# Representation of Z2 acting on wavefunctions

• A
QFT1995
If I have a wavefunction ##\Psi(X)## that is invariant under the group ##Z_2##, what specifically does that mean? There can be several operators that are representations of the group ##Z_2##, for example the operators

$$\mathbb{Z}_2=\{ \mathbb{I}, -\mathbb{I} \},$$
or
$$\mathbb{Z}_2=\{ \mathbb{I}, \hat{\Pi} \},$$
where ##\hat{\Pi}## is the parity operators with the action ##\hat{\Pi}: \Psi(X) \mapsto \Psi(-X)##. My question is, which one is it? Also, say if I have the 2 component wave-function, the following also produce a representation of ##Z_2##
$$\mathbb{Z}_2^A=\bigg\{\mathbb{I},\;\;\begin{pmatrix} \hat{\Pi} & 0\\ 0& \hat{\Pi} \end{pmatrix} \bigg\},$$
or
$$\mathbb{Z}_2^B=\bigg\{\mathbb{I},\;\;\begin{pmatrix} 0 & \hat{\Pi}\\ \hat{\Pi}&0 \end{pmatrix} \bigg\}.$$
My question is, how do you determine which one is correct when people state that the wavefunction is invariant under ##Z_2## since a wavfunction invariant under ##Z_2^A## isn't necessarily invariant under ##Z_2^B##, however they are both representations of ##Z_2##.

Gold Member
2022 Award
It depends upon which symmetry you want to describe. The symmetry group ##\{\mathbb{I},\hat{\Pi} \}## describes symmetry under spatial reflections. For your spin-1/2 example ##\mathbb{Z}_2^A## is also a usual spatial reflection, while ##\mathbb{A}_2^B## is a spatial reflection combined with a spin flip.

QFT1995
Okay, but people say things like, the theory is invariant under SU(3) yet they provide no extra details. What is usually meant by that?