Construction

Let ${\displaystyle E_{jk}}$ be the matrix with 1 in the jk-th entry and 0 elsewhere. Consider the space of ${\displaystyle d\times d}$ complex matrices, ${\displaystyle \mathbb {C} ^{d\times d}}$, for a fixed ${\displaystyle d}$.

Define the following matrices,

${\displaystyle f_{k,j}^{\,\,\,\,\,d}={\begin{cases}E_{kj}+E_{jk}&{\text{for }}k<j,\\-i(E_{jk}-E_{kj})&{\text{for }}k>j.\end{cases}}}$

and

${\displaystyle h_{k}^{\,\,\,d}={\begin{cases}I_{d}&{\text{for }}k=1,\\h_{k}^{\,\,\,d-1}\oplus 0&{\text{for }}1<k<d,\\{\sqrt {\tfrac {2}{d(d-1)}}}\left(h_{1}^{d-1}\oplus (1-d)\right)={\sqrt {\tfrac {2}{d(d-1)}}}\left(I_{d-1}\oplus (1-d)\right)&{\text{for }}k=d\end{cases}}}$

The collection of matrices defined above without the identity matrix are called the generalized Gell-Mann matrices, in dimension ${\displaystyle d}$.^[2][3] 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 ${\displaystyle \mathbb {C} ^{d\times d}}$. By dimension count, one sees that they span the vector space of ${\displaystyle d\times d}$ complex matrices, ${\displaystyle {\mathfrak {gl}}(d,\mathbb {C} )}$. They then provide a Lie-algebra-generator basis acting on the fundamental representation of ${\displaystyle {\mathfrak {su}}(d)}$.

In dimensions ${\displaystyle d}$ = 2 and 3, the above construction recovers the Pauli and Gell-Mann matrices, respectively.

Framing

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

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

The goal now is to extend the above to higher dimensions, ${\displaystyle d}$.

Construction: The clock and shift matrices

Fix the dimension ${\displaystyle d}$ as before. Let ${\displaystyle \omega =\exp(2\pi i/d)}$, a root of unity. Since ${\displaystyle \omega ^{d}=1}$ and ${\displaystyle \omega \neq 1}$, 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 ${\displaystyle \sigma _{1}}$ and ${\displaystyle \sigma _{3}}$, 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^[7][8][9] as formulated by Hermann Weyl, and they find routine applications in numerous areas of mathematical physics.^[10] The clock matrix amounts to the exponential of position in a "clock" of ${\displaystyle d}$ 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 ${\displaystyle d}$-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 ${\displaystyle W}$, 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,^[11] still denoted by ${\displaystyle W}$ 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 ${\displaystyle W}$ 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, ${\displaystyle W}$, a Vandermonde matrix, arrays the eigenvectors of ${\displaystyle \Sigma _{1}}$, which has the same eigenvalues as ${\displaystyle \Sigma _{3}}$.

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

Definition

The complete family of ${\displaystyle d^{2}}$ 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 ${\displaystyle {\mathfrak {gl}}(d,\mathbb {C} )}$, known as "nonions" ${\displaystyle {\mathfrak {gl}}(3,\mathbb {C} )}$, "sedenions" ${\displaystyle {\mathfrak {gl}}(4,\mathbb {C} )}$, etc...^[12][13]

This basis can be systematically connected to the above Hermitian basis.^[14] (For instance, the powers of ${\displaystyle \Sigma _{3}}$, the Cartan subalgebra, map to linear combinations of the ${\displaystyle h_{k}^{\,\,\,d}}$ matrices.) It can further be used to identify ${\displaystyle {\mathfrak {gl}}(d,\mathbb {C} )}$, as ${\displaystyle d\to \infty }$, with the algebra of Poisson brackets.

Properties

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 ${\displaystyle \sigma _{k,j}}$.