Intrinsic parity

In quantum mechanics, the intrinsic parity is a phase factor that arises as an eigenvalue of the parity operation ${\displaystyle x_{i}\rightarrow x_{i}'=-x_{i}}$ (a reflection about the origin).[1] To see that the parity's eigenvalues are phase factors, we assume an eigenstate of the parity operation (this is realized because the intrinsic parity is a property of a particle species) and use the fact that two parity transformations leave the particle in the same state, thus the new wave function can differ by only a phase factor, i.e.: ${\displaystyle P^{2}\psi =e^{i\phi }\psi }$ thus ${\displaystyle P\psi =\pm e^{i\phi /2}\psi }$, since these are the only eigenstates satisfying the above equation.

The intrinsic parity's phase is conserved for non-weak interactions (the product of the intrinsic parities is the same before and after the reaction). As ${\displaystyle [P,H]=0}$ the Hamiltonian is invariant under a parity transformation. The intrinsic parity of a system is the product of the intrinsic parities of the particles, for instance for noninteracting particles we have ${\displaystyle P(|1\rangle |2\rangle )=(P|1\rangle )(P|2\rangle )}$. Since the parity commutes with the Hamiltonian and ${\displaystyle {\frac {dP}{dt}}=0}$ its eigenvalue does not change with time, therefore the intrinsic parities phase is a conserved quantity.

A consequence of the Dirac equation is that the intrinsic parity of fermions and antifermions obey the relation ${\displaystyle P_{\bar {f}}P_{f}=-1}$, so particles and their antiparticles have the opposite parity. Single leptons can never be created or destroyed in experiments, as lepton number is a conserved quantity. This means experiments are unable to distinguish the sign of a leptons parity, so by convention it is chosen that leptons have intrinsic parity +1, antileptons have ${\displaystyle P=-1}$. Similarly the parity of the quarks is chosen to be +1, and antiquarks is -1.[2]