Generalizations of Pauli matrices

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,

f_k,j^d =

E_kj + E_jk, for k < j .

−i (E_jk − E_kj), for k > j .

h_k^d =

I_d, the identity matrix, for k = 1,

h_k^d−1 ⊕ 0, for 1 < k < d .

${\displaystyle {\sqrt {\tfrac {2}{d(d-1)}}}\left(h_{1}^{d-1}\oplus (1-d)\right)={\sqrt {\tfrac {2}{d(d-1)}}}\left(I_{d-1}\oplus (1-d)\right),}$ for k = d.

The collection of matrices defined above without the identity matrix are called the generalized Gell-Mann matrices, in dimension d.[2] 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, ${\displaystyle {\mathfrak {gl}}}$(d,ℂ). They then provide a Lie-algebra-generator basis acting on the fundamental representation of ${\displaystyle {\mathfrak {su}}}$(d ).

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

The Pauli matrices ${\displaystyle \sigma _{1}}$ and ${\displaystyle \sigma _{3}}$ satisfy the following:

${\displaystyle \sigma _{1}^{2}=\sigma _{3}^{2}=I,\quad \sigma _{1}\sigma _{3}=-\sigma _{3}\sigma _{1}=e^{\pi i}\sigma _{3}\sigma _{1}.}$

The so-called Walsh–Hadamard conjugation matrix is

${\displaystyle W={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}.}$

Like the Pauli matrices, W is both Hermitian and unitary. ${\displaystyle \sigma _{1},\;\sigma _{3}}$ and W satisfy the relation

${\displaystyle \;\sigma _{1}=W\sigma _{3}W^{*}.}$

The goal now is to extend the above to higher dimensions, d.

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:

${\displaystyle 1+\omega +\cdots +\omega ^{d-1}=0.}$

Integer indices may then be cyclically identified mod d.

Now define, with Sylvester, the shift matrix

${\displaystyle \Sigma _{1}={\begin{bmatrix}0&0&0&\cdots &0&1\\1&0&0&\cdots &0&0\\0&1&0&\cdots &0&0\\0&0&1&\cdots &0&0\\\vdots &\vdots &\vdots &\ddots &\vdots &\vdots \\0&0&0&\cdots &1&0\\\end{bmatrix}}}$

and the clock matrix,

${\displaystyle \Sigma _{3}={\begin{bmatrix}1&0&0&\cdots &0\\0&\omega &0&\cdots &0\\0&0&\omega ^{2}&\cdots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\cdots &\omega ^{d-1}\end{bmatrix}}.}$

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[6][7][8] as formulated by Hermann Weyl, and they find routine applications in numerous areas of mathematical physics.[9] 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, so the exponential of the momentum. They are (finite-dimensional) representations of the corresponding elements of the Weyl-Heisenberg group on a d-dimensional Hilbert space.

The following relations echo and generalize those of the Pauli matrices:

${\displaystyle \Sigma _{1}^{d}=\Sigma _{3}^{d}=I}$

and the braiding relation,

${\displaystyle \Sigma _{3}\Sigma _{1}=\omega \Sigma _{1}\Sigma _{3}=e^{2\pi i/d}\Sigma _{1}\Sigma _{3},}$

the Weyl formulation of the CCR, and can be rewritten as

${\displaystyle \Sigma _{3}\Sigma _{1}\Sigma _{3}^{d-1}\Sigma _{1}^{d-1}=\omega ~.}$

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

${\displaystyle W={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&1\\1&\omega ^{2-1}\end{bmatrix}}={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&1\\1&\omega ^{d-1}\end{bmatrix}}.}$

Define, again with Sylvester, the following analog matrix,[10] still denoted by W in a slight abuse of notation,

${\displaystyle W={\frac {1}{\sqrt {d}}}{\begin{bmatrix}1&1&1&\cdots &1\\1&\omega ^{d-1}&\omega ^{2(d-1)}&\cdots &\omega ^{(d-1)^{2}}\\1&\omega ^{d-2}&\omega ^{2(d-2)}&\cdots &\omega ^{(d-1)(d-2)}\\\vdots &\vdots &\vdots &\ddots &\vdots \\1&\omega &\omega ^{2}&\cdots &\omega ^{d-1}\end{bmatrix}}~.}$

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

${\displaystyle \Sigma _{1}=W\Sigma _{3}W^{*}~,}$

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 discrete Fourier transform matrix, converting position coordinates to momentum coordinates and vice versa.

The complete family of d² unitary (but non-Hermitian) independent matrices ${\displaystyle \{\sigma _{k,j}\}_{k,j=1}^{d}}$ is defined as follows:

This provides Sylvester's well-known trace-orthogonal basis for ${\displaystyle {\mathfrak {gl}}}$(d,ℂ), known as "nonions" ${\displaystyle {\mathfrak {gl}}}$(3,ℂ), "sedenions" ${\displaystyle {\mathfrak {gl}}}$(4,ℂ), etc...[11][12]

This basis can be systematically connected to the above Hermitian basis.[13] (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 ${\displaystyle {\mathfrak {gl}}}$(d,ℂ), as d → ∞, with the algebra of Poisson brackets.

With respect to the Hilbert–Schmidt inner product on operators, ${\displaystyle \langle A,B\rangle _{\text{HS}}=\operatorname {Tr} (A^{*}B)}$, Sylvester's generalized Pauli operators are orthogonal and normalized to ${\displaystyle {\sqrt {d}}}$:

${\displaystyle \langle \sigma _{k,j},\sigma _{k',j'}\rangle _{\text{HS}}=\delta _{kk'}\delta _{jj'}\|\sigma _{k,j}\|_{\text{HS}}^{2}=d\delta _{kk'}\delta _{jj'}}$.

This can be checked directly from the above definition of ${\displaystyle \sigma _{k,j}}$.