The Källén–Lehmann spectral representation gives a general expression for the two-point function of an interacting quantum field theory as a sum of free propagators. It was discovered by Gunnar Källén and Harry Lehmann independently. This can be written as
Δ
(
p
)
=
∫
0
∞
d
μ
2
ρ
(
μ
2
)
1
p
2
−
μ
2
+
i
ϵ
,
where
ρ
(
μ
2
)
is the spectral density function that should be positive definite. In a gauge theory, this latter condition cannot be granted but nevertheless a spectral representation can be provided. This belongs to non-perturbative techniques of quantum field theory.
In order to derive a spectral representation for the propagator of a field
Φ
(
x
)
, one consider a complete set of states
{
|
n
⟩
}
so that, for the two-point function one can write
⟨
0
|
Φ
(
x
)
Φ
†
(
y
)
|
0
⟩
=
∑
n
⟨
0
|
Φ
(
x
)
|
n
⟩
⟨
n
|
Φ
†
(
y
)
|
0
⟩
.
We can now use Poincaré invariance of the vacuum to write down
⟨
0
|
Φ
(
x
)
Φ
†
(
y
)
|
0
⟩
=
∑
n
e
−
i
p
n
⋅
(
x
−
y
)
|
⟨
0
|
Φ
(
0
)
|
n
⟩
|
2
.
Let us introduce the spectral density function
ρ
(
p
2
)
θ
(
p
0
)
(
2
π
)
−
3
=
∑
n
δ
4
(
p
−
p
n
)
|
⟨
0
|
Φ
(
0
)
|
n
⟩
|
2
.
We have used the fact that our two-point function, being a function of
p
μ
, can only depend on
p
2
. Besides, all the intermediate states have
p
2
≥
0
and
p
0
>
0
. It is immediate to realize that the spectral density function is real and positive. So, one can write
⟨
0
|
Φ
(
x
)
Φ
†
(
y
)
|
0
⟩
=
∫
d
4
p
(
2
π
)
3
∫
0
∞
d
μ
2
e
−
i
p
⋅
(
x
−
y
)
ρ
(
μ
2
)
θ
(
p
0
)
δ
(
p
2
−
μ
2
)
and we freely interchange the integration, this should be done carefully from a mathematical standpoint but here we ignore this, and write this expression as
⟨
0
|
Φ
(
x
)
Φ
†
(
y
)
|
0
⟩
=
∫
0
∞
d
μ
2
ρ
(
μ
2
)
Δ
′
(
x
−
y
;
μ
2
)
being
Δ
′
(
x
−
y
;
μ
2
)
=
∫
d
4
p
(
2
π
)
3
e
−
i
p
⋅
(
x
−
y
)
θ
(
p
0
)
δ
(
p
2
−
μ
2
)
.
From CPT theorem we also know that holds an identical expression for
⟨
0
|
Φ
†
(
x
)
Φ
(
y
)
|
0
⟩
and so we arrive at the expression for the chronologically ordered product of fields
⟨
0
|
T
Φ
(
x
)
Φ
†
(
y
)
|
0
⟩
=
∫
0
∞
d
μ
2
ρ
(
μ
2
)
Δ
(
x
−
y
;
μ
2
)
being now
Δ
(
p
;
μ
2
)
=
1
p
2
−
μ
2
+
i
ϵ
a free particle propagator. Now, as we have the exact propagator given by the chronologically ordered two-point function, we have obtained the spectral decomposition.
