Why the Dirac matrices are four dimensional

For particles with spin-1/2 we would expect (on the basis of nonrelativistic quantum mechanics) that spinors with two components would be sufficient. But the Dirac spinors have to be (at least) four-dimensional. A mathematical reason lies in the nature of the algebraic properties that have to be satisfied by the Dirac matrices a and 0 if the Dirac equation should satisfy the relativistic energy-momentum relation in the sense described above, see (6). 

As the matrix product is an elementary example of a noncommutative product, it is very natural to look for n x n matrices in order to satisy the anticommutation relations (5). Here we show that matrices a and 0 obeying the anti-commutation relations have to be at least four-dimensional. 

It is fairly clear that the matrices Oj would have to be Hermitian, otherwise the free Dirac operator (4) has no chance to be self-adjoint. The Hermiticity implies that all eigenvalues must be real. From = 1 we conclude that the only possible eigenvalues of aj are +1 and —1. 

Next we compute the trace of the matrix aj. For a given j choose k with j and write, using first a1 = i and then OjUh = —akaj.  

In this calculation we have used the well-known properties of the trace (in particular, tr (—A) = —tr A and trAB = tr BA). The result tells us that tr aj must be equal to its negative, and hence the trace of aj must be zero. 

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