Quantum Mechanics for Mathematicians

a basis for the complexified dual phase spaceM⊗C. Note that these coordinates
provide a decomposition
M⊗C=C⊕C

of the complexified dual phase space into subspaces spanned byzand byz.
The Lie bracket is the Poisson bracket, extended by complex linearity. The
only non-zero bracket between basis elements is given by

{z,z}=i

Quantization by annihilation and creation operators produces a Lie algebra
representation by

Γ′(1) =−i 1 , Γ′(z) =−ia†, Γ′(z) =−ia (22.5)

with the operator relation
[a,a†] = 1

equivalent to the Lie algebra homomorphism property

[Γ′(z),Γ′(z)] = Γ′({z,z})

We have now seen two different unitarily equivalent realizations of this Lie
algebra representation: the Schr ̈odinger version Γ′Son functions ofq, where

a=

1

√

2

(

q+

d

dq

)

, a†=

1

√

2

(

q−

d

dq

)

and the Bargmann-Fock version Γ′BFon functions ofz, where

a=

d

dz

, a†=z

Note that while annihilation and creation operators give a representation
of the complexified Heisenberg Lie algebrah 3 ⊗C, this representation is only
unitary on the real Lie subalgebrah 3. This corresponds to the fact that general
complex linear combinations ofaanda†are not self-adjoint, to get something
self-adjoint one must take real linear combinations of

a+a† and i(a−a†)

22.5 For further reading

All quantum mechanics books should have a similar discussion of the harmonic
oscillator, with a good example the detailed one in chapter 7 of Shankar [81].
One source for a detailed treatment of the Bargmann-Fock representation is
[26].