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The requirement is that $\chi^{(2)}$ be non-centrosymmetric. That's a bit different than having a particular parity. The states involved must be neither odd nor even; the parity must be mixed. That way the dipole matrix element exists between all three intermediate states involved in calculation of the susceptibility.


There is a mistake in your definition of time reversal as $x$ is fixed under that transformation, the remaining transformations being correct. With this correct version of T, the Hamiltonian you study is PT symmetric.


In most of your derivations, you have used the symbol $\psi$ for the genuine field operator (operator distribution). But your equation $$ \gamma^5 \psi = -\psi$$ clearly doesn't work for any Dirac field. This equation an operator equation equivalent to $$ (1+\gamma^5) \psi = 0$$ which says that one-half of the components of $\psi$ are zero as operators. But ...


A basic postulate in elementary particle theories is CPT invariance. Also the weak interaction is the only fundamental interaction that breaks parity-symmetry, and similarly, the only one to break CP-symmetry. ...... The laws of nature were long thought to remain the same under mirror reflection, the reversal of one spatial axis. The results of an ...

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