Quantum Mechanics for Mathematicians

For more general choices ofJwe start by taking

Γ′J(0,c) =−ic 1 (26.14)

which is chosen so that it commutes with all other operators of the represen-
tation, and forcreal gives a skew-adjoint transformation and thus a unitary
representation. We would like to construct Γ′J(u+,0) as a linear combination of
creation operators and Γ′J(u−,0) as a linear combination of annihilation opera-
tors. The compatibility condition of equation 26.5 will ensure that the Γ′J(u+,0)
will commute, since ifu+ 1 ,u+ 2 ∈M+J, by 26.13 we have

[Γ′J(u+ 1 ,0),Γ′J(u+ 2 ,0)] = Γ′J(0,Ω(u+ 1 ,u+ 2 )) =−iΩ(u+ 1 ,u+ 2 ) 1

and
Ω(u+ 1 ,u+ 2 ) = Ω(Ju+ 1 ,Ju+ 2 ) = Ω(iu+ 1 ,iu+ 2 ) =−Ω(u+ 1 ,u+ 2 ) = 0

The Γ′J(u−,0) will commute with each other by essentially the same argument.
To see the necessity of the positivity condition 26.10 onJ, recall that the
annihilation and creation operators satisfy (ford= 1)

[a,a†] = 1

a condition which corresponds to

[−ia,−ia†] = [Γ′J 0 (z,0),Γ′J 0 (z,0)] = Γ′J 0 (0,{z,z}) = Γ′J 0 (0,−i) =− 1

Use of the opposite sign for the commutator would correspond to interchanging
the role ofaanda†, with the state| 0 〉now satisfyinga†| 0 〉= 0 and no state
|ψ〉in the state space satisfyinga|ψ〉= 0. In order to have a state| 0 〉that is
annihilated by all annihilation operators and a total number operator with non-
negative eigenvalues (and thus a Hamiltonian with a positive energy spectrum),
we need all the commutators [aj,a†j] to have the positive sign.

For any choice of a potential basis elementuofM+J, by 26.13 and 26.14 we
have,
[Γ′J(u,0),Γ′J(u,0)] = Γ′J(0,Ω(u,u)) =−iΩ(u,u) 1 (26.15)

and the positivity condition 26.10 onJ will ensure that quantizing such an
element by a creation operator will give a representation with non-negative
number operator eigenvalues. We have the following general result about the
Bargmann-Fock construction for suitableJ:

Theorem.Given a positive compatible complex structureJonM, there is a
basiszJj ofM+J such that a representation ofh 2 d+1⊗C, unitary for the real
subalgebrah 2 d+1, is given by

Γ′J(zJj,0) =−ia†j, Γ′J(zJj,0) =−iaj, Γ′J(0,c) =−ic 1

whereaj,a†jsatisfy the conventional commutation relations, andzJj is the com-

plex conjugate ofzJj.