Construction of real gamma matrices

  • #1

Summary:

Can I construct real gamma matrices in 2+1 dimensions?
Hi!

Is it possible to construct gamma matrices satisfying the Clifford algebra ##\{\gamma^\mu, \gamma^\nu \} = \eta^{\mu \nu}## that are *real*, for ##\eta = diag(-1,1,1)##?

I know that I can construct them in principle from sigma matrices, but I don't know how to construct real gamma matrices..

And also, do the gamma matrices necessarily have to be 3-dimensional for d=2+1?
 

Answers and Replies

  • #2
king vitamin
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Summary:: Can I construct real gamma matrices in 2+1 dimensions?

Is it possible to construct gamma matrices satisfying the Clifford algebra {γμ,γν}=ημν that are *real*, for η=diag(−1,1,1)?
Yes, viz:
[tex]
\gamma^0 = i \sigma^y, \qquad \gamma^1 = \sigma^x, \qquad \gamma^2 = \sigma^x.
[/tex]

Summary:: Can I construct real gamma matrices in 2+1 dimensions?

And also, do the gamma matrices necessarily have to be 3-dimensional for d=2+1?
The gamma matrices are always even dimensional. In d space-time dimensions, the smallest representation has dimension 2^floor(d/2).
 
  • #3
Yes, viz:
[tex]
\gamma^0 = i \sigma^y, \qquad \gamma^1 = \sigma^x, \qquad \gamma^2 = \sigma^x.
[/tex]


The gamma matrices are always even dimensional. In d space-time dimensions, the smallest representation has dimension 2^floor(d/2).
Hi and thank you very much for responding.

But then I will still have i's in the matrices such that they are not real?

May I also ask, on the right hand side the metric is still a 3x3 matrix, but on left hand side the matrices are 2x2, can we really have that?
 
  • #4
vanhees71
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The matrices are all real:
$$\gamma^0=\mathrm{i} \sigma_2=\begin{pmatrix}0 & 1 \\ -1 & 0 \end{pmatrix}, \quad \gamma^1=\sigma_1=\begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \gamma^2=\sigma_3 = \begin{pmatrix}1 & 0 \\-1 & 0 \end{pmatrix}.$$
They form the Clifford algebra of the space with the fundamental form of signature ##(2,1)##, i.e.,
$$\{\sigma^{\mu},\sigma^{\nu} \}=2 \eta^{\mu \nu}=\mathrm{diag}(-1,1,1).$$
These are all ##2 \times 2## matrices of course.
 
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  • #5
Thank you very much for clarifying. Mabye stupid question, but is it really possible to have a LHS and RHS of different matrix dimension?
 
  • #6
vanhees71
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All matrices involved are ##2 \times 2## matrices.
 
  • #7
George Jones
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Thank you very much for clarifying. Mabye stupid question, but is it really possible to have a LHS and RHS of different matrix dimension?
Think of it like this:

$$\{\sigma^{\mu},\sigma^{\nu} \} = 2 \eta^{\mu \nu} I,$$

where ##I## is the ##2 \times 2## identity matrix, and the ##\eta^{\mu \nu}## are the components of the ##3 \times 3## matrix ##\eta = \mathrm{diag}(-1,1,1)##.
 

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