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## Main Question or Discussion Point

This has been bugging me for a while, but feel to tell me if it's a nonsensical or silly question..

Suppose there were 4 spatial dimensions instead of 3. How would we go about constructing the Pauli matrices?

Assuming each matrix still only has 2 eigenvectors, we require 4, 2x2 mutually orthogonal matrices satisfying the commutation relations. As well as that, the eigenvectors of any matrix must be expressible as linear combinations of the eigenvectors of any other, ie. each set of eigenvalues forms a basis.

It seems to me that the only reason we are able to do this in 3 dimensions is through the use of imaginary numbers. Without imaginary numbers we can only form the two bases {(1,0), (0,1)} and {(1,1), (1,-1)}. Is there perhaps some more general extension of complex numbers that is necessary to extend the Pauli matrices to dimensions >3?

Suppose there were 4 spatial dimensions instead of 3. How would we go about constructing the Pauli matrices?

Assuming each matrix still only has 2 eigenvectors, we require 4, 2x2 mutually orthogonal matrices satisfying the commutation relations. As well as that, the eigenvectors of any matrix must be expressible as linear combinations of the eigenvectors of any other, ie. each set of eigenvalues forms a basis.

It seems to me that the only reason we are able to do this in 3 dimensions is through the use of imaginary numbers. Without imaginary numbers we can only form the two bases {(1,0), (0,1)} and {(1,1), (1,-1)}. Is there perhaps some more general extension of complex numbers that is necessary to extend the Pauli matrices to dimensions >3?