Quantum Mechanics for Mathematicians

unitary representation ofu(d) on Fock spaceFd, but after exponentiation this

is a representation not of the groupU(d), but of a double cover we callU ̃(d).
One could instead quantize using normal ordered operators, taking

zjzj→−ia†jaj

The definition of normal ordering in section 24.3 generalizes simply, since the
order of annihilation and creation operators with different values ofjis imma-
terial. Using this normal ordered choice, the usual quantized operators of the
Bargmann-Fock representation are shifted by a scalar^12 for eachj, and after
exponentiation the state spaceH=Fdprovides a representation ofU(d), with
no need for a double cover. As au(d) representation however, this does not
extend to a representation ofsp(2d,R), since commutation ofa^2 jwith (a†j)^2 can
land one on the unshifted operators.
Since the normal ordering doesn’t change the commutation relations obeyed
by products of the forma†jak, the quadratic expression forμAcan be quantized

using normal ordering, and get quadratic combinations of theaj,a†kwith the
same commutation relations as in theorem 25.1. Letting

UA′ =

∑

j,k

a†jAjkak (25.6)

we have

Theorem 25.2.ForA∈gl(d,C)adbydcomplex matrix

[UA′,UA′′] =U[′A,A′]

As a result
A∈gl(d,C)→UA′

is a Lie algebra representation ofgl(d,C)onH=C[z 1 ,...,zd], the harmonic
oscillator state space inddegrees of freedom.
In addition (for column vectorsawith componentsa 1 ,...,ad)

[UA′,a†] =ATa†, [UA′,a] =−Aa (25.7)

Proof.Essentially the same proof as 25.1.

ForA∈u(d) the Lie algebra representationUA′ ofu(d) exponentiates to give
a representation ofU(d) onH=C[z 1 ,...,zd] by operators

UeA=eU

′A

These satisfy

UeAa†(UeA)−^1 =eA

T

a†, UeAa(UeA)−^1 =eA

T

a (25.8)

(the relations 25.7 are the derivative of these). This shows that theUeAare
intertwining operators for aU(d) action on annihilation and creation operators