Quantum Mechanics for Mathematicians

and one creation operator, operators which annihilate| 0 〉F. Non-zero pairs of
two creation operators act non-trivially on| 0 〉F, corresponding to the fact that
elements ofSO(2d) not in theU(d) subgroup take| 0 〉Fto a different state in
the spinor representation.
Given any group element

g 0 =eA⊂U(d)

acting on the fermionic dual phase space preservingJand the inner product, we
can use exactly the same method as in theorems 25.1 and 25.2 to construct its
action on the fermionic state space by the second of the above representations.
ForAa skew-adjoint matrix we have a fermionic moment map

A∈u(d)→μA=

∑

j,k

θjAjkθk

satisfying
{μA,μA′}+=μ[A,A′]

and
{μA,θ}+=ATθ, {μA,θ}+=ATθ=−Aθ (31.5)
The Lie algebra representation operators are the

UA′ =

∑

j,k

a†FjAjkaFk

which satisfy (see theorem 27.1)

[UA′,UA′′] =U[A,A′]

and
[UA′,a†F] =ATa†F, [UA′,aF] =ATaF
Exponentiating these gives the intertwining operators, which act on the an-
nihilation and creation operators as

UeAa†F(UeA)−^1 =eA

T

a†F, UeAaF(UeA)−^1 =eA

T

aF

For the simplest example, consider theU(1)⊂U(d)⊂SO(2d) that acts by

θj→eiφθj, θj→e−iφθj

corresponding toA=iφ 1. The moment map will be

μA=−φh

where

h=−i

∑d

j=1

θjθj