13.12 Example 1: Clifford Algebra

Now that we have QuadraticForm available, let’s put it to use. Given some quadratic form Q described by an n by n matrix over a field K, the domain CliffordAlgebra(n, K, Q) defines a vector space of dimension 2n over K. This is an interesting domain since complex numbers, quaternions, exterior algebras and spin algebras are all examples of Clifford algebras.

The basic idea is this: the quadratic form Q defines a basis e1,e2…,en for the vector space Kn, the direct product of K with itself n times. From this, the Clifford algebra generates a basis of 2n elements given by all the possible products of the ei in order without duplicates, that is,

1, e1, e2, e1e2, e3, e1e3, e2e3, e1e2e3, and so on.

The algebra is defined by the relations

\[\]

eiei=Q(ei)eiej=-ejeifori≠j

Now look at the snapshot of its definition given below. Lines 9-10 show part of the definitions of the Exports. A Clifford algebra over a field K is asserted to be a ring, an algebra over K, and a vector space over K. Its explicit exports include e(n), which returns the n-th unit element.

NNI ==> NonNegativeInteger
PI  ==> PositiveInteger
CliffordAlgebra(n,K,q): Exports == Implementation where
    n: PI
    K: Field
    q: QuadraticForm(n, K)
    Exports == Join(Ring,Algebra(K),VectorSpace(K)) with
      e: PI -> $
          ...
    Implementation == add
      Qeelist :=
        [q.unitVector(i::PI) for i in 1..n]
      dim     :=  2^n
      Rep     := PrimitiveArray K
      New ==> new(dim, 0$K)$Rep
      x + y ==
        z := New
        for i in 0..dim-1 repeat z.i := x.i + y.i
        z
      addMonomProd: (K, NNI, K, NNI, $) -> $
      addMonomProd(c1, b1, c2, b2, z) ==  ...
      x * y ==
        z := New
        for ix in 0..dim-1 repeat
          if x.ix \\notequal{} 0 then for iy in 0..dim-1 repeat
            if y.iy \\notequal{} 0
            then addMonomProd(x.ix,ix,y.iy,iy,z)
          z
           ...

Part of the CliffordAlgebra domain.

The Implementation part begins by defining a local variable Qeelist to hold the list of all q.v where v runs over the unit vectors from 1 to the dimension n. Another local variable dim is set to 2n, computed once and for all. The representation for the domain is PrimitiveArray(K), which is a basic array of elements from domain K. Line 18 defines New as shorthand for the more lengthy expression new(dim,0$K)$Rep, which computes a primitive array of length 2n filled with 0’s from domain K.

Lines 19-22 define the sum of two elements x and y straightforwardly. First, a new array of all 0’s is created, then filled with the sum of the corresponding elements. Indexing for primitive arrays starts at 0. The definition of the product of x and y first requires the definition of a local function addMonomProd. FriCAS knows it is local since it is not an exported function. The types of all local functions must be declared.