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Consider a general Hamiltonian that is made up of three terms

$\mathcal{H}$ = term I + term II + term III .

Suppose the combination of charge conjugation and parity (CP) is a symmetry of this Hamiltonian, provided an additional operation A is done on the third term. For instance, suppose the third term is a Yukawa interaction in which the scalar field does not have definite parity; instead it smoothly transitions from a vacuum with even-parity to a higher-energy configuration that is odd under parity. Suppose $\widetilde{CP}$ represents this simultaneous action of CP and A, i.e.

$(\widetilde{CP})\;\mathcal{H}\;(\widetilde{CP})^{-1} = \mathcal{H}\;$.

What is the correct mathematical way to write $\;\widetilde{CP}\;$in terms of CP and A?

Does it involve something like a direct product or is it done differently? Also, given the specific example mentioned above, can A be regarded as an additional $\mathcal{Z}_2$ operation on the scalar field?

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    $\begingroup$ It seems rather vague what you're asking, can you give a specific example? $\endgroup$ – knzhou Feb 11 at 21:20
  • $\begingroup$ @knzhou: Question edited. $\endgroup$ – Optimus Prime Feb 12 at 5:20

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