Pauli matrices via Clifford algebra

To cite the clifford package in publications please use Hankin (2022). 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 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 (Hankin 2023). 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 (Hankin 2023) 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:

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

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

##                       [,1]                    [,2]
## [1,] -1.4000435+0.6215527i -2.437263611-1.8218177i
## [2,]  0.2553171+1.1484116i -0.005571287-0.2473253i

matrix_to_clifford(M)

## Element of a Clifford algebra, equal to
## - 0.7028074 - 1.090973e_1 + 1.485115e_2 + 0.434439e_12 - 0.6972361e_3 +
## 1.34629e_13 - 0.336703e_23 + 0.1871137e_123

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

## Element of a Clifford algebra, equal to
## + 2.220446e-16e_12 + 8.326673e-17e_3 + 2.220446e-16e_23 - 8.326673e-17e_123

Mod(diff)

## [1] 3.353719e-16

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

## Element of a Clifford algebra, equal to
## + 155 + 60e_1 - 90e_2 + 74e_12 - 63e_3 + 10e_13 + 27e_23 - 27e_123

clifford_to_matrix(C)

##        [,1]     [,2]
## [1,] 92+47i  50+117i
## [2,] 70-63i 218-101i

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

clifford_to_matrix(matrix_to_clifford(M)) - M

##      [,1]                        [,2]
## [1,] 0+0i  0.000000e+00+2.220446e-16i
## [2,] 0+0i -7.372575e-17+2.775558e-17i

matrix_to_clifford(clifford_to_matrix(C))- C

## Element of a Clifford algebra, equal to
## the zero clifford element (0)

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)

##      [,1] [,2]
## [1,] 0+0i 0+0i
## [2,] 0+0i 0+0i

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

##             [,1]        [,2]
## [1,] -0.05+0.00i -0.74-0.11i
## [2,] -0.74+0.11i -0.56+0.00i

M2

##             [,1]       [,2]
## [1,] -0.56+0.00i 0.75-1.92i
## [2,]  0.75+1.92i 0.19+0.00i

p1 <- (M1 %*% M2 + M2 %*% M1)/2
p1 - ht(p1)  # zero for Hermitian matrices

##      [,1] [,2]
## [1,] 0+0i 0+0i
## [2,] 0+0i 0+0i

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

## Element of a Clifford algebra, equal to
## - 0.383 - 0.09185e_1 - 0.60595e_2 + 0.0672e_3

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)

## A disord object with hash dc0ef121a24b8ab8c67bbfc6468e439b5622e81e and elements
## [1] 0 1 1 1
## (in some order)

References

Hankin, R. K. S. 2022. “Clifford Algebra in R.” arXiv. https://doi.org/10.48550/ARXIV.2209.13659.

———. 2023. “Jordan Algebra in R.” arXiv. https://doi.org/10.48550/arXiv.2303.06062.

Robin K. S. Hankin

Implementation

R implementation

Closure

References