Basic definitions
We consider a many-body theory with field operator (annihilation operator written in the position basis)
.
The Heisenberg operators can be written in terms of Schrödinger operators as
and the creation operator is
, where
is the grand-canonical Hamiltonian.
Similarly, for the imaginary-time operators, 
[Note that the imaginary-time creation operator
is not the Hermitian conjugate of the annihilation operator
.]
In real time, the
-point Green function is defined by
where we have used a condensed notation in which
signifies
and
signifies
. The operator
denotes time ordering, and indicates that the field operators that follow it are to be ordered so that their time arguments increase from right to left.
In imaginary time, the corresponding definition is
where
signifies
. (The imaginary-time variables
are restricted to the range from
to the inverse temperature
.)
Note regarding signs and normalization used in these definitions: The signs of the Green functions have been chosen so that Fourier transform of the two-point (
) thermal Green function for a free particle is
and the retarded Green function is
where
is the Matsubara frequency.
Throughout,
is
for bosons and
for fermions and
denotes either a commutator or anticommutator as appropriate.
(See below for details.)
Two-point functions
The Green function with a single pair of arguments (
) is referred to as the two-point function, or propagator. In the presence of both spatial and temporal translational symmetry, it depends only on the difference of its arguments. Taking the Fourier transform with respect to both space and time gives
where the sum is over the appropriate Matsubara frequencies (and the integral involves an implicit factor of
, as usual).
In real time, we will explicitly indicate the time-ordered function with a superscript T: 
The real-time two-point Green function can be written in terms of 'retarded' and 'advanced' Green functions, which will turn out to have simpler analyticity properties. The retarded and advanced Green functions are defined by
and
respectively.
They are related to the time-ordered Green function by
where
is the Bose–Einstein or Fermi–Dirac distribution function.
Imaginary-time ordering and β-periodicity
The thermal Green functions are defined only when both imaginary-time arguments are within the range
to
. The two-point Green function has the following properties. (The position or momentum arguments are suppressed in this section.)
Firstly, it depends only on the difference of the imaginary times:
The argument
is allowed to run from
to
.
Secondly,
is (anti)periodic under shifts of
. Because of the small domain within which the function is defined, this means just
for
. Time ordering is crucial for this property, which can be proved straightforwardly, using the cyclicity of the trace operation.
These two properties allow for the Fourier transform representation and its inverse, 
Finally, note that
has a discontinuity at
; this is consistent with a long-distance behaviour of
.
Spectral representation
The propagators in real and imaginary time can both be related to the spectral density (or spectral weight), given by
where |α⟩ refers to a (many-body) eigenstate of the grand-canonical Hamiltonian H − μN, with eigenvalue Eα.
The imaginary-time propagator is then given by
and the retarded propagator by
where the limit as
is implied.
The advanced propagator is given by the same expression, but with
in the denominator.
The time-ordered function can be found in terms of
and
. As claimed above,
and
have simple analyticity properties: the former (latter) has all its poles and discontinuities in the lower (upper) half-plane.
The thermal propagator
has all its poles and discontinuities on the imaginary
axis.
The spectral density can be found very straightforwardly from
, using the Sokhatsky–Weierstrass theorem
where P denotes the Cauchy principal part. This gives 
This furthermore implies that
obeys the following relationship between its real and imaginary parts:
where
denotes the principal value of the integral.
The spectral density obeys a sum rule,
which gives
as
.
The similarity of the spectral representations of the imaginary- and real-time Green functions allows us to define the function
which is related to
and
by
and
A similar expression obviously holds for
.
The relation between
and
is referred to as a Hilbert transform.
Proof of spectral representation
We demonstrate the proof of the spectral representation of the propagator in the case of the thermal Green function, defined as 
Due to translational symmetry, it is only necessary to consider
for
, given by
Inserting a complete set of eigenstates gives 
Since
and
are eigenstates of
, the Heisenberg operators can be rewritten in terms of Schrödinger operators, giving
Performing the Fourier transform then gives 
Momentum conservation allows the final term to be written as (up to possible factors of the volume)
which confirms the expressions for the Green functions in the spectral representation.
The sum rule can be proved by considering the expectation value of the commutator,
and then inserting a complete set of eigenstates into both terms of the commutator: 
Swapping the labels in the first term then gives
which is exactly the result of the integration of ρ.
Non-interacting case
In the non-interacting case,
is an eigenstate with (grand-canonical) energy
, where
is the single-particle dispersion relation measured with respect to the chemical potential. The spectral density therefore becomes 
From the commutation relations,
with possible factors of the volume again. The sum, which involves the thermal average of the number operator, then gives simply
, leaving 
The imaginary-time propagator is thus
and the retarded propagator is 
Zero-temperature limit
As β → ∞, the spectral density becomes
where α = 0 corresponds to the ground state. Note that only the first (second) term contributes when ω is positive (negative).