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What do we mean by 'Equivalent Projective representation ?

  1. Dec 18, 2012 #1
    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. jcsd
  3. Dec 21, 2012 #2


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    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.
  4. Dec 21, 2012 #3
    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

    [tex]Z=\{cI_n~\vert~c\in \mathbb{R}\}[/tex]

    Then we define a projective representation as a group homomorphism

    [tex]\rho: G\rightarrow GL_n(\mathbb{R})/Z[/tex]

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

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

    [tex]\rho(g)=U\cdot \rho^\prime(g)\cdot U^{-1}[/tex]

    for all [itex]g\in G[/itex].

    The complex case is similar.
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