Quantum Mechanics for Mathematicians

31.1.1 Quantization of the pseudo-classical spin

As an example, one can consider the quantization of the pseudo-classical spin
degree of freedom of section 30.3.1. In that case Γ+takes values in Cliff(3, 0 ,R),
for which an explicit identification with the algebraM(2,C) of two by two
complex matrices was given in section 28.2. One has

Γ+(ξj) =

1

√

2

γj=

1

√

2

σj

and the Hamiltonian operator is

−iH= Γ+(h) =Γ+(B 12 ξ 1 ξ 2 +B 13 ξ 1 ξ 3 +B 23 ξ 2 ξ 3 )

=

1

2

(B 12 σ 1 σ 2 +B 13 σ 1 σ 3 +B 23 σ 2 σ 3 )

=i

1

2

(B 1 σ 1 +B 2 σ 2 +B 3 σ 3 )

This is nothing but our old example from chapter 7 of a fixed spin particle in a
magnetic field.
The pseudo-classical equation of motion
d
dt

ξj(t) =−{h,ξj}+

after quantization becomes the Heisenberg picture equation of motion for the
spin operators (see equation 7.3)

d

dt

SH(t) =−i[SH·B,SH]

for the case of Hamiltonian
H=−μ·B

(see equation 7.2) and magnetic moment operator

μ=S

Here the state space isH=C^2 , with an explicit choice of basis given by our
chosen identification of Cliff(3, 0 ,R) with two by two complex matrices. In the
next sections we will consider the case of an even dimensional fermionic phase
space, but there provide a basis-independent construction of the state space and
the action of the Clifford algebra on it.

31.2 The Schr ̈odinger representation for fermions: ghosts

We would like to construct representations of Cliff(r,s,R) and thus fermionic
state spaces by using analogous constructions to the Schr ̈odinger and Bargmann-
Fock ones in the bosonic case. The Schr ̈odinger construction took the state