444
12
Fields in Interaction
the Lorentzinvariant functions
ˆ
and
, separating it from the kinematics, which is
captured by the phasespace element
d
(
n
)
.
In this section we are going to express
these quantities in terms of the interaction Hamiltonian
H
I
.
12.3.1 Interaction Representation
As anticipated in the introduction, in perturbation theory the Hamiltonian of the
interacting system is computed on free fields, namely on fields evolving according
to
H
0
.
Inthe
Schrödingerpicture
,seeSect.
9.3.2
,operators,includingtheHamiltonian,are
constant while states

ψ(
t
)
S
evolve in time according to the Schrödinger equation:
i
∂
∂
t

ψ(
t
)
S
=
ˆ
H

ψ(
t
)
S
=
(
ˆ
H
0
+
ˆ
H
I
)

ψ(
t
)
S
.
(12.31)
Both
ˆ
H
0
and
ˆ
H
I
can be expressed in terms of Hamiltoniandensity operators
ˆ
H
0
=
d
3
x
H
0
,
ˆ
H
I
=
d
3
x
H
I
,
(12.32)
H
0
and
H
I
being functions of freefield operators and their derivatives computed at
some fixed time
t
=
0, and thus not evolving
H
0
=
H
0
(
ˆ
φ
0
(
0
,
x
), ∂
μ
ˆ
φ
0
(
0
,
x
)),
H
I
=
H
I
(
ˆ
φ
0
(
0
,
x
), ∂
μ
ˆ
φ
0
(
0
,
x
)).
(12.33)
Clearly, the Schrödinger picture does not provide a relativisticallycovariant descrip
tion of the interaction since the freefield operators are all computed at
t
=
0
.
The time evolution of states was described, in
Sect.9.3.2
of
Chap.9
, in terms of
a timeevolution operator
U
(
t
,
t
0
)
, defined by property (
9.73
):

ψ
;
t
S
=
U
(
t
,
t
0
)

ψ
;
t
0
S
.
(12.34)
Let us recall the main properties of
U
(
t
,
t
0
)
discussed in
Sect.9.3.2
. The inverse of
U
(
t
,
t
0
)
is the operator which maps the state at
t
back to
t
0
:
U
(
t
,
t
0
)
−
1
=
U
(
t
0
,
t
).
Substituting (
12.34
) into (
12.31
) we find that
U
(
t
,
t
0
)
is solution to the following
equation:
i
d
dt
U
(
t
,
t
0
)
=
ˆ
HU
(
t
,
t
0
),
(12.35)
with the initial condition
U
(
t
0
,
t
0
)
=
1
.
From hermiticity of
ˆ
H
it follows that
U
(
t
,
t
0
)
is unitary. Indeed let us first show that
U
(
t
,
t
0
)
†
U
(
t
,
t
0
)
is constant:
d
dt
(
U
(
t
,
t
0
)
†
U
(
t
,
t
0
))
=
d
dt
U
(
t
,
t
0
)
†
U
(
t
,
t
0
)
+
U
(
t
,
t
0
)
†
d
dt
U
(
t
,
t
0
)
=
i
U
(
t
,
t
0
)
†
ˆ
H
†
U
(
t
,
t
0
)
+
U
(
t
,
t
0
)
†
−
i
ˆ
HU
(
t
,
t
0
)
=
iU
(
t
,
t
0
)
†
ˆ
HU
(
t
,
t
0
)
−
iU
(
t
,
t
0
)
†
ˆ
HU
(
t
,
t
0
)
=
0
.
(12.36)