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I'm confused by the CPT theorem. It states (more or less) that a Lorentz invariant quantum field theory needs to be CPT invariant. But what does it actually mean for a QFT to be CPT invariant? It surely means that it's Lagrangian is. What about the Hamiltonian? Does the invariance get inherited by it as well? What about other observables that I can measure in an experiment? Are all of those also invariant under CPT (even though they might not be Lorentz invariant)?

Related to this: If I have a Lagrangian (density) that transforms in a specific way under C, P, and T, and I derive a low energy Hamiltonian from it, does this one necessarily inherit the same transformation properties? And if yes, is the reverse true? If a Hamiltonian (of an isolated system) is odd under C, P, or T, will this necessarily correspond to violations of the same symmetries at high energies?

user
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The WP statement is that

Any Lorentz invariant local quantum field theory with a Hermitian Hamiltonian must have CPT symmetry.

The theory involves the vacuum state, the Lagrangian, and the Hamiltonian.

What about other observables that I can measure in an experiment? Are all of those also invariant under CPT?

They don't need to, in principle: that's how you might measure a violation; but they turn out to be, confirming CPT is a sacrosanct symmetry.

Re: your second paragraph. P and T are violated by the weak interactions, the first maximally and the second (CP) by a little, e.g. in neutron decay. CPT is still conserved there, low energies, just as in higher ones! The weak QFT Hamiltonian preserves CPT at all energies!

Cosmas Zachos
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