# Mathematical representation of Symmetry Transformation

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?

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