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.