Complex arithmetic using Clifford algebra

Robin K. S. Hankin

To cite the clifford package in publications please use Hankin (2022). This short document shows how complex arithmetic may be implemented using Clifford algebra (of course, if one really wants to use complex numbers, base R is much more efficient and uses nicer idiom than the methods presented here). Recall that complex numbers are a two-dimensional algebra over the reals, with \((a,b)\cdot(c,d)=(ac-bd,ad+bc)\); we usually write \((a,b)\) as \(a+bi\). There are two natural ways to identify complex numbers with Clifford objects; but because they use different signatures it is more convenient to treat them separately.

First method

We use \(\operatorname{Cl}(0,1)\), so \(e_1^2=-1\). Package idiom is straightforward; to coerce complex numbers to Clifford objects and vice versa, we need a couple of functions:

signature(0,1)
options(maxdim=1) # paranoid-level safety measure
complex_to_clifford <- function(z){Re(z) + e(1)*Im(z)}
clifford_to_complex <- function(C){const(C) + 1i*getcoeffs(C,1)}
clifford_to_complex <- function(C){const(C) + 1i*coeffs(Im(C))}

Then numerical verification is immediate. First we choose some complex numbers:

z1 <- 35 + 67i
z2 <- -2 + 12i

Then, for example:

z1

## [1] 35+67i

complex_to_clifford(z1)

## Element of a Clifford algebra, equal to
## + 35 + 67e_1

Checking that the coercion is a homomorphism is easy:

complex_to_clifford(z1) * complex_to_clifford(z2) == complex_to_clifford(z1*z2)

## [1] TRUE

Above, note that the * on the left is the geometric product, while the * on the right is the usual complex multiplication. And because the map is invertible we can check the other way too:

(C1 <- 23 + 7*e(1))

## Element of a Clifford algebra, equal to
## + 23 + 7e_1

clifford_to_complex(C1)

## [1] 23+7i

C2 <- 2  - 8*e(1)
clifford_to_complex(C1)*clifford_to_complex(C2) == clifford_to_complex(C1*C2)

## [1] TRUE

Second method

We use \(\operatorname{Cl}(2)\), so \(e_1^2=e_2^2=1\), and identify the imaginary unit \(i\) with \(e_{12}\) (thus \(e_{12}^2=e_{12}e_{12}=e_{1212}=-e_{1122}=-e_1^2e_2^2=-1\)). A general complex number \(z=x+iy\) maps to Clifford object \(x + ye_{12}\).

options(maxdim=2)  # paranoid-level safety measure
signature(2)
complex_to_clifford <- function(z){Re(z) + e(1:2)*Im(z)}
clifford_to_complex <- function(C){const(C) + 1i*coeffs(Im(C))}

Then numerical verification:

z1 <- 35 + 67i
z2 <- -2 + 12i
complex_to_clifford(z1) * complex_to_clifford(z2) == complex_to_clifford(z1*z2)

## [1] TRUE

C1 <- 23 + 7*e(1:2)
C2 <- 2  - 8*e(1:2)
clifford_to_complex(C1)*clifford_to_complex(C2) == clifford_to_complex(C1*C2)

## [1] TRUE

Note

The identification \(x+iy\longrightarrow x+ye_{12}\) is a homomorphism whenever \(e_1^2e_2^2=1\); above we used \(\operatorname{Cl}(2,0)\) so \(e_1^2=e_2^2=1\). However, the relation is also satisfied if \(e_1^2=e_2^2=-1\), so we can equally well use \(\operatorname{Cl}(0,2)\):

signature(0,2)
c(
complex_to_clifford(z1)*complex_to_clifford(z2) == complex_to_clifford(z1*z2),
clifford_to_complex(C1)*clifford_to_complex(C2) == clifford_to_complex(C1*C2)
)

## [1] TRUE TRUE

Default

It is best to return the signature and maxdim to their default values in order to prevent interference with other vignettes:

options(maxdim=NULL)
signature(Inf)

References

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