# What is a rigorous and general definition of the parity operator?

Is there a rigorous definition of the parity operator?

I see parity come up in the context of angular momentum, magnetic fields, quantum spin/particles. It is also related to the Levi-Civita symbol vs tensor discussion. Likewise it's related to tensor densities. It's related to cross products and bivectors.

It seems like you "just have to know" whether a certain quantity has been deemed to be a psuedo- or non-pseudo tensor, and furthermore, it seems like the rules for the parity of different types of physical objects are assigned in a way which is somewhat arbitrary. It's not entirely arbitrary, the parity rules do have some sort of self-consistency. But, it seems like the only thing we gain from the rules for parity is self-consistency of the parity of different objects. It doesn't seem like parity adds any intuition otherwise and the entire concept could be dispensed with.

That said, I might change my mind about this if someone could present me with a rigorous mathematical definition of the parity operator (perhaps in the context of differentiable manifolds and coordinates) that is general and can help clarify its utility.

I'm seeking a mathematically motivated answer rather than stuff like "what happens when you look in a mirror"

Parity is a discrete spacetime symmetry under which the spatial coordinates $$x^i$$ transform as $$x^i \rightarrow -x^i$$ and the time coordinate transforms as $$t \rightarrow t$$.
Various fields also transform under parity. Ultimately the origin of these transformations is that, in a theory with parity invariance, the Lagrangian should be invariant under parity. For example, if a spin-0 field $$\phi$$ couples to a fermion $$\psi$$ via the coupling $$\phi \bar\psi \psi$$, then $$\phi$$ must be a scalar for the Lagrangian to be invariant. On the other hand, if $$\phi$$ couples via $$\phi \bar\psi \gamma^5 \psi$$, then $$\phi$$ must be a pseudoscalar. The transformation laws of the fields are not written on stone tablets, but come from finding transformations that leave the Lagrangian invariant.