Quantum Mechanics for Mathematicians

andE(3) will act on Ψ(ψ) by

Ψ(ψ)→(a,R)·Ψ(ψ) =

∫

R^3

Ψ(x)ψ(R−^1 (x−a))d^3 x

=

∫

R^3

Ψ(Rx+a)ψ(x)d^3 x

(using invariance of the integration measure underE(3) transformations).

More generally, if elements ofH 1 are multi-component functionsψj(for

instance in the case of spin^12 wavefunctions), the (double cover of) the

E(3) group may act by

ψj(x)→

∑

k

Ωjkψk(R−^1 (x−a))

on wavefunctions, and

Ψj(x)→

∑

k

(Ω−^1 )jkΨk(Rx+a)

on distributional fields (see section 38.3.3).

• The action ofE(3) onH 1 is a linear map preserving the symplectic struc-
  ture. We thus expect by the general method of section 20.2 to be able to
  construct intertwining operators, by taking the quadratic functions given
  by the moment map, quantizing to get a Lie algebra representation, and
  exponentiating to get a unitary representation ofE(3). More specifically,
  we will use Bargmann-Fock quantization, and the method carried out for
  a finite dimensional phase space in section 25.3. We end up with a rep-
  resentation ofE(3) on the quantum field theory state spaceH, given by
  unitary operatorsU(a,R).

It is the last of these that we want to examine here, and as usual for quantum
field theory, we don’t want to try and explicitly construct the multi-particle state
spaceHand see theE(3) action on that construction, but instead want to use
the analog of the Heisenberg picture in the time-translation case, taking the
group to act on operators. For each (a,R)∈E(3) we want to find operators
U(a,R) that will be built out of the field operators, and act on the field operators
as
Ψ(̂x)→U(a,R)Ψ(̂x)U(a,R)−^1 =Ψ(̂Rx+a) (38.9)

38.3.1 Spatial translations

For spatial translations, we want to construct momentum operatorsP̂such
that the−iP̂give a unitary Lie algebra representation of the translation group.
Exponentiation will then give the unitary representation

U(a, 1 ) =e−ia·

P̂