Is chirality operator defined in odd dimensions?

  • Thread starter arroy_0205
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  • #1
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As far as I remember, I heard from someone that the matrix
[tex]
\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3
[/tex]
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
[tex]
\Gamma^c=i\gamma^0\gamma^1\gamma^2\gamma^3\gamma^4
[/tex]
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?
 

Answers and Replies

  • #2
305
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:

[tex]
i = \gamma_0\gamma_1\gamma_2\gamma_3 = -\gamma^0\gamma^1\gamma^2\gamma^3
[/tex]

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:

[tex]
i\gamma^0\gamma^1\gamma^2\gamma^3 = -\gamma^0
[/tex].

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?
 
  • #3
samalkhaiat
Science Advisor
Insights Author
1,718
992
As far as I remember, I heard from someone that the matrix
[tex]
\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3
[/tex]
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
[tex]
\Gamma^c=i\gamma^0\gamma^1\gamma^2\gamma^3\gamma^4
[/tex]
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
In any dimensions, (n = 2p, or n = 2p + 1), a "generilised [itex]\gamma_{5}[/itex]" can be defined by

[tex]\Gamma_{n + 1} \equiv \Gamma_{1}\Gamma_{2}...\Gamma_{n}[/tex]

From the algebra

[tex]\{\Gamma_{a}, \Gamma_{b}\} = 2 \eta_{ab} \ I[/tex]

it follows that [itex]\Gamma_{n+1}[/itex] anticommutes with all [itex]\Gamma_{a}[/itex] for even dimensions ( n = 2p ), i.e.,

[tex]\{\Gamma_{n+1}, \Gamma_{a}\} = 0 \ \ \forall a = 1,2,..,2p[/tex]

but, for odd dimensions (n = 2p +1), it COMMUTES with all [itex]\Gamma_{a}[/itex], i.e.,

[tex][\Gamma_{n+1},\Gamma_{a}] = 0, \ \ \forall a = 1,2,..,2p+1[/tex]

Therefore in odd dimensions, by Schur's lemma, [itex]\Gamma_{n+1}[/itex] is a multiple of the unit matrix. This fact is valid in any representation you choose for the gamma matrices.

regards

sam
 

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