Quantum Mechanics for Mathematicians

intertwining operators we are looking for. The fermionic analog of 20.2 is

[UL′,Γ+(ξ)] = Γ+(L·ξ)

whereL∈so(r,s,R) andLacts onV∗as an infinitesimal orthogonal transfor-
mation. In terms of basis vectors ofV∗

ξ=







ξ 1

..

.

ξn







this says
[UL′,Γ+(ξ)] = Γ+(LTξ)
Just as in the bosonic case, theUL′ can be found by looking first at the
pseudo-classical case, where one has theorem 30.1 which says

{μL,ξ}+=LTξ

where

μL=

1

2

ξ·Lξ=

1

2

∑

j,k

Ljkξjξk

One then takes

UL′= Γ+(μL) =

1

4

∑

j,k

Ljkγjγk

For the positive definite cases= 0 and a rotation in thejkplane, with
L=jkone recovers formulas 29.4 and 29.5 from chapter 29, with
[
−

1

2

γjγk,γ(v)

]

=γ(jkv)

the infinitesimal action of a rotation on theγmatrices, and

γ(v)→e−

θ 2 γjγk

γ(v)e

θ 2 γjγk

=γ(eθjkv)

the group version. Just as in the symplectic case, exponentiating theUL′ only
gives a representation up to sign, and one needs to go to the double cover of
SO(n) to get a true representation. As in that case, the necessity of the double
cover is best seen by use of a complex structure and an analog of the Bargmann-
Fock construction, an example will be given in section 31.4.
In order to have a full construction of a quantization of a pseudo-classical
system, we need to construct the Γ+(ξj) as linear operators on a state space.
As mentioned in section 28.2, it can be shown that the real Clifford algebras
Cliff(r,s,R) are isomorphic to either one or two copies of the matrix algebras
M(2l,R),M(2l,C), orM(2l,H), with the powerldepending onr,s. The irre-
ducible representations of such a matrix algebra are just the column vectors of
dimension 2l, and there will be either one or two such irreducible representa-
tions for Cliff(r,s,R) depending on the number of copies of the matrix algebra.
This is the fermionic analog of the Stone-von Neumann uniqueness result in the
bosonic case.