- #1
ismaili
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Dear guys,
I read a derivation of the dimension of gamma matrices in a d dimension space, which I don't quite understand.
First of all, in d dimension, where d is even.
One assumes the dimension of gamma matrices which satisfy
\{ \gamma^\mu , \gamma^\nu \} = 2\eta^{\mu\nu} \quad\quad\cdots(*)
is m.
A general a m by m matrix with complex arguments should have 2m^2 independent components.
Now, eq(*) gives m^2 constraints. (<= I don't quite understand this.)
So, the independent components of a single gamma matrix should be m^2.
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,
\bar{\psi}\gamma^{\mu\nu}\psi \rightarrow \Lambda^\mu{}_\rho\Lambda^{\nu}{}_\sigma\bar{\psi}\gamma^{\rho\sigma}\psi
where \gamma^{\mu\nu} = \gamma^{[\mu}\gamma^{\nu]}
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:
C^d_0 + C^d_1 +C^d_2 + \cdots + C^d_d = 2^d
Now we match the two independent components we calculated (Why?! why they should match?!)
m^2 = 2^d
This concludes that m = 2^{d/2}.
Now, for d = 2k+1 being odd, one can easily add \gamma^{2k} \sim \gamma^0\gamma^1\cdots\gamma^{2k-1}, together with the original \gamma^0,\gamma^1,\cdots,\gamma^{2k-1} to form gamma matrices in d = 2k+1.
Since the anti-symmetric tensors has a linear relation,
\gamma^{\mu_0\mu_1\cdots\mu_r} = \epsilon^{\mu_0\mu_1\cdots\mu_{2k}}\gamma_{\mu_{r+1}\cdots\mu_{2k}}.
So there are actually 2^{d}/2 independent components for odd d.
Hence, the dimension of gamma matrices in odd spacetime dimension should be 2^{\frac{d-1}{2}}.
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 2^{d/2}.
I read a derivation of the dimension of gamma matrices in a d dimension space, which I don't quite understand.
First of all, in d dimension, where d is even.
One assumes the dimension of gamma matrices which satisfy
\{ \gamma^\mu , \gamma^\nu \} = 2\eta^{\mu\nu} \quad\quad\cdots(*)
is m.
A general a m by m matrix with complex arguments should have 2m^2 independent components.
Now, eq(*) gives m^2 constraints. (<= I don't quite understand this.)
So, the independent components of a single gamma matrix should be m^2.
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,
\bar{\psi}\gamma^{\mu\nu}\psi \rightarrow \Lambda^\mu{}_\rho\Lambda^{\nu}{}_\sigma\bar{\psi}\gamma^{\rho\sigma}\psi
where \gamma^{\mu\nu} = \gamma^{[\mu}\gamma^{\nu]}
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:
C^d_0 + C^d_1 +C^d_2 + \cdots + C^d_d = 2^d
Now we match the two independent components we calculated (Why?! why they should match?!)
m^2 = 2^d
This concludes that m = 2^{d/2}.
Now, for d = 2k+1 being odd, one can easily add \gamma^{2k} \sim \gamma^0\gamma^1\cdots\gamma^{2k-1}, together with the original \gamma^0,\gamma^1,\cdots,\gamma^{2k-1} to form gamma matrices in d = 2k+1.
Since the anti-symmetric tensors has a linear relation,
\gamma^{\mu_0\mu_1\cdots\mu_r} = \epsilon^{\mu_0\mu_1\cdots\mu_{2k}}\gamma_{\mu_{r+1}\cdots\mu_{2k}}.
So there are actually 2^{d}/2 independent components for odd d.
Hence, the dimension of gamma matrices in odd spacetime dimension should be 2^{\frac{d-1}{2}}.
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 2^{d/2}.
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