In manybody theory, the term Green's function (or Green function) is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators.
The name comes from the Green's functions used to solve inhomogeneous differential equations, to which they are loosely related. (Specifically, only twopoint 'Green's functions' are Green's functions in the mathematical sense; the linear operator that they invert is the part of the Hamiltonian operator that is quadratic in the fields.)
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We consider a manybody theory with field operator (annihilation operator written in the position basis) .
The Heisenberg operators can be written in terms of Schrödinger operators as
and , where K = H − μN is the grandcanonical Hamiltonian.
Similarly, for the imaginarytime operators,
(Note that the imaginarytime creation operator is not the Hermitian conjugate of the annihilation operator .)
In real time, the 2npoint Green function is defined by
where we have used a condensed notation in which j signifies and j' signifies . The operator T denotes time ordering, and indicates that the field operators that follow it are to be ordered so that the their time arguments increase from right to left.
In imaginary time, the corresponding definition is
where j signifies . (The imaginarytime variables τ_{j} are restricted to the range 0 to β.)
Note regarding signs and normalization used in these definitions: The signs of the Green functions have been chosen so that Fourier transform of the twopoint (n = 1) thermal Green function for a free particle is
and the retarded Green function is
(See below for details.)
Throughout, ζ is + 1 for bosons and − 1 for fermions and denotes either a commutator or anticommutator as appropriate.
The Green function with a single pair of arguments (n = 1) is referred to as the twopoint 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 (2π) ^{− d}, as usual).
In real time, we will explicitly indicate the timeordered function with a superscript T:
The realtime twopoint 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 timeordered Green function by
where
is the BoseEinstein or FermiDirac distribution function.
The thermal Green functions are defined only when both imaginarytime arguments are within the range 0 to β. The twopoint 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 τ − τ' is allowed to run from − β to β.
Secondly, is periodic under shifts of β. Because of the small domain within which the function is defined, this means just
for 0 < τ < β. (Note that the function is antiperiodic for fermions.) 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 τ = 0; this is consistent with a longdistance behaviour of .
The propagators in real and imaginary time can both be related to the spectral density (or spectral weight), given by
where refers to a (manybody) eigenstate of the grandcanonical Hamiltonian H − μN, with eigenvalue E_{α}.
The imaginarytime 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 − iη in the denominator. The timeordered function can be found in terms of G^{R} and G^{A}. As claimed above, G^{R}(ω) and G^{A}(ω) have simple analyticity properties: the former (latter) has all its poles and discontinuities in the lower (upper) halfplane. The thermal propagator has all its poles and discontinuities on the imaginary ω_{n} axis.
The spectral density can be found very straightforwardly from G^{R}, 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 P 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 realtime Green functions allows us to define the function
which is related to and G^{R} by
and
A similar expression obviously holds for G^{A}.
The relation between and is referred to as a Hilbert transform.
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 τ > 0, given by
Inserting a complete set of eigenstates gives
Since and are eigenstates of H − μN, 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.
In the noninteracting case, is an eigenstate with (grandcanonical) energy , where is the singleparticle 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 imaginarytime propagator is thus
and the retarded propagator is
As , 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).
We can use `field operators' as above, or creation and annihilation operators associated with other singleparticle states, perhaps eigenstates of the (noninteracting) kinetic energy. We then use
where ψ_{α} is the annihilation operator for the singleparticle state α and is that state's wavefunction in the position basis. This gives
with a similar expression for G^{(n)}.
These depend only on the difference of their time arguments, so that
and
We can again define retarded and advanced functions in the obvious way; these are related to the timeordered function in the same way as above.
The same periodicity properties as described in above apply to . Specifically,
and
for τ < 0.
In this case,
where m and n are manybody states.
The expressions for the Green functions are modified in the obvious ways:
and
Their analyticity properties are identical. The proof follows exactly the same steps, except that the two matrix elements are no longer complex conjugates.
If the particular singleparticle states that are chosen are `singleparticle energy eigenstates', ie,
then for an eigenstate:
so is :
and so is :
We therefore have
where
We then rewrite
and use the fact that the thermal average of the number operator gives the BoseEinstein or FermiDirac distribution function.
Finally, the spectral density simplifies to give
so that the thermal Green function is
and the retarded Green function is
Note that the noninteracting Green function is diagonal, but this will not be true in the interacting case.
