quantum mechanics - Generator for parity?

The unitary translation operator, $\hat{T}(\lambda) = e^{i\hat{p}\lambda/\hbar}$, is generated from the Hermitian operator $\hat{p}$.

The unitary rotation operator, $\hat{R}_z(\alpha)=e^{-i\hat{L_z}\alpha/\hbar}$, is generated from the Hermitian operator $\hat{L}_z$.

The unitary parity operator $\hat{P}: \Psi(x) \rightarrow \hat{P}\Psi(x)=\Psi(-x)$, is generated from which Hermitian operator?

Answer

The parity operator does not have a generator in the way that the translation or rotation operators do.

Notice how you gave the translation operators and the rotation operators a parameter, like $\hat{T}(\lambda)$. There isn't just one translation operator, but a whole family of translation operators that form a group. What's more, the operator family is continuous in the sense that $\hat{T}(\lambda)$ and $\hat{T}(\lambda')$ are close if $\lambda$ and $\lambda'$ are close. In this case, Stone's theorem proves that we can take the derivative of this family: $$ \hat{t} \equiv i\times \lim_{h\rightarrow 0} \frac{\hat{T}(h)-I}{\lambda}$$ and that this "derivative operator" $\hat{t}$ is a hermitian operator, and that $\hat{T}(\lambda) = \exp(-i\lambda \hat{t})$. We call $\hat{t}$ the generator of the group $\hat{T}(\lambda)$

On the other hand, the parity operator is just a single operator. There is no continuous family of parity operators, and hence no way to take a derivative or define a generator.

In the language of group theory, a group of transformations that is continuous like translations or rotations forms a "Lie group." Lie groups have all sorts of special structure because of their continuity, including Lie group generators, where we can say things like $g(\lambda) = \exp(\lambda X)$ where $g$ is an element of a group and $X$ is an element of a different mathematical object called a Lie algebra.

Discrete groups, like the combined group of CPT, or the discrete symmetries of a crystal, do not have Lie group generators, or the concept of the exponential, or Lie algebras. Group theory does talk about "generators" for these groups. In this case the generators are elements of the group that can be multiplied to make any other element of the group. For example, 90 degree clockwise rotations can be chained together to make a 180 degree rotation and a 90 degree counterclockwise rotation.