Parity Transformation

Parity refers to a transformation of the coordinate system, known as inversion.

An inversion is nothing but a reflection followed by a $180^o$ rotation (about the horizontal axis, by convention).

A parity operator, when applied to a process, carries every point to the diametrically opposite location through the origin i.e. if we apply $P$ to a vector $\textbf{a}$, it produces a vector pointing in the opposite direction: $P(\textbf{a}) = -\textbf{a}$.

If we repeat this operation, we get the original vector $\textbf{a}$.

Now, consider a vector, $\textbf{c}$, which is the cross product of $\textbf{a}$ and another vector $\textbf{b}$

i.e. $\textbf{c} = \textbf{a} \times \textbf{b}$.

What would happen if we apply $P$ to $\textbf{c}$?

Well, if $P$ changes the sign of $\textbf{a}$ and $\textbf{b}$ then $\textbf{c}$ evidently doesn’t change sign: $P(\textbf{c}) = \textbf{c}$. 

Type of Vectors

There are two kinds of vectors:
Polar vectors: The vectors that change signs under parity. Ex: velocity ($\textbf{v}$), momentum($\textbf{p}$), etc.

Pseudovectors: The vectors that do not change signs under parity. Ex: angular momentum ($\textbf{L}$), magnetic field ($\textbf{B}$), etc. 

Note that the cross product of a polar vector with a pseudovector would be a polar vector.

Similarly, there are two types of scalars, “ordinary”, which does not change signs, and pseudoscalars, which do. If we apply the parity operator twice, we get the original vector/scalar.

$P^{2}$ = I

Evidently, the eigenvalues of $P$ are $\pm 1$. $+1$ corresponds to scalars and pseudovectors, whereas $-1$ corresponds to pseudoscalars and vectors.

Points to remember

1. The hadrons are eigenstates of $P$ and can be classified according to their eigenvalue.

2. The parity of a fermion is opposite to that of its corresponding antiparticle, according to Quantum Field Theory. Whereas the parity of a boson is the same as its antiparticle.

3. The parity of quarks is positive, and that of antiquarks is negative. This is an arbitrary choice and is just a matter of convention.

4. The parity of a composite system in its ground state is the product of the parity of its constituents.

5. Parity is a multiplicative quantum number, as opposed to charge, strangeness, etc. which are additive.