Quantum Mechanics for Mathematicians

28.1.3 Multiple degrees of freedom

For a larger number of degrees of freedom, one can generalize the above and
define Weyl and Clifford algebras as follows:

Definition(Complex Weyl algebras). The complex Weyl algebra forddegrees
of freedom is the algebraWeyl(2d,C)generated by the elements 1 ,aBj,aB†j,
j= 1,...,dsatisfying the CCR

[aBj,aB†k] =δjk 1 , [aBj,aBk] = [aB†j,aB†k] = 0
Weyl(2d,C) can be identified with the algebra of polynomial coefficient dif-
ferential operators indcomplex variablesz 1 ,z 2 ,...,zd. The subspace of complex
linear combinations of the elements

1 ,zj,

∂

∂zj

,zjzk,

∂^2

∂zj∂zk

,zj

∂

∂zk

is closed under commutators and provides a representation of the complexifica-
tion of the Lie algebrah 2 d+1osp(2d,R) built out of the Heisenberg Lie algebra
forddegrees of freedom and the Lie algebra of the symplectic groupSp(2d,R).
Recall that this is the Lie algebra of polynomials of degree at most 2 on the
phase spaceR^2 d, with the Poisson bracket as Lie bracket. The complex Weyl
algebra could also be defined by taking complex linear combinations of products
of generators 1,Pj,Qj, subject to the Heisenberg commutation relations.
For Clifford algebras one has:

Definition(Complex Clifford algebras, using annihilation and creation oper-
ators). The complex Clifford algebra for ddegrees of freedom is the algebra
Cliff(2d,C)generated by 1 ,aFj,aF†jforj= 1, 2 ,...,dsatisfying the CAR

[aFj,aF†k]+=δjk 1 , [aFj,aFk]+= [aF†j,aF†k]+= 0

or, alternatively, one has the following more general definition that also works
in the odd dimensional case:

Definition(Complex Clifford algebras).The complex Clifford algebra innvari-
ables is the algebraCliff(n,C)generated by 1 ,γj forj= 1, 2 ,...,nsatisfying
the relations
[γj,γk]+= 2δjk
We won’t try and prove this here, but one can show that, abstractly as
algebras, the complex Clifford algebras are something well known. Generalizing
the cased= 1 where we saw that Cliff(2,C) was isomorphic to the algebra of 2
by 2 complex matrices, one has isomorphisms

Cliff(2d,C)↔M(2d,C)

in the even dimensional case, and in the odd dimensional case

Cliff(2d+ 1,C)↔M(2d,C)⊕M(2d,C)

Two properties of Cliff(n,C) are