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According to the CPT thoerem, the choice of positive parity for particles and negative parity for antiparticles is just as arbitrary as the choice of positive charge for protons and negative charge for electrons.


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Because in an isotropic space x is equivalent to -x, this is just Noether theorem. If you want to see how parity violation is explained in the standard model look here


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If the parity operator commuted with the Hamiltonian, you'd be able to simultaneously diagonalize the parity operator and the Hamiltonian. That would mean you could find a complete set of eigenstates of the Hamiltonian that were also eigenstates of the parity operator. The eigenstates of the parity operator are just even/odd functions, but odd potentials ...



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