taking the boundary value asτ→0. This fixes the branch of the square root in
23.12 and one finds (see for instance section 7.6.7 of [107]) that the square root
factor needs to be taken to be
√
ω
i 2 πsin(ωT)
=e−i
π 4
e−in
π 2
√
ω
2 π|sin(ωT)|
forωT∈[nπ,(n+ 1)π].
23.5 The Bargmann transform
The Stone von-Neumann theorem implies the existence of:
Definition.Bargmann transform
There is a unitary map called the Bargmann transform
B:HS→F
intertwining the Schr ̈odinger representation and the Bargmann-Fock represen-
tation, i.e., with operators satisfying the relation
Γ′S(X) =B−^1 Γ′BF(X)B
forX∈h 3.
In practice, knowingBexplicitly is often not needed, since the representation
independent relation
a=
1
√
2
(Q+iP)
can be used to express operators either purely in terms ofaanda†, which have
a simple expression
a=
d
dz
, a†=z
in the Bargmann-Fock representation, or purely in terms ofQandPwhich have
a simple expression
Q=q, P=−i
d
dq
in the Schr ̈odinger representation.
To compute the Bargmann transform one uses equation 23.11, for non-
normalizable continuous basis states|δu〉, to get
〈q|δu〉=
(ω
π
)^14
e−
u 22
e−
ω 2 q 2
e
√
2 ωuq
and
〈δu|q〉=
(ω
π
)^14
e−
u 22
e−
ω 2 q 2
e
√
2 ωuq (23.13)