of this.

Answer 3

I'd say it's a theorem, or at least a physics theorem. The standard proofs (e.g., Weinberg Vol I) always have loopholes in them, but they cover the typical cases pretty well. Generally if you have a relativistic QFT with local degrees of freedom that violates CPT, you expect it to have trouble with causality. Maybe one also wants perturbativity hypotheses, so that one can safely reduce to the free fields.

EDIT: Misunderstood what OP meant by multiplets. When people are writing down examples, as in your SUSY example, they insert CPT conjugates to avoid these problems. I think if you apply the argument from Weinbergt to the free version of the SUSY model without CPT conjugates, you'll see that the space-like commutators don't vanish.

Answer 4

CPT invariance is specifically a theorem in quantum field theory (your second statement is of course much better). Basically it says that antiparticles (the T part in the Feynman-Stuckelberg interpretation) have opposite charge and parity to the corresponding particles. There is a better statement, an outline argument, and numerous references at this wikipedia page