Note that these are not the momentum operatorsPthat act onH 1 , but are
operators in the quantum field theory that will be built out of quadratic com-
binations of the field operators. By equation 38.9 we want
e−ia·P̂Ψ(̂x)eia·P̂=Ψ(̂x+a)
or the derivative of this equation
[−iP̂,Ψ(̂x)] =∇Ψ(̂x) (38.10)
Such an operatorP̂can be constructed in terms of quadratic combinations of
the field operators by our moment map methods. We find (generalizing theorem
25.1) that the quadratic expression
μ−∇=i
∫
R^3
Ψ(x)(−∇)Ψ(x)d^3 x
is real (since∇is skew-adjoint) and satisfies
{μ−∇,Ψ(x)}=∇Ψ(x), {μ−∇,Ψ(x)}=∇Ψ(x)
Using the Poisson bracket relations, this can be checked by computing for in-
stance (we’ll do this just ford= 1)
{μ−dxd,Ψ(x)}=i
{∫
Ψ(y)(−
d
dy
)Ψ(y)dy,Ψ(x)
}
=−i
∫
{Ψ(y),Ψ(x)}
d
dy
Ψ(y)dy
=
∫
δ(x−y)
d
dy
Ψ(y)dy=
d
dx
Ψ(x)
Quantization replaces Ψ,Ψ byΨ̂†,Ψ and gives the self-adjoint expression̂
P̂=
∫
R^3
Ψ̂†(x)(−i∇)Ψ(̂x)d^3 x (38.11)
for the momentum operator. In chapter 37 we saw that, in terms of momentum
space annihilation and creation operators, this operator is
P̂=
∫
R^3
pa†(p)a(p)d^3 p
which is the integral over momentum space of the momentum times the number-
density operator in momentum space.
38.3.2 Spatial rotations
For spatial rotations, we found in chapter 19 that these had as generators the
angular momentum operators
L=X×P=X×(−i∇)