Quantum Mechanics for Mathematicians

44.2.1 Translations

For time translations, we have already found the Hamiltonian operatorĤ, which
gives the infinitesimal translation action on fields by

∂̂Φ

∂t

= [Φ̂,−iĤ]

The behavior of the field operator under time translation is given by the stan-
dard Heisenberg picture relation for operators

Φ(̂t+a 0 ) =eia^0 ĤΦ(̂t)e−ia^0 Ĥ

For the infinitesimal action of spatial translations onH 1 , the momentum
operator is the usual
P=−i∇

(the convention for the Hamiltonian is the opposite signH=i∂t∂). On fields the

infinitesimal action will be given by an operatorP̂satisfying the commutation
relations
[−iP̂,Φ] =̂ ∇̂Φ

(see the discussion for the non-relativistic case in section 38.3 and equation
38.10). Finite spatial translations byawill act by

Φ(̂x)→e−ia·P̂Φ(̂x)eia·̂P=̂Φ(x+a)

The operator needed is the quadratic operator

P̂=

∫

R^3

pa†(p)a(p)d^3 p (44.7)

which in terms of fields is given by

P̂=−

∫

R^3

:Π(̂x)∇Φ(̂x):d^3 x

One can see that this is the correct operator by showing that it satisfies the
commutation relation 38.10 witĥΦ, using the canonical commutation relations
forΦ and̂ ̂π.
Note that here again moment map methods could have been used to find
the expression for the momentum operator. This is a similar calculation to that
of section 38.3 although one needs to keep track of a factor of−icaused by the
fact that the basic Poisson bracket relations are

{Φ(x),Π(x)}=δ^3 (x−x′) versus {Ψ(x),Ψ(x)}=iδ^3 (x−x′)