- #1

ismaili

- 160

- 0

Dear guys,

I read a derivation of the dimension of gamma matrices in a [tex]d[/tex] dimension space, which I don't quite understand.

First of all, in [tex]d[/tex] dimension, where [tex]d[/tex] is even.

One assumes the dimension of gamma matrices which satisfy

[tex] \{ \gamma^\mu , \gamma^\nu \} = 2\eta^{\mu\nu} \quad\quad\cdots(*)[/tex]

is [tex] m [/tex].

A general a m by m matrix with complex arguments should have [tex]2m^2[/tex] independent components.

Now, eq(*) gives [tex]m^2[/tex] constraints.

So, the independent components of a single gamma matrix should be [tex] m^2 [/tex].

On the other hand, one finds that the anti-symmetrization of gamma matrices can produce space-time tensors under Lorentz transformation, i.e. for example,

[tex] \bar{\psi}\gamma^{\mu\nu}\psi \rightarrow \Lambda^\mu{}_\rho\Lambda^{\nu}{}_\sigma\bar{\psi}\gamma^{\rho\sigma}\psi[/tex]

where [tex] \gamma^{\mu\nu} = \gamma^{[\mu}\gamma^{\nu]}[/tex]

Now, the various antisymmetric tensors decompose the Lorentz group into different pieces which do not mix. We now calculate the independent components of each anti-symmetric tensor, and add it up:

[tex] C^d_0 + C^d_1 +C^d_2 + \cdots + C^d_d = 2^d [/tex]

[tex] m^2 = 2^d [/tex]

This concludes that [tex] m = 2^{d/2} [/tex].

Now, for d = 2k+1 being odd, one can easily add [tex]\gamma^{2k} \sim \gamma^0\gamma^1\cdots\gamma^{2k-1}[/tex], together with the original [tex]\gamma^0,\gamma^1,\cdots,\gamma^{2k-1}[/tex] to form gamma matrices in d = 2k+1.

Since the anti-symmetric tensors has a linear relation,

[tex]\gamma^{\mu_0\mu_1\cdots\mu_r} = \epsilon^{\mu_0\mu_1\cdots\mu_{2k}}\gamma_{\mu_{r+1}\cdots\mu_{2k}}[/tex].

So there are actually [tex] 2^{d}/2 [/tex] independent components for odd [tex]d[/tex].

Hence, the dimension of gamma matrices in odd spacetime dimension should be [tex]2^{\frac{d-1}{2}}[/tex].

Anyone help me go through the puzzles? thanks so much!

ismaili

----

Oh, by the way, I found that from the Dirac representation method which I described in another nearby thread titled "spinors in various dimensions", one can easily realize the dimension of gamma matrices in even dimension d should be [tex] 2^{d/2} [/tex].

I read a derivation of the dimension of gamma matrices in a [tex]d[/tex] dimension space, which I don't quite understand.

First of all, in [tex]d[/tex] dimension, where [tex]d[/tex] is even.

One assumes the dimension of gamma matrices which satisfy

[tex] \{ \gamma^\mu , \gamma^\nu \} = 2\eta^{\mu\nu} \quad\quad\cdots(*)[/tex]

is [tex] m [/tex].

A general a m by m matrix with complex arguments should have [tex]2m^2[/tex] independent components.

Now, eq(*) gives [tex]m^2[/tex] constraints.

*(<= I don't quite understand this.)*So, the independent components of a single gamma matrix should be [tex] m^2 [/tex].

On the other hand, one finds that the anti-symmetrization of gamma matrices can produce space-time tensors under Lorentz transformation, i.e. for example,

[tex] \bar{\psi}\gamma^{\mu\nu}\psi \rightarrow \Lambda^\mu{}_\rho\Lambda^{\nu}{}_\sigma\bar{\psi}\gamma^{\rho\sigma}\psi[/tex]

where [tex] \gamma^{\mu\nu} = \gamma^{[\mu}\gamma^{\nu]}[/tex]

Now, the various antisymmetric tensors decompose the Lorentz group into different pieces which do not mix. We now calculate the independent components of each anti-symmetric tensor, and add it up:

[tex] C^d_0 + C^d_1 +C^d_2 + \cdots + C^d_d = 2^d [/tex]

*Now we match the two independent components we calculated*(Why?! why they should match?!)[tex] m^2 = 2^d [/tex]

This concludes that [tex] m = 2^{d/2} [/tex].

Now, for d = 2k+1 being odd, one can easily add [tex]\gamma^{2k} \sim \gamma^0\gamma^1\cdots\gamma^{2k-1}[/tex], together with the original [tex]\gamma^0,\gamma^1,\cdots,\gamma^{2k-1}[/tex] to form gamma matrices in d = 2k+1.

Since the anti-symmetric tensors has a linear relation,

[tex]\gamma^{\mu_0\mu_1\cdots\mu_r} = \epsilon^{\mu_0\mu_1\cdots\mu_{2k}}\gamma_{\mu_{r+1}\cdots\mu_{2k}}[/tex].

So there are actually [tex] 2^{d}/2 [/tex] independent components for odd [tex]d[/tex].

Hence, the dimension of gamma matrices in odd spacetime dimension should be [tex]2^{\frac{d-1}{2}}[/tex].

*My question is that, isn't the linear relation between anti-symmetric tensors also hold in d = even spacetime dimension?*Anyone help me go through the puzzles? thanks so much!

ismaili

----

Oh, by the way, I found that from the Dirac representation method which I described in another nearby thread titled "spinors in various dimensions", one can easily realize the dimension of gamma matrices in even dimension d should be [tex] 2^{d/2} [/tex].

Last edited: