In many-body quantum theory, the Lehmann representation
is an alternative way to write the Green’s functions,
obtained by expanding in the many-particle eigenstates
under the assumption of a time-independent Hamiltonian .
First, we write out the greater Green’s function ,
and then expand its expected value (at thermodynamic equilibrium)
into a sum of many-particle basis states :
Where , and is the grand partition function
(see grand canonical ensemble);
the operator gives the weight of each term at equilibrium.
Since is an eigenstate of with energy ,
this gives us a factor of .
Furthermore, we are in the Heisenberg picture,
so we write out the time-dependence of and :
Where we used that the trace
is invariant under cyclic permutations of .
The form a basis of eigenstates of ,
so we insert an identity operator :
Note that now only depends on the time difference ,
because is time-independent.
Next, we take the Fourier transform
Here, we recognize the integral
as a Dirac delta function ,
thereby introducing a factor of ,
and arriving at the Lehmann representation of :
We now go through the same process for the lesser Green’s function :
Where is for bosons, and for fermions.
Fourier transforming yields the following:
We swap and , leading to the following
Lehmann representation of :
Due to the delta function ,
each term is only nonzero for ,
so we write:
Therefore, we arrive at the following useful relation
between and :
Moving on, let us do the same for
the retarded Green’s function , given by:
We take the Fourier transform, but to ensure convergence,
we must introduce an infinitesimal positive to the exponent
(and eventually take the limit):
Leading us to the following Lehmann representation
of the retarded Green’s function :
Finally, we go through the same steps for the advanced Green’s function :
For the Fourier transform, we must again introduce
(although note the sign):
Therefore, the Lehmann representation of
the advanced Green’s function is as follows:
As a final note, let us take the complex conjugate of this expression:
Note the subscripts and .
Comparing this to gives us another useful relation:
- H. Bruus, K. Flensberg,
Many-body quantum theory in condensed matter physics,