Quantum Mechanics for Mathematicians

Recall (theorem 25.2) that fornbynmatricesXandY



∑n

j,k=1

a†jXjkak,

∑n

j,k=1

a†jYjkak



=

∑n

j,k=1

a†j[X,Y]jkak

So, for eachXin the Lie algebragl(n,C), quantization will give us a represen-
tation ofgl(n,C) whereXacts as the operator

∑n

j,k=1

a†jXjkak

When the matricesXare chosen to be skew-adjoint (Xjk=−Xkj) this con-
struction will give us a unitary representation ofu(n).
As in theU(1) case, one gets an operator in the quantum field theory by
integrating over quadratic combinations of thea(p),a†(p) in momentum space,

or the field operatorsΨ(̂x),Ψ̂†(x) in configuration space, finding for eachX∈
u(n) an operator

X̂=

∫+∞

−∞

∑n

j,k=1

Ψ̂†j(x)XjkΨ̂k(x)dx=

∫+∞

−∞

∑n

j,k=1

a†j(p)Xjkak(p)dp (38.5)

This satisfies
[X,̂Ŷ] =[̂X,Y] (38.6)

and, acting on operators

[X,̂Ψ̂j(x)] =−

∑n

k=1

XjkΨ̂k(x), [X,̂Ψ̂†j(x)] =

∑n

k=1

XkjΨ̂†k(x) (38.7)

X̂provides a Lie algebra representation ofu(n) on the multi-particle state space.
After exponentiation, this representation takes

eX∈U(n)→U(eX) =eX̂=e

∫+∞

−∞

∑n

b,c=1̂Ψ†j(x)XjkΨ̂k(x)dx

The construction of the operatorX̂above is an infinite dimensional example
of our standard method of creating a Lie algebra representation by quantizing
moment map functions. In this case the quadratic moment map function on the
space of solutions of the Schr ̈odinger equation is

μX=i

∫+∞

−∞

∑n

j,k=1

Ψj(x)XjkΨk(x)dx

which (generalizing the finite dimensional case of theorem 25.1) satisfies the
Poisson bracket relations
{μX,μY}=μ[X,Y]