Quantum Mechanics for Mathematicians

that preserves the canonical commutation relations. Here the use of normal
ordered operators means thatUA′ is a representation ofu(d) that differs by a
constant from the metaplectic representation, andUeAdiffers by a phase-factor.
This does not affect the commutation relations withUA′ or the conjugation
action ofUeA. The representation constructed this way differs in two ways from
the metaplectic representation. It acts on the same spaceH=Fd, but it is a
true representation ofU(d), no double cover is needed. It also does not extend
to a representation of the larger groupSp(2d,R).
The operatorsUA′ andUeAcommute with the Hamiltonian operator for the
harmonic oscillator (the quantization of equation 25.5). For physicists this is
quite useful, as it provides a decomposition of energy eigenstates into irreducible
representations ofU(d). For mathematicians, the quantum harmonic oscillator
state space provides a construction of a large class of irreducible representations
ofU(d), by considering the energy eigenstates of a given energy.

25.4 Examples ind= 2and 3

25.4.1 Two degrees of freedom andSU(2)

In the cased= 2, the action of the groupU(2)⊂Sp(4,R) discussed in section
25.3 commutes with the standard harmonic oscillator Hamiltonian and thus acts
as symmetries on the quantum harmonic oscillator state space, preserving en-
ergy eigenspaces. Restricting to the subgroupSU(2)⊂U(2), we’ll see that we
can recover our earlier (see section 8.2) construction ofSU(2) representations
in terms of homogeneous polynomials, in a new context. This use of the energy
eigenstates of a two dimensional harmonic oscillator appears in the physics liter-
ature as the “Schwinger boson method” for studying representations ofSU(2).
The state space for thed= 2 Bargmann-Fock representation, restricting to
finite linear combinations of energy eigenstates, is

H=F 2 fin=C[z 1 ,z 2 ]

the polynomials in two complex variablesz 1 ,z 2. Recall from ourSU(2) discus-
sion that it was useful to organize these polynomials into finite dimensional sets
of homogeneous polynomials of degreenforn= 0, 1 , 2 ,...

H=H^0 ⊕H^1 ⊕H^2 ⊕···

There are four annihilation or creation operators

a† 1 =z 1 , a† 2 =z 2 , a 1 =

∂

∂z 1

, a 2 =

∂

∂z 2

acting onH. These are the quantizations of complexified phase space coordi-
natesz 1 ,z 2 ,z 1 ,z 2 , with quantization the Bargmann-Fock construction of the
representation Γ′BFofh 2 d+1=h 5

Γ′BF(1) =−i 1 , Γ′BF(zj) =−ia†j, Γ′BF(zj) =−iaj