Parity
Definition: Parity
Parity $P$ is a symmetry transformation $$P: (t,\vec{x}) \to (t,-\vec{x}).$$ It is part of the Lorentz group, since it leaves $s^2$ invariant. However it is not smoothly connected to the identity.
Since Spinors are not representations of the Lorentz group $O(1,3)$ but only its universal cover $SL(2,C)$, the action of $P$ on spinors is not neccessarily defined. Therefore we will define the action of $P$ on Spinors.
$P^2=1$ should always hold.
# Scalars
The action of parity on scalar should leave the free Lagrangian $$\mathcal{L}=-\frac{1}{2}\phi\Box\phi-\frac{1}{2}m^2\phi^2,$$ invariant, which is true if $P^2=1$. So we can choose $$P\phi\to\pm\phi(t,-\vec{x}).$$ Which sign is realised depends on the intrinsic parity of a particle. There are particles with both, even parity (+1) and odd parity (-1).
Since in the action one integrates over all $\vec{x}$, the change $\vec{x}\to -\vec{x}$ does not change anything.
For a complex scalar there can be a complex phase $\eta$: $$P\phi\to\eta\phi(t,-\vec{x})$$ Therefore $P^2\phi = \eta^2\phi$ introduces a phase as well. However one can argue, that this phase can be “absorbed” into the global charge symmetry and one can define $P$ in a way so that $P\phi \to \pm\phi(t,-\vec{x})$ also for complex fields.
# Vectors
For a vector $V_\mu$, we require $$P(V_0)\to\pm V_0(t,-\vec{x}),\qquad P(V_i)\to \mp V_i(t,-\vec{x}),$$
- Vector: If a vector field transforms as $P(V_i)\to -V_i$ we say it has parity $-1$ and call it Vector.
- Pseudovector: If a vector field transforms as $P(V_i)\to V_i$ we say it has parity $+1$ and call it Pseudovector.
Massless Vectors like the photon have to have parity $-1$ to leave the interaction term invariant under Parity.
# Spinors
For a Spinor, Parity transforms as $$P(\psi)\to \gamma_0\psi(t,-\vec{x}).$$
Note:
- There is a phase ambiguity like in the scalar case, which can be set to $1$.
- A chiral theory (without $L\leftrightarrow R$ symmetry) cannot be invariant under parity.
- Some transformations: $$P(\bar{\psi}\psi)\to \bar{\psi}\psi(t,-\vec{x})$$ $$P(\bar{\psi}\gamma_0\psi)\to\bar{\psi}\gamma_0\psi(t,-\vec{x})$$ $$P(\bar{\psi}\gamma_i\psi)\to -\bar{\psi}\gamma_i\psi(t,-\vec{x})$$
- Since $\bar{\psi}\gamma_\mu\psi$ transforms like a Vector, the Dirac Lagrangian in invariant under parity.
- Inserting a $\gamma_5$ like in the Weak interaction gives the opposite behavior under parity.