Quantum Mechanics for Mathematicians

by using the Clifford algebra relations, or by noting that this is the special case
of equation 29.5 forv=el. That equation shows that commuting by−^12 γjγk
acts by the infinitesimal rotationjkin thejkcoordinate plane.
For 30.2, the Clifford algebra relations can again be used to show
[
1
2

γjγk,

1

2

γlγm

]

=δkl

1

2

γjγm−δjl

1

2

γkγm+δkm

1

2

γlγj−δjm

1

2

γlγk

One could instead use the commutation relations for theso(n) Lie algebra sat-
isfied by the basis elementsjkcorresponding to infinitesimal rotations. One
must get identical commutation relations for the−^12 γjγkand can show that
these are the relations needed for commutators of Γ+(ξjξk) and Γ+(ξlξm).

Note that here we are not introducing the factors ofiinto the definition of
quantization that in the bosonic case were necessary to get a unitary represen-
tation of the Lie group corresponding to the real Heisenberg Lie algebrah 2 d+1.
In the bosonic case we worked with all complex linear combinations of powers
of theQj,Pj(the complex Weyl algebra Weyl(2d,C)), and thus had to identify
the specific complex linear combinations of these that gave unitary represen-
tations of the Lie algebrah 2 d+1osp(2d,R). Here we are not complexifying
for now, but working with the real Clifford algebra Cliff(r,s,R), and it is the
irreducible representations of this algebra that provide an analog of the unique
interesting irreducible representation ofh 2 d+1. In the Clifford algebra case, the
representations of interest are not just Lie algebra representations and may be
on real vector spaces. There is no analog of the unitarity property of theh 2 d+1
representation.
In the bosonic case we found thatSp(2d,R) acted on the bosonic dual phase
space, preserving the antisymmetric bilinear form Ω that determined the Lie al-
gebrah 2 d+1, so it acted on this Lie algebra by automorphisms. We saw (see
chapter 20) that intertwining operators there gave us a representation of the
double cover ofSp(2d,R) (the metaplectic representation), with the Lie alge-
bra representation given by the quantization of quadratic functions of theqj,pj
phase space coordinates. There is a closely analogous story in the fermionic case,
whereSO(r,s,R) acts on the fermionic phase spaceV, preserving the symmet-
ric bilinear form (·,·) that determines the Clifford algebra relations. Here a
representation of the spin groupSpin(r,s,R) double coveringSO(r,s,R) is
constructed using intertwining operators, with the Lie algebra representation
given by quadratic combinations of the quantizations of the fermionic coordi-
natesξj. The case ofr= 3,s= 1 will be of importance later in our discussion
of special relativity (see chapter 41), giving the spinor representation of the
Lorentz group.
The fermionic analog of 20.1 is

UkΓ+(ξ)Uk−^1 = Γ+(φk 0 (ξ)) (31.1)

Herek 0 ∈SO(r,s,R),ξ∈V∗=Rn(n=r+s),φk 0 is the action ofk 0 on
V∗. TheUkfork= Φ−^1 (k 0 )∈Spin(r,s) (Φ is the 2-fold covering map) are the