Clif.Tutorial

Clifford algebras (or geometric algebras) are itself mathematically interesting objects, but also a useful tool for vector algebra. This library attempts to make the Clifford algebraic computations easy and at least somewhat computationally efficient, while keeping the implementation as general as possible.

Since definitions and terminology vary greatly, here is a (non-rigorous) summary of terms that will be used in this tutorial:

• A vector is just an element of some set (or type). Note that this set may in principle be infinite.
• A basis is a set of vectors along with a bilinear form which maps a pair of them to some number (see Clif.Basis).
• A blade is any finite concatenation (free product) of vectors which contains each vector at most once, possibly multiplied by a scalar.
• A scalar is just a number, or a number multiplying the empty product.
• A Clif is a collection (direct sum) of the empty product, distinct blades and their multipliers (often called a multivector).
• The Clifford algebra is the set (type) of Clifs with the Clifford (geometric) product and direct summation.

cf. https://en.wikipedia.org/wiki/Clifford_algebra

Getting started

To begin, we just need to import the main module of the library, Clif.

import Clif

This provides us with

• Constructors for the type Clif with Num, Eq and other instances that implement the Clifford algebra,
• Constructors for Euclidean and Lorentzian basis vectors,
• The typeclass Basis for constructing our own algebras,
• Operations of the Clifford algebra defined in Clif.Algebra.

Defining our algebra

The type Clif b a joins the type b (for basis) and some field (or ring) Num a together to form a Clifford algebra. Only Clifs with matching types b and a can be multiplied directly. To generate the Clifford product between any types b and a, we need to specify the bilinear form between them. This is done by providing an instance of the type class Basis b a. Let us do that for Basis Char Double:

instance Basis Char Double where
    metric 't' 't' = -1
    metric  a   b  = if a == b then 1 else 0

The minimal complete definition for Basis is the function metric, which we have here defined to be a diagonal quadratic form on Char. This is all we need to define the reasonably high-dimensional Clifford algebra Cl(1,n)(R) where n is the number of Unicode codepoints represented by Char, with the signature (-, +, +, ...) for ('t', 'a', 'b', ...).

Few notes:

• We could have used the provided instance for the newtype Lorentzian to wrap Char with a similar metric:

instance (Ord b, Num a) => Basis (Lorentzian b) a where
   metric (T a) (T b) = if a == b then -1 else 0
   metric (S a) (S b) = if a == b then  1 else 0
   metric _ _         = 0

In that case, T a for any Char a would have the same signature as the character 't' in our definition. However, defining a metric for the plain Char is useful for demonstration, since it provides us with pretty printouts.

• Note that as in the definition for Basis (Lorentzian b) a, we do not actually need to fix the field ə apart from the Num constraint while defining the basis.
• If the metric is not diagonal (metric a b /= 0 for some a /= b), we need to replace the default implementations of the functions canonical and basisMul in the instance with more general implementations. See Clif.Basis for details.

Construction of Clif values

Using our Char basis, we can start constructing values of type Clif Char Double using the provided constructors. We can start by introducing the vectors needed to represent Cl(1,3)(R):

t = vec 't' 1
x = vec 'x' 1
y = vec 'y' 1
z = vec 'z' 1

To make it specific that we are working in Cl(1,3)(R), we can define the pseudoscalar txyz. All the following definitions are equivalent:

i = t * x * y * z

i = blade "txyz" 1

i = 1 *: "txyz"

Note that the last form using the infix operator '(*:)' is used for the Show instance to produce concise output. p Due to the Num instance, we do not usually need to explicitly embed scalars. If we want to be specific, we can write

answer = 42 :: Clif Char Double

Computation

We can now use any of the available operations to calculate on the Clif values, such as

• Simple multivector algebra:

>>> 2 * x * y - y * x
1.0 *: "xy" 

• Wedge products:

>>> x /\ y
1.0 *: "xy" 

• Reversion:

>>> rev i
1.0 *: "zyxt"

• Since Doubles have inverses, so do Clifs for which v * v is scalar. Trivial example is vectors:

>>> x * y * z / y
-1.0 *: "xy"

Copyright

This tutorial is licensed under a Creative Commons Attribution 4.0 International License