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ismaili
ismaili is offline
#6
Aug15-10, 10:50 PM
P: 161
Quote Quote by ismaili View Post
I reconsidered this recently.
I think for my first question, the reason that reduces the independent components [tex]2m^2[/tex] into [tex]m^2[/tex] is the (anti-)hermitian properties of the gamma matrices, not the Clifford algebra. Then, the antisymmetrized products of gamma matrices form a basis for the algebra, hence, by matching independent components [tex]m^2[/tex] and the number of basis [tex]2^d[/tex], we see that [tex] m = 2^{d/2} [/tex] for even dimension [tex]d[/tex].

The solution to the second equation is due to the role of [tex]\gamma_5[/tex].
That's why only in odd dimension, those basis are related by Levi-Civita tensor,
and the number of basis is reduced to [tex]2^{(d-1)/2} [/tex] for odd [tex] d [/tex].
For your information.
I found this reference which deals with Clifford algebra in a mathematical rigorous way.
http://arxiv.org/abs/hep-th/9811101
But at least those [tex]B,C[/tex] matrices are not suddenly popped.