Quantum Mechanics for Mathematicians

where theRcomponent is the constant functions. The Lie bracket is the Poisson
bracket. Complexifying gives

h 2 d+1⊗C= (M⊕R)⊗C= (M⊗C)⊕C=M+J⊕M−J⊕C

so elements ofh 2 d+1⊗Ccan be written as pairs (u,c) = (u++u−,c) where

u∈M⊗C, u+∈M+J, u−∈M−J, c∈C

This complexified Lie algebra is still a Lie algebra, with the Lie bracket relations

[(u 1 ,c 1 ),(u 2 ,c 2 )] = (0,Ω(u 1 ,u 2 )) (26.12)

and antisymmetric bilinear form Ω onMextended from the real Lie algebra by
complex linearity.
For eachJ, we would like to find a quantization that takes elements of
M+J to linear combinations of creation operators, elements ofM−J to linear
combinations of annihilation operators. This will give a representation of the
complexified Lie algebra

Γ′J: (u,c)∈h 2 d+1⊗C→Γ′J(u,c)

if it satisfies the Lie algebra homomorphism property

[Γ′J(u 1 ,c 1 ),Γ′J(u 2 ,c 2 )] = Γ′J([(u 1 ,c 1 ),(u 2 ,c 2 )]) = Γ′J(0,Ω(u 1 ,u 2 )) (26.13)

Since we can write
(u,c) = (u+,0) + (u−,0) + (0,c)

whereu+∈M+J andu−∈M−J, we have

Γ′J(u,c) = Γ′J(u+,0) + Γ′J(u−,0) + Γ′J(0,c)

Note that we only expect Γ′Jto be a unitary representation (with Γ′J(u,c)
skew-adjoint operators) for (u,c) in the real Lie subalgebrah 2 d+1(meaning
u+=u−, c∈R).
For the case ofJ =J 0 , the Lie algebra representation is given on basis
elements
zj∈M+J 0 , zj∈M−J 0

by

Γ′J 0 (0,1) =−i 1 , Γ′J 0 (zj,0) =−ia†j=−izj, Γ′J 0 (zj,0) =−iaj=−i

∂

∂zj

and is precisely the Bargmann-Fock representation Γ′BF(see equation 22.5), Note
that the operatorsajanda†jare not skew-adjoint, so Γ′J 0 is not unitary on the
full Lie algebrah 2 d+1⊗C, but only on the real subspaceh 2 d+1of real linear
combinations ofqj,pj,1.