The name comes from the Green's functions used to solve inhomogeneous differential equations, to which they are loosely related.
(Specifically, only two-point "Green's functions" in the case of a non-interacting system are Green's functions in the mathematical sense; the linear operator that they invert is the Hamiltonian operator, which in the non-interacting case is quadratic in the fields.)
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
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 thermal Green functions are defined only when both imaginary-time arguments are within the range
Firstly, it depends only on the difference of the imaginary times:
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,
where |α⟩ refers to a (many-body) eigenstate of the grand-canonical Hamiltonian H − μN, with eigenvalue Eα.
have simple analyticity properties: the former (latter) has all its poles and discontinuities in the lower (upper) half-plane.
obeys the following relationship between its real and imaginary parts:
We demonstrate the proof of the spectral representation of the propagator in the case of the thermal Green function, defined as
Inserting a complete set of eigenstates 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
is the single-particle dispersion relation measured with respect to the chemical potential.
The sum, which involves the thermal average of the number operator, then gives simply
Note that only the first (second) term contributes when ω is positive (negative).
The expressions for the Green functions are modified in the obvious ways:
The proof follows exactly the same steps, except that the two matrix elements are no longer complex conjugates.
If the particular single-particle states that are chosen are 'single-particle energy eigenstates', i.e.
and the fact that the thermal average of the number operator gives the Bose–Einstein or Fermi–Dirac distribution function.
Finally, the spectral density simplifies to give
Note that the noninteracting Green function is diagonal, but this will not be true in the interacting case.