Construction of real gamma matrices

[John Greger]

TL;DR 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?

 

[king vitamin]

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:
$$\gamma^0 = i \sigma^y, \qquad \gamma^1 = \sigma^x, \qquad \gamma^2 = \sigma^x.$$

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).

 

[John Greger]

Yes, viz:
$$\gamma^0 = i \sigma^y, \qquad \gamma^1 = \sigma^x, \qquad \gamma^2 = \sigma^x.$$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?

 

[vanhees71]

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.

 

[John Greger]

Thank you very much for clarifying. Mabye stupid question, but is it really possible to have a LHS and RHS of different matrix dimension?

 

[vanhees71]

All matrices involved are $2 \times 2$ matrices.

 

[George Jones]

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 $\eta = \mathrm{diag}(-1,1,1)$.