Generalized Pauli matrices

In and, in particular , the term generalized Pauli matrices refers to families of matrices which generalize the (linear algebraic) properties of the. Here, a few classes of such matrices are summarized.

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,


 * $f_{k,j}^{d} =$, for $E_{kj} + E_{jk}$.
 * $k < j$, for $−i (E_{jk} − E_{kj})$.
 * $k > j$, the identity matrix, for $h_{k}^{d} =$,.
 * $I_{d}$, for $k = 1$.
 * $$\sqrt{\tfrac{2}{d(d-1)}} \left( h_1^{d-1} \oplus (1 - d)\right) = \sqrt{\tfrac{2}{d(d-1)}} \left( I_{d-1} \oplus (1 - d)\right),$$ for $h_{k}^{d−1} ⊕ 0$.
 * $1 < k < d$, for $k = d$.
 * $$\sqrt{\tfrac{2}{d(d-1)}} \left( h_1^{d-1} \oplus (1 - d)\right) = \sqrt{\tfrac{2}{d(d-1)}} \left( I_{d-1} \oplus (1 - d)\right),$$ for $ℂ^{d×d}$.

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  above) means.

The generalized Gell-Mann matrices are and  by construction, just like the Pauli matrices. One can also check that they are orthogonal in the  on $d × d$. By dimension count, one sees that they span the vector space of $ω = exp(2πi/d)$ complex matrices, $$\mathfrak{gl}$$($d$,ℂ). They then provide a Lie-algebra-generator basis acting on the fundamental representation of $$\mathfrak{su}$$($d$ ).

In dimensions $d$ = 2 and 3, the above construction recovers the Pauli and, respectively.

A non-Hermitian generalization of Pauli matrices
The Pauli matrices $$\sigma _1$$ and $$\sigma _3$$ satisfy the following:



\sigma _1 ^2 = \sigma _3 ^2 = I, \; \sigma _1 \sigma _3 = - \sigma _3 \sigma _1 = e^{\pi i} \sigma _3 \sigma_1. $$

The so-called is



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

Like the Pauli matrices, W is both and. $$\sigma _1, \; \sigma _3$$ and W satisfy the relation


 * $$\; \sigma _1 = W \sigma _3 W^* .$$

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

Construction: The clock and shift matrices
Fix the dimension $d$ as before. Let $ω^{d} = 1$,  a root of unity. Since $ω ≠ 1$  and  $Σ_{1}$,  the sum  of all roots annuls:


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

Integer indices may then be cyclically identified mod $d$.

Now define, with Sylvester, the shift matrix

\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,

\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, 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, 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, so the exponential of the momentum. They are (finite-dimensional) representations of the corresponding elements of the  on a d-dimensional Hilbert space.

The following relations echo and generalize those of the Pauli matrices:
 * $$\Sigma _ 1 ^d = \Sigma _ 3 ^d = I$$

and the braiding relation,
 * $$\; \Sigma_3 \Sigma _1 = \omega \Sigma_1 \Sigma _3 = e^{2 \pi i / d} \Sigma_1 \Sigma _3 ,$$

the, and can be rewritten as
 * $$\; \Sigma_3 \Sigma _1   \Sigma _3^{d-1}  \Sigma_1 ^{d-1}      = \omega ~.$$

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

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, still denoted by W in a slight abuse of notation,

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
 * $$\; \Sigma_1 = W \Sigma_3 W^*   ~,$$

which is the desired analog result. Thus, $W$, a , arrays the eigenvectors of  $Σ_{3}$, which has the same eigenvalues as   $Σ_{3}$.

When d = 2k,  W * is precisely the matrix of the , 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 $$\mathfrak{gl}$$(d,ℂ), known as "nonions" $$\mathfrak{gl}$$(3,ℂ), "sedenions" $$\mathfrak{gl}$$(4,ℂ), etc...

This basis can be systematically connected to the above Hermitian basis. (For instance, the powers of  $h_{k}^{d}$, the , map to linear combinations of the $d → ∞$s.) It can further be used to identify $$\mathfrak{gl}$$(d,ℂ) , as ᙭᙭᙭᙭᙭, with the algebra of.