Chapter 28

Weyl and Clifford Algebras

We have seen that just changing commutators to anticommutators takes the
harmonic oscillator quantum system to a very different one (the fermionic os-
cillator), with this new system having in many ways a parallel structure. It
turns out that this parallelism goes much deeper, with every aspect of the har-
monic oscillator story having a fermionic analog. We’ll begin in this chapter by
studying the operators of the corresponding quantum systems.

28.1 The Complex Weyl and Clifford algebras

In mathematics, a “ring” is a set with addition and multiplication laws that are
associative and distributive (but not necessarily commutative), and an “algebra”
is a ring that is also a vector space over some field of scalars. The canonical
commutation and anticommutation relations define interesting algebras, called
the Weyl and Clifford algebras respectively. The case of complex numbers as
scalars is simplest, so we’ll start with that, before moving on to the real number
case.

28.1.1 One degree of freedom, bosonic case

Starting with the one degree of freedom case (corresponding to two operators
Q,P, which is why the notation will have a 2) we can define:

Definition(Complex Weyl algebra, one degree of freedom).The complex Weyl
algebra in the one degree of freedom case is the algebraWeyl(2,C)generated by
the elements 1 ,aB,a†B, satisfying the canonical commutation relations:

[aB,a†B] = 1, [aB,aB] = [a†B,a†B] = 0

In other words, Weyl(2,C) is the algebra one gets by taking arbitrary prod-
ucts and complex linear combinations of the generators. By repeated use of the
commutation relation
aBa†B= 1 +a†BaB