Pauli matrices via Clifford algebra

In clifford: Arbitrary Dimensional Clifford Algebras

knitr::opts_chunk$set(echo = TRUE)
library("clifford")
library("onion")
library("jordan")
library("emulator")

knitr::include_graphics(system.file("help/figures/clifford.png", package = "clifford"))
knitr::include_graphics(system.file("help/figures/jordan.png", package = "jordan"))

To cite the clifford package in publications please use @hankin2025_clifford_rmd. This short document shows how the Pauli matrices, often used in quantum mechanics, can be calculated using Clifford algebra as implemented by the clifford R package. The Pauli matrices are a set of three $2\times 2$ matrices with complex entries. They represent observables corresponding to measuring spin along the $x$, $y$, and $z$ axes. They are also useful when considering polarized light. The Pauli matrices have a pleasing relationship with Jordan algebra [@hankin2023_jordan]. In component form, they are:

[ \sigma_0=\left(\begin{matrix}1&0\0&1\end{matrix}\right)\qquad \sigma_x=\left(\begin{matrix}0&1\1&0\end{matrix}\right)\qquad \sigma_y=\left(\begin{matrix}0&-i\i&0\end{matrix}\right)\qquad \sigma_z=\left(\begin{matrix}1&0\0&-1\end{matrix}\right) ]

We observe that $\sigma_x\sigma_y=i\sigma_z$, $\sigma_y\sigma_z=i\sigma_x$, and $\sigma_z\sigma_x=i\sigma_y$, and further that $\sigma_x^2=\sigma_y^2=\sigma_z^2=-i\sigma_x\sigma_y\sigma_z=\sigma_0$.

The non-identity Pauli matrices [that is, $\sigma_x,\sigma_y,\sigma_z$] are subject to the following commutation relations:

[ \left[\sigma_x,\sigma_y\right]=2i\sigma_z\qquad \left[\sigma_y,\sigma_z\right]=2i\sigma_x\qquad \left[\sigma_z,\sigma_x\right]=2i\sigma_y]

(here, $\left[x,y\right]=xy-yx$). We also have the following anticommutation relations:

[ \left\lbrace\sigma_x,\sigma_y\right\rbrace=2i\sigma_z\qquad \left\lbrace\sigma_y,\sigma_z\right\rbrace=2i\sigma_x\qquad \left\lbrace\sigma_z,\sigma_x\right\rbrace=2i\sigma_y]

(here, $\left\lbrace x,y\right\rbrace=xy+yx$).

Because any $2\times 2$ Hermitian matrix may be expressed as $A\sigma_0+B\sigma_x+C\sigma_y+D\sigma_z$ for $A,B,C,D\in\mathbb{R}$, we observe that the anticommutation relations imply that the Pauli matrices are closed under the Jordan operator $x\circ y=(xy+yx)/2$. For more details, see the jordan package [@hankin2023_jordan] which implements this operation in a more general context. The Jordan multiplication rule is

[ \sigma_a\sigma_b=\delta_{ab}I_2 + i\epsilon_{abc}\sigma_c ]

which suggests the following identification:

\begin{aligned} \sigma_0&\longleftrightarrow 1\ \sigma_x&\longleftrightarrow e_1\ \sigma_y&\longleftrightarrow e_2\ \sigma_z&\longleftrightarrow e_3\ \end{aligned}

Then we make the formal identifications:

\begin{aligned} i\sigma_x&\longleftrightarrow e_2e_3\ i\sigma_y&\longleftrightarrow e_3e_1\ i\sigma_z&\longleftrightarrow e_1e_2\ \end{aligned}

and so we recover the Pauli matrix relations from the Clifford algebra.

Implementation

Let us start with the Pauli matrices:

[ \sigma_0=\left(\begin{matrix}1&0\0&1\end{matrix}\right)\qquad \sigma_x=\left(\begin{matrix}0&1\1&0\end{matrix}\right)\qquad \sigma_y=\left(\begin{matrix}0&-i\i&0\end{matrix}\right)\qquad \sigma_z=\left(\begin{matrix}1&0\0&-1\end{matrix}\right) ]

[ i\sigma_0=\left(\begin{matrix}i&0\0&i\end{matrix}\right)\qquad i\sigma_x=\left(\begin{matrix}0&i\i&0\end{matrix}\right)\qquad i\sigma_y=\left(\begin{matrix}0&1\-1&0\end{matrix}\right)\qquad i\sigma_z=\left(\begin{matrix}i&0\0&-i\end{matrix}\right) ]

Given a general complex matrix

[ \left(\begin{matrix} \alpha +\beta i & \gamma+\delta i\ \epsilon+\zeta i & \eta+\theta i
\end{matrix}\right) ]

we see that

\begin{eqnarray} \sigma_0&=(\alpha+\eta)/2\qquad i\sigma_0=(\beta+\theta)/2\ \sigma_x&=(\gamma+\epsilon)/2\qquad i\sigma_x=(\delta+\xi)/2\ \sigma_y&=(\gamma-\epsilon)/2\qquad i\sigma_x=(\delta-\xi)/2\ \sigma_z&=(\alpha-\eta)/2\qquad i\sigma_z=(\beta-\theta)/2\ \end{eqnarray}

R implementation

s0 <- matrix(c(1,0,0,1),2,2)
sx <- matrix(c(0,1,1,0),2,2)
sy <- matrix(c(0,1i,-1i,0),2,2)
sz <- matrix(c(1,0,0,-1),2,2)

Given a general complex matrix M, we may coerce this to Clifford form as follows:

matrix_to_clifford <- function(M){
      (Re(M[1,1] + M[2,2]))/2             + 
      (Re(M[1,1] - M[2,2]))/2*e(c(  3  )) +
      (Im(M[1,1] + M[2,2]))/2*e(c(1,2,3)) + 
      (Im(M[1,1] - M[2,2]))/2*e(c(1,2  )) +

      (Re(M[2,1] + M[1,2]))/2*e(c(1    )) + 
      (Re(M[2,1] - M[1,2]))/2*e(c(1,  3)) +
      (Im(M[2,1] + M[1,2]))/2*e(c(  2,3)) + 
      (Im(M[2,1] - M[1,2]))/2*e(c(  2  ))
}

and then test it as follows:

rmat <- function(...){matrix(rnorm(4),2,2) + 1i*matrix(rnorm(4),2,2)}
M <- rmat()
M
matrix_to_clifford(M)

We can now test whether matrix_to_clifford() is a group homomorphism:

M1 <- rmat()
M2 <- rmat()

diff <- matrix_to_clifford(M1)*matrix_to_clifford(M2) - matrix_to_clifford(M1 %*% M2)
diff
Mod(diff)

We see agreement to numerical precision. Now we can coerce from a Clifford to a matrix:

`clifford_to_matrix` <- function(C){
   return(
                          const(C)*s0 + getcoeffs(C,list(1))*sx 
  +           getcoeffs(C,list(2))*sy + getcoeffs(C,list(3))*sz
  + getcoeffs(C,list(c(1,2,3)))*1i*s0 + getcoeffs(C,list(c(  2,3)))*1i*sx 
  - getcoeffs(C,list(c(1,  3)))*1i*sy + getcoeffs(C,list(c(1,2  )))*1i*sz
  )
} 

rc <- function(...){rcliff(100,d=3,g=3)}
C <- 104 + rc()
C
clifford_to_matrix(C)

Now test that the two coercion functions are inverses of one another:

clifford_to_matrix(matrix_to_clifford(M)) - M 
matrix_to_clifford(clifford_to_matrix(C))- C

Now we can establish that clifford_to_matrix() is a homomorphism:

C1 <- 222 + rc()
C2 <- 333 + rc()
clifford_to_matrix(C1*C2) - clifford_to_matrix(C1)%*%clifford_to_matrix(C2)

Closure

The reason that Pauli matrices are useful in physics is that they are closed under the Jordan operation $x\circ y=(xy+yx)/2$, which we will verify for matrices and their Clifford representation.

M1 <- as.1matrix(rchm(1,2))
M2 <- as.1matrix(rchm(1,2))
M1
M2
p1 <- (M1 %*% M2 + M2 %*% M1)/2
p1 - ht(p1)  # zero for Hermitian matrices

Above, see how $M_1\circ M_2$ is Hermitian. Now, in Clifford form:

C1 <- matrix_to_clifford(M1)
C2 <- matrix_to_clifford(M2)
p2 <- (C1 * C2 + C2 * C1)/2
p2

above, see how the clifford product p2 is a pure Pauli matrix as its only nonzero coefficients are those of the scalar and the grade-one blades:

grades(p2)

References

clifford documentation built on July 5, 2026, 5:07 p.m.