Chapter 29

Clifford Algebras and

Geometry

The definitions given in chapter 28 of Weyl and Clifford algebras were purely
algebraic, based on a choice of generators and relations. These definitions do
though have a more geometrical formulation, with the definition in terms of
generators corresponding to a specific choice of coordinates. For the Weyl alge-
bra, the geometry involved is symplectic geometry, based on a non-degenerate
antisymmetric bilinear form. We have already seen that in the bosonic case
quantization of a phase spaceR^2 ddepends on the choice of a non-degenerate an-
tisymmetric bilinear form Ω which determines the Poisson brackets and thus the
Heisenberg commutation relations. Such a Ω also determines a groupSp(2d,R),
which is the group of linear transformations ofR^2 dpreserving Ω.
The Clifford algebra also has a coordinate invariant definition, based on
a more well known structure on a vector spaceRn, that of a non-degenerate
symmetric bilinear form, i.e., an inner product. In this case the group that
preserves the inner product is an orthogonal group. In the symplectic case
antisymmetric forms require an even number of dimensions, but this is not true
for symmetric forms, which also exist in odd dimensions.

29.1 Non-degenerate bilinear forms

In the case ofM=R^2 d, the dual phase space, the Poisson bracket determines
an antisymmetric bilinear form onM, which, for a basisqj,pjand two vectors
u,u′∈M
u=cq 1 q 1 +cp 1 p 1 +···+cqdqd+cpdpd∈M
u′=c′q 1 q 1 +c′p 1 p 1 +···+c′qdqd+c′pdpd∈M

is given explicitly by