Higher Dimensional Dirac Matricies

[y35dp]

Homework Statement

If D =7 and the metric g^{[tex]\mu[/tex][tex]\nu[/tex]}=diag(+------), Using the outer product of matrices, A [tex]\otimes[/tex] B construct a suitable set of [tex]\gamma[/tex] matrices from the 2 x 2 [tex]\sigma[/tex]-matrices

Homework Equations

[tex]\sigma[/tex]¹=(0, 1 ) [tex]\sigma[/tex]²=(0, -i)
(1, 0) (i, 0)
[tex]\o[/tex]³=(1, 0)
(0, -1)
we need only refer to the basic properties of the sigma matrices

[tex]\sigma[/tex]_i[tex]\sigma[/tex]_j = i [tex]\epsilon[/tex]_ijk[tex]\sigma[/tex]_k + [tex]\delta[/tex]_ijI₂

and

[tex]\sigma[/tex]₁^T=[tex]\sigma[/tex]₁, [tex]\sigma[/tex]₂^T=[tex]\sigma[/tex]₂, [tex]\sigma[/tex]₃^T=-[tex]\sigma[/tex]₃, [tex]\sigma[/tex]₁^*=[tex]\sigma[/tex]₁, [tex]\sigma[/tex]₂^*=[tex]\sigma[/tex]₃^*=-[tex]\sigma₃[/tex]

The Attempt at a Solution

As of yet I have found no [tex]\gamma[/tex]-matrices that satisfy {[tex]\gamma[/tex]^{[tex]\mu[/tex]}, [tex]\gamma[/tex]^{[tex]\nu[/tex]}} = 2g^{[tex]\mu[/tex][tex]\nu[/tex]}. The closest I have come is a set which satisfy {[tex]\gamma[/tex]^{[tex]\mu[/tex]}, [tex]\gamma[/tex]^{[tex]\nu[/tex]}} = 2[tex]\delta[/tex]_{[tex]\mu[/tex][tex]\nu[/tex]}I₇

 

[y35dp]

Sorry those [tex]\sigma[/tex]¹,[tex]\sigma[/tex]²,[tex]\sigma[/tex]³ are supposed to be the Pauli matrices, pretty poor attempt at making matrices on my part

 

[fzero]

If you have a matrix that satisfies [tex]M^2 = I[/tex], there's always a matrix [tex]N[/tex] that's a scalar multiple of [tex]M[/tex] such that [tex]N^2 = -I[/tex].