# Representation of Z2 acting on wavefunctions

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• QFT1995
In summary, the conversation discusses the concept of a wavefunction being invariant under the group ##Z_2## and the different operators that can represent this group. The question is raised about which operator is correct when stating that the wavefunction is invariant under ##Z_2##, as there are multiple representations of this group. The response explains that it depends on the specific symmetry being described, such as spatial reflections or spin flips. The conversation then touches on the use of the group SU(3) in physics, and how it can have different meanings depending on the context. The summary concludes by emphasizing that simply stating the group of a symmetry does not necessarily provide the full understanding of its physical meaning.
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##.

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.

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?

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.

## 1. What is the representation of Z2 acting on wavefunctions?

The representation of Z2 acting on wavefunctions refers to the mathematical description of how the group Z2 (the cyclic group of order 2) affects or transforms wavefunctions in quantum mechanics. This representation is important in understanding the symmetries of physical systems and can be used to simplify calculations and analyze the behavior of wavefunctions.

## 2. How is Z2 represented in quantum mechanics?

In quantum mechanics, Z2 is represented as a group of operators that act on wavefunctions. These operators can be thought of as transformations that leave the physical properties of the system unchanged. The Z2 group has two elements, the identity element and the element that flips the sign of the wavefunction. These operators can be combined to create more complex transformations.

## 3. What is the significance of Z2 acting on wavefunctions?

The significance of Z2 acting on wavefunctions lies in its relationship to symmetries in quantum systems. By understanding how Z2 acts on wavefunctions, we can identify symmetries in physical systems and use them to simplify calculations and predict the behavior of the system. This representation is also useful in the study of topological phases of matter.

## 4. How is Z2 acting on wavefunctions related to time-reversal symmetry?

Z2 acting on wavefunctions is closely related to time-reversal symmetry, which is a fundamental symmetry in quantum mechanics. In fact, the Z2 group can be thought of as the group of operators that represent time-reversal symmetry. This means that understanding how Z2 acts on wavefunctions can give us insight into the behavior of systems under time-reversal transformations.

## 5. Are there other groups that can act on wavefunctions?

Yes, there are other groups besides Z2 that can act on wavefunctions in quantum mechanics. Some examples include the rotation group, the translation group, and the permutation group. Each of these groups represents a different type of symmetry and has its own set of operators that act on wavefunctions. Understanding how these groups act on wavefunctions is essential in studying the symmetries and properties of physical systems.

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