Graduate Representation of Z2 acting on wavefunctions

[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 $Z_2$.

 

[vanhees71]

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?

 

[vanhees71]

Of course, you must get the context. SU(3) can mean a lot of things. E.g., the symmetry group of the 3D symmetric harmonic oscillator is SU(3), i.e., there's a set of combinations of annihilation and creation operators that build an su(3) Lie algebra representation.

It could also be an approximate chiral flavor symmetry of QCD with 3 quarks (u, d, s).

Or it's the (exact) local gauge group of QCD with the quarks and antiquark states transforming according to the two fundamental irreducible representations of SU(3).

As I said before, just telling the group of a symmetry doesn't necessarily tell you the physics behind it. More precisely in quantum physics you deal with unitary (ray) representations of groups. The physical meaning is given by how the representation is realized.