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Is chirality operator defined in odd dimensions?

  1. Jul 31, 2008 #1
    As far as I remember, I heard from someone that the matrix
    also known as the chirality operator in 3+1 dimension is not defined in odd dimensions. I do not understand why that should be the case. Suppose I am in the 4+1 dimension and I choose one more gamma matrix suitably to close the Clifford algebra in five dimensions and then define analogously to the 4 dimensional case, the operator
    as my chirality operator. Will that be a mistake?
    What about six dimensions? Can I define my chirality operator by multiplying the required no. of basic gamma matrices in that dimension?
  2. jcsd
  3. Jul 31, 2008 #2
    The STA basis vectors, which is also a clifford algebra, (http://en.wikipedia.org/wiki/Space-time_algebra), I believe map to the matrix respresentation you are using. With that basis, the value:

    i = \gamma_0\gamma_1\gamma_2\gamma_3 = -\gamma^0\gamma^1\gamma^2\gamma^3

    where i is the usual pseudoscalar for the space, and i^2 = -1. Not knowing exactly what your matrix representation is, I'd guess you have:

    i\gamma^0\gamma^1\gamma^2\gamma^3 = -\gamma^0

    If that's the case, then it doesn't appear to me that this is a very useful operator in 3+1 dimensions.

    Before thinking about the 4+1 case, what matrixes are you using for [itex]\gamma^{\mu}[/itex] in the 3+1 dimensional case?
  4. Jul 31, 2008 #3


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