Complex arithmetic using Clifford algebra
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) ⋅ (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 Cl (0, 1), so e12 = −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:
Then, for example:
## [1] 35+67i## Element of a Clifford algebra, equal to
## + 35 + 67e_1Checking that the coercion is a homomorphism is easy:
## [1] TRUEAbove, 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:
## Element of a Clifford algebra, equal to
## + 23 + 7e_1## [1] 23+7i## [1] TRUESecond method
We use Cl (2), so e12 = e22 = 1, and identify the imaginary unit i with e12 (thus e122 = e12e12 = e1212 = −e1122 = −e12e22 = −1). A general complex number z = x + iy maps to Clifford object x + ye12.
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] TRUEC1 <- 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] TRUENote
The identification x + iy → x + ye12 is a homomorphism whenever e12e22 = 1; above we used Cl (2, 0) so e12 = e22 = 1. However, the relation is also satisfied if e12 = e22 = −1, so we can equally well use 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