# What do we mean by 'Equivalent Projective representation ?

1. Dec 18, 2012

### S_klogW

What do we mean by 'Equivalent Projective representation"?

I know that we say two representations R and R' of a group G is equivalent if there exists a unitary matrix U such that URU^(-1)=R'.
But what do we mean by equivalent projective rerpesentations?
I've heard of the theorem that the SO(3) group has only 2 inequivalent projective representations. But what does that exactly mean?
I am very interested in projective representation because it's projective representation rather than ordinary representation that represents symmetry in Quantum Mechanics since the vector A and exp(id)A represent the same physical state.
So does anyone know if there are some books that can serve as an introduction to projective representations?

2. Dec 21, 2012

### Fredrik

Staff Emeritus
Re: What do we mean by 'Equivalent Projective representation"?

Since you haven't received any replies, I will mention that "Geometry of quantum theory" by Varadarajan covers projective representations and their relevance to quantum mechanics. I hesitate to recommend it because I find it very hard to read, but I don't know a better option.

3. Dec 21, 2012

### micromass

Staff Emeritus
Re: What do we mean by 'Equivalent Projective representation"?

I've never done anything about projective representations before, so this post is just a guess. But it would make sense to define first

$$Z=\{cI_n~\vert~c\in \mathbb{R}\}$$

Then we define a projective representation as a group homomorphism

$$\rho: G\rightarrow GL_n(\mathbb{R})/Z$$

This last group is often called $PGL_n(\mathbb{R})$, or the projective general linear group.

Given, $\rho,\rho^\prime$ projective representations, it would make sense to define them equivalent if there exist $U\in O_n(\mathbb{R})/Z$ such that

$$\rho(g)=U\cdot \rho^\prime(g)\cdot U^{-1}$$

for all $g\in G$.

The complex case is similar.