--- title: "Pauli matrices via Clifford algebra" author: "Robin K. S. Hankin" output: html_vignette bibliography: clifford.bib link-citations: true vignette: > %\VignetteEngine{knitr::rmarkdown} %\VignetteIndexEntry{Pauli matrices via Clifford algebra} %\usepackage[utf8]{inputenc} --- ```{r setup, include=FALSE} knitr::opts_chunk$set(echo = TRUE) library("clifford") library("onion") library("jordan") library("emulator") ``` ```{r out.width='15%', out.extra='style="float:right; padding:10px"',echo=FALSE} 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 ```{r} 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: ```{r} 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: ```{r} 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: ```{r} 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: ```{r} `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 ) } ``` ```{r} 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: ```{r} 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: ```{r} 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. ```{r} 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: ```{r} 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: ```{r} grades(p2) ``` ## References