Questions: CPT Theorem

Question 1

The weak interaction violates C, P, and CP individually. How is this consistent with the CPT theorem?

• A. It is not consistent — the CPT theorem must be wrong
• B. The CPT theorem says only the combined operation CPT is a symmetry — each individual operation (C, P, T, or any pair like CP) can be violated, as long as the triple product CPT is conserved
• C. The weak interaction is not a local quantum field theory
• D. CPT is only approximately conserved in the weak interaction

Answer: (B)

The CPT theorem constrains only the triple combination. The weak interaction maximally violates P (only left-handed fermions couple to W bosons) and C (particle and antiparticle couplings differ). It also violates CP (observed in kaon and B-meson decays). But CPT remains exact: if CP is violated, then T must also be violated in exactly the compensating way to preserve CPT. This has been verified experimentally — direct T violation in the kaon system matches the CP violation as predicted by CPT invariance.

Question 2

The CPT theorem implies that a particle and its antiparticle have exactly the same mass. The most precise test of this is the comparison of the proton and antiproton masses, which agree to better than one part in 10^{10}.

• T. True
• F. False

Answer: True

CPT transforms a particle at rest into its antiparticle at rest. Since CPT is a symmetry of the Hamiltonian, both states must have the same energy, which means the same mass. The CERN BASE experiment has measured the proton-to-antiproton charge-to-mass ratio to a precision of 16 parts per trillion. Any difference would signal CPT violation and require abandoning one of the axioms of quantum field theory (locality, Lorentz invariance, or Hermiticity of the Hamiltonian). No deviation has been found.

Question 3

If CPT is violated, at least one of the foundational axioms of quantum field theory must be wrong. Which axioms does the CPT theorem depend on?

• A. Only Lorentz invariance
• B. Locality (interactions occur at spacetime points), Lorentz invariance (the theory is the same in all inertial frames), and unitarity (probability is conserved, equivalently the Hamiltonian is Hermitian)
• C. Only unitarity and gauge invariance
• D. Only the spin-statistics connection

Answer: (B)

The CPT theorem is derived from three axioms: (1) the theory is a local quantum field theory (fields at spacelike separations commute or anticommute), (2) the theory is Lorentz invariant, and (3) the Hamiltonian is Hermitian (ensuring unitary time evolution). If any of these fails, CPT invariance is not guaranteed. Some theories of quantum gravity suggest that Lorentz invariance or locality might be violated at the Planck scale, which would potentially allow CPT violation. This is why experimental tests of CPT are so important — they probe the deepest foundations of physics.

Question 4

Explain why CPT invariance, combined with observed CP violation, implies that T (time reversal) must also be violated in nature.

Model answer: CPT invariance means the combined transformation is an exact symmetry: (CPT)^{-1} H (CPT) = H. If CP is violated, meaning (CP)^{-1} H (CP) != H, then T must also be violated to compensate: (CPT) = (CP)(T), so if CP is broken, T must be broken in precisely the way that makes the product CPT still a symmetry. Concretely, if a process has different rates under CP transformation (like certain kaon decays), then the time-reversed process must also show an asymmetry, and the two asymmetries must be equal. This has been confirmed experimentally: direct T violation has been observed in the kaon and B-meson systems, and it matches the CP violation as required by CPT.

This argument makes T violation a prediction, not just an observation. Given the measured CP violation in the CKM matrix of the Standard Model, the CPT theorem predicts exactly how much T violation there must be. The experimental confirmation of this predicted T violation is a stringent test of CPT.