Chapter 30

Anticommuting Variables

and Pseudo-classical

Mechanics

The analogy between the algebras of operators in the bosonic (Weyl algebra) and
fermionic (Clifford algebra) cases can be extended by introducing a fermionic
analog of phase space and the Poisson bracket. This gives a fermionic ana-
log of classical mechanics, sometimes called “pseudo-classical mechanics”, the
quantization of which gives the Clifford algebra as operators, and spinors as
state spaces. In this chapter we’ll introduce “anticommuting variables”ξjthat
will be the fermionic analogs of the variablesqj,pj. These objects will become
generators of the Clifford algebra under quantization, and will later be used in
the construction of fermionic state spaces, by analogy with the Schr ̈odinger and
Bargmann-Fock constructions in the bosonic case.

30.1 The Grassmann algebra of polynomials on

anticommuting generators

Given a phase spaceM=R^2 d, one gets classical observables by taking poly-
nomial functions onM. These are generated by the linear functionsqj,pj,j=
1 ,...,d, which lie in the dual spaceM=M∗. One can instead start with a
real vector spaceV=Rnwithnnot necessarily even, and again consider the
spaceV∗of linear functions onV, but with a different notion of multiplication,
one that is anticommutative on elements ofV∗. Using such a multiplication, an
anticommuting analog of the algebra of polynomials onVcan be generated in
the following manner, beginning with a choice of basis elementsξjofV∗:

Definition(Grassmann algebra). The algebra over the real numbers generated