Quantum Mechanics for Mathematicians

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∇)