Quantum Mechanics for Mathematicians

any element of this algebra can be written as a sum of elements in normal order,
of the form
cl,m(a†B)lamB

with all annihilation operatorsaBon the right, for some complex constantscl,m.
As a vector space overC, Weyl(2,C) is infinite dimensional, with a basis

1 , aB, a†B, a^2 B, a†BaB,(a†B)^2 , a^3 B, a†Ba^2 B,(a†B)^2 aB,(a†B)^3 ,...

This algebra is isomorphic to a more familiar one. Setting

a†B=z, aB=

d

dz

one sees that Weyl(2,C) can be identified with the algebra of polynomial coef-
ficient differential operators on functions of a complex variablez. As a complex
vector space, the algebra is infinite dimensional, with a basis of elements

zl

dm
dzm
In our study of quantization by the Bargmann-Fock method, we saw that
the subset of such operators consisting of complex linear combinations of

1 , z,

d

dz

, z^2 ,

d^2

dz^2

, z

d

dz

is closed under commutators, and is a representation of a Lie algebra of complex
dimension 6. This Lie algebra includes as subalgebras the Heisenberg Lie algebra
h 3 ⊗C(first three elements) and the Lie algebrasl(2,C) =sl(2,R)⊗C(last
three elements). Note that here we are allowing complex linear combinations,
so we are getting the complexification of the real six dimensional Lie algebra
that appeared in our study of quantization.
Since theaBanda†Bare defined in terms ofP andQ, one could of course
also define the Weyl algebra as the one generated by 1,P,Q, with the Heisenberg
commutation relations, taking complex linear combinations of all products of
these operators.

28.1.2 One degree of freedom, fermionic case

Changing commutators to anticommutators, one gets a different algebra, the
Clifford algebra:

Definition(Complex Clifford algebra, one degree of freedom). The complex
Clifford algebra in the one degree of freedom case is the algebraCliff(2,C)gen-
erated by the elements 1 ,aF,a†F, subject to the canonical anticommutation rela-
tions (CAR)
[aF,a†F]+= 1, [aF,aF]+= [a†F,a†F]+= 0