Quantum Mechanics for Mathematicians

38.1 Unitary transformations onH 1

The single-particle state spaceH 1 of non-relativistic quantum field theory can
be parametrized by either wavefunctionsψ(x) or their Fourier transformsψ ̃(p),
and carries a Hermitian inner product

〈ψ 1 ,ψ 2 〉=

∫

ψ 1 (x)ψ 2 (x)dx=

∫

ψ ̃ 1 (p)ψ ̃ 2 (p)dp

As a dual phase space, the symplectic structure is given by the imaginary part
of this

Ω(ψ 1 ,ψ 2 ) =

1

2 i

∫

(ψ 1 (x)ψ 2 (x)−ψ 2 (x)ψ 1 (x))dx

There is an infinite dimensional symplectic group that acts onH 1 by linear
transformations that preserve Ω. It has an infinite dimensional unitary sub-
group, those transformations preserving the full inner product. In this chapter
we’ll consider various finite dimensional groupsGthat are subgroups of this uni-
tary group, and see how they are represented on the quantum field theory state
space. Note that there are also groups that act as symplectic but not unitary
transformations ofH 1 , after quantization acting by a unitary representation on
the multi-particle state space. Such actions change particle number, and the
vacuum state| 0 〉in particular will not be invariant. For some indications of
what happen in this more general situation, see sections 25.5 and 39.4.
The finite dimensional version of the case of unitary transformations ofH 1
was discussed in detail in sections 25.2 and 25.3 where we saw that the moment
map for theU(d) action was given by

μA=i

∑

j,k

zjAjkzk

forAa skew-adjoint matrix. Quantization took this quadratic function on phase
space to the quadratic combination of annihilation and creation operators

∑

j,k

a†jAjkak

and exponentiation of these operators gave the unitary representation on state
space.
In the quantum field theory case with dual phase spaceH 1 , the generalization
of the finite dimensional case will be

indexj→xor p

zj→Ψ(x) or α(p), zj→Ψ(x) orα(p)

a†j→Ψ̂†(x) or a†(p), aj→Ψ(̂x) or a(p)