Chapter 31

Fermionic Quantization and

Spinors

In this chapter we’ll begin by investigating the fermionic analog of the notion
of quantization, which takes functions of anticommuting variables on a phase
space with symmetric bilinear form (·,·) and gives an algebra of operators with
generators satisfying the relations of the corresponding Clifford algebra. We
will then consider analogs of the constructions used in the bosonic case which
there gave us the Schr ̈odinger and Bargmann-Fock representations of the Weyl
algebra on a space of states.
We know that for a fermionic oscillator withddegrees of freedom, the al-
gebra of operators will be Cliff(2d,C), the algebra generated by annihilation
and creation operatorsaFj,aF†j. These operators will act onHF=Fd+, a com-
plex vector space of dimension 2d, and this will provide a fermionic analog of the
bosonic Γ′BFacting onFd. Since the spin group consists of invertible elements of
the Clifford algebra, it has a representation onFd+. This is known as the “spinor
representation”, and it can be constructed by analogy with the construction of
the metaplectic representation in the bosonic case. We’ll also consider the ana-
log in the fermionic case of the Schr ̈odinger representation, which turns out to
have a problem with unitarity, but finds a use in physics as “ghost” degrees of
freedom.

31.1 Quantization of pseudo-classical systems

In the bosonic case, quantization was based on finding a representation of the
Heisenberg Lie algebra of linear functions on phase space, or more explicitly,
for basis elementsqj,pjof this Lie algebra finding operatorsQj,Pjsatisfying
the Heisenberg commutation relations. In the fermionic case, the analog of
the Heisenberg Lie algebra is not a Lie algebra, but a Lie superalgebra, with
basis elements 1,ξj,j= 1,...,nand a Lie superbracket given by the fermionic