Quantum Mechanics for Mathematicians

This can be shown using

|δw〉〈δw|=

(∞

∑

n=0

wn

√

n!

|n〉

)(∞

∑

m=0

zm

√

m!

〈m|

)

as well as

1 =

∑∞

n=0

|n〉〈n|

and the orthogonality relations 23.4. Note that the normalized coherent states
similarly provide an over-complete basis, with

1 =

1

π

∫

C

|α〉〈α|d^2 α (23.7)

To avoid confusion over the various ways in which complex variableszand
wappear here, note that this is just the analog of what happens in the position
space representation, whereqis variously a coordinate on classical phase space,
an argument of a wavefunction, a label of a position operator eigenstate, and
a multiplication operator. The analog of the position operatorQhere isa†,
which is multiplication byz(unlike Q, not self-adjoint). The conjugate com-
plex coordinatezis analogous to the momentum coordinate, quantized to a
differentiation operator. One confusing aspect of this formalism is that complex
conjugation takes elements ofH(holomorphic functions) to antiholomorphic
functions, which are in a different space. The quantization ofzis not the
complex-conjugate ofz, but the adjoint operator.

23.3 The Heisenberg group action on operators

The representation operators

Γ(α,γ) =D(α)e−iγ

act not just on states, but also on operators, by the conjugation action

D(α)aD(α)−^1 =a−α, D(α)a†D(α)−^1 =a†−α

(on operators the phase factors cancel). These relations follow from the fact
that the commutation relations

[αa†−αa,a] =−α, [αa†−αa,a†] =−α

are the derivatives with respect totof

D(tα)aD(tα)−^1 =a−tα, D(tα)a†D(tα)−^1 =a†−tα (23.8)

Att= 0 this is just equation 5.1, but it holds for alltsince multiple commutators
vanish.