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Dirac Matrices in Higher Dimensions

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It is possible to define Dirac matrices for dimensions higher than four using Kronecker products of Pauli matrices. For dimension , there are Dirac matrices, and the dimension of the matrices is .

Contributed by: Enrique Zeleny (August 2008)
Open content licensed under CC BY-NC-SA

Details

A. Pais, "On Spinors in n Dimensions," Journal of Mathematical Physics, 3(6), 1962.

Starting with Pauli matrices , , , with , define

,

,

,

where .

For odd dimensions, the additional generator (a generalization of ) is

.

The Dirac matrices satisfy canonical anti-commutation relation .

The above definition corresponds to the so-called "chiral basis," where Dirac matrices are block anti-diagonal. Other bases are possible, and are related to the chiral basis by rotations.

The Dirac matrices generate Euclidean Clifford algebra.

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