Generalizations of Pauli matrices

Construction

Let E_jk be the matrix with 1 in the jk-th entry and 0 elsewhere. Consider the space of d×d complex matrices, ℂ^d×d, for a fixed d.

Define the following matrices,

The collection of matrices defined above without the identity matrix are called the generalized Gell-Mann matrices, in dimension d. The symbol ⊕ (utilized in the Cartan subalgebra above) means matrix direct sum.

The generalized Gell-Mann matrices are Hermitian and traceless by construction, just like the Pauli matrices. One can also check that they are orthogonal in the Hilbert–Schmidt inner product on ℂ^d×d. By dimension count, one sees that they span the vector space of d × d complex matrices, g l (d,ℂ). They then provide a Lie-algebra-generator basis acting on the fundamental representation of s u (d ).

In dimensions d=2 and 3, the above construction recovers the Pauli and Gell-Mann matrices, respectively.

A non-Hermitian generalization of Pauli matrices

The Pauli matrices σ 1 and σ 3 satisfy the following:

The so-called Walsh–Hadamard conjugation matrix is

Like the Pauli matrices, W is both Hermitian and unitary. σ 1 , σ 3 and W satisfy the relation

The goal now is to extend the above to higher dimensions, d, a problem solved by J. J. Sylvester (1882).

Construction: The clock and shift matrices

Fix the dimension d as before. Let ω = exp(2πi/d), a root of unity. Since ω^d = 1 and ω ≠ 1, the sum of all roots annuls:

Integer indices may then be cyclically identified mod d.

Now define, with Sylvester, the shift matrix

and the clock matrix,

These matrices generalize σ₁ and σ₃, respectively.

Note that the unitarity and tracelessness of the two Pauli matrices is preserved, but not Hermiticity in dimensions higher than two. Since Pauli matrices describe quaternions, Sylvester dubbed the higher-dimensional analogs "nonions", "sedenions", etc.

These two matrices are also the cornerstone of quantum mechanical dynamics in finite-dimensional vector spaces as formulated by Hermann Weyl, and find routine applications in numerous areas of mathematical physics. The clock matrix amounts to the exponential of position in a "clock" of d hours, and the shift matrix is just the translation operator in that cyclic vector space, and so it can be considered the exponential of the momentum operator. The group generated by the clock and shift matrices and the phase factor ω I is sometimes called the Weyl–Heisenberg or Heisenberg–Weyl group (not to be confused with the continuous Heisenberg group).

Each of the Pauli matrices have order 2, and the clock and shift matrices in dimension d generalize that property:

Furthermore, they satisfy the braiding relation,

which was Weyl's reformulation of the quantum canonical commutation relation for finite-dimensional spaces. This can also be written

On the other hand, to generalize the Walsh–Hadamard matrix W, note

Define, again with Sylvester, the following analog matrix, still denoted by W in a slight abuse of notation,

It is evident that W is no longer Hermitian, but is still unitary. Direct calculation yields

which is the desired analog result. Thus, W , a Vandermonde matrix, arrays the eigenvectors of Σ₁, which has the same eigenvalues as Σ₃.

When d = 2^k, W * is precisely the matrix of the discrete Fourier transform, converting position coordinates to momentum coordinates and vice versa.

The complete family of d² unitary (but non-Hermitian) independent matrices

provides Sylvester's well-known trace-orthogonal basis for g l (d,ℂ), known as "nonions" g l (3,ℂ), "sedenions" g l (4,ℂ), etc...

This basis can be systematically connected to the above Hermitian basis. (For instance, the powers of Σ₃, the Cartan subalgebra, map to linear combinations of the h_k^ds.) It can further be used to identify g l (d,ℂ) , as d → ∞, with the algebra of Poisson brackets.