Pauli Matrices in higher dimensions

[Kazza_765]

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?

 

[Phlogistonian]

Look up quaternions.

Or you could take Dirac's approach. He solved the problem in 4 spacetime dimensions by using 4x4 matrices. For that, look up gamma matrices.

 

[AllanGoff]

The Dirac matrices do not answer the question, they are for Minkowski space, 3 dimensions of space and 1 of time, not the 4 spatial dimensions that are at issue. The question is a deeper one than it appears. For position and momentum only 2 conjugate bases are assumed, connected by a Fourier transform. For discrete systems, 3 and more conjugate bases are possible. A theorem by Wisemen (? of quantum cryptography fame) has shown that the number of mutually conjugate bases scale as the log of the state space. An 8 state system should have 4 mutually conjugate bases. Therefore, there should be an unlimited number of mutually conjugate bases for the infinite dimension case of position and momentum, but I've never seen them discussed.

A case can be made that the mathematics used in quantum mechanics are overkill, they have more degrees of freedom than the physics does. This means they could be a red herring, providing false clues about the fundamentals of quantum mechanics. The fact that Pauli spin matrices can be constructed for 3 spatial dimensions but not 4 suggests they might be a convenient accident. The observation that they cannot be easily extended to 4 spatial dimensions is rather insightful.

To see this, tensor multiply each base by itself. Two of them will yield bases conjugate to each other applicable to a 4-state system, but the third one (the one with imaginary components) yields a matrix that looks like a basis (orthonormal vectors) but is not Hermitian, and therefore does not represent a physically realizable measurement.