Is chirality operator defined in odd dimensions?

[arroy_0205]

As far as I remember, I heard from someone that the matrix
<br /> \gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3<br />
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
<br /> \Gamma^c=i\gamma^0\gamma^1\gamma^2\gamma^3\gamma^4<br />
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?

 

[Peeter]

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:

<br /> i = \gamma_0\gamma_1\gamma_2\gamma_3 = -\gamma^0\gamma^1\gamma^2\gamma^3<br />

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:

<br /> i\gamma^0\gamma^1\gamma^2\gamma^3 = -\gamma^0<br />.

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 \gamma^{\mu} in the 3+1 dimensional case?

 

[samalkhaiat]

As far as I remember, I heard from someone that the matrix
<br /> \gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3<br />
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
<br /> \Gamma^c=i\gamma^0\gamma^1\gamma^2\gamma^3\gamma^4<br />
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 \gamma_{5}" can be defined by

\Gamma_{n + 1} \equiv \Gamma_{1}\Gamma_{2}...\Gamma_{n}

From the algebra

\{\Gamma_{a}, \Gamma_{b}\} = 2 \eta_{ab} \ I

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

\{\Gamma_{n+1}, \Gamma_{a}\} = 0 \ \ \forall a = 1,2,..,2p

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

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

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

regards

sam