What Is a Green Function?

A Green function is an inverse kernel with a boundary prescription. In quantum mechanics, the phrase appears in several related roles:

as the kernel of an inverse operator such as $(z-H)^{-1}$ ;
as the response to a localized source;
as a spectral tool whose poles and branch cuts encode energies;
as a time-domain object selected by retarded, advanced, or time-ordered boundary conditions.

There is no single object called “the Green function” until the operator, variables, boundary conditions, and convention are specified.

The general differential-equation idea is developed in Green Functions. This page explains why the same idea becomes central in quantum dynamics.

Green Functions as Inverse Operators

In finite dimensions, if a matrix $A$ is invertible, the equation

A u=f

has solution

u=A^{-1}f.

The matrix entries of $A^{-1}$ tell how a unit source in component $j$ contributes to the response in component $i$ :

u_i=\sum_j (A^{-1})_{ij}f_j.

In coordinate space, a Green function plays the same role. If

L\psi=\eta,

then a Green function $G(x,x')$ satisfies

L_xG(x,x')=\delta(x-x'),

and the solution is written schematically as

\psi(x)=\int dx'\,G(x,x')\eta(x').

The boundary conditions are part of the inverse. Changing from outgoing to incoming scattering conditions, from Dirichlet to periodic boundaries, or from retarded to advanced time behavior changes the Green function even when the local differential expression looks the same.

Quantum Resolvents

For a time-independent Hamiltonian, the central energy-domain inverse is the resolvent

G(z)=(z-H)^{-1},

defined for complex $z$ away from the spectrum of $H$ . It obeys

(z-H)G(z)=I.

In an energy eigenbasis with discrete eigenstates,

G(z) = \sum_n \frac{|n\rangle\langle n|}{z-E_n}.

This formula shows why Green functions are spectral tools. Singularities occur where the inverse fails: at eigenvalues for bound states, and along branch cuts or boundary values for continuous spectra.

The coordinate-space energy Green function is the kernel

G(x,x';z) = \langle x|(z-H)^{-1}|x'\rangle.

It solves

(z-H_x)G(x,x';z)=\delta(x-x'),

with a boundary condition fixed by how $z$ approaches the spectrum.

Response to Sources

Green functions also answer source-response questions. Consider a driven Schrödinger-type equation

\left( i\hbar\frac{\partial}{\partial t} -H \right)\psi(t) = \eta(t).

A time-domain Green function satisfies

\left( i\hbar\frac{\partial}{\partial t} -H \right)G(t,t') = \delta(t-t')I,

with a specified time boundary condition. A retarded Green function vanishes for $t\lt t'$ , so the response occurs only after the source acts. An advanced Green function vanishes for $t\gt t'$ . A time-ordered Green function follows a different ordering convention, especially important in perturbation theory and field theory.

The boundary condition is not a technical afterthought; it states the physical question.

Time-Domain and Energy-Domain Versions

The time-evolution kernel

K(x_f,t_f;x_i,t_i) = \langle x_f|U(t_f,t_i)|x_i\rangle

evolves an initial wavefunction forward in time. It is a propagator kernel in the sense developed in Propagator Kernel.

An energy-domain Green function is instead an inverse of $z-H$ or $E-H\pm i0$ . Its coordinate kernel, source equation, and free examples are developed in Energy Green Function. The two are related by Fourier or Laplace transforms, but they are not interchangeable. Schematically,

(z-H)^{-1} \quad\text{is an energy-domain inverse,}

while

e^{-iH(t-t')/\hbar} \quad\text{is a time-evolution operator.}

Different texts may call both objects “propagators.” In a careful calculation, state which object is meant and which convention is being used.

Boundary Prescriptions

On the spectrum of $H$ , the inverse $(E-H)^{-1}$ is singular. One often defines boundary values

G^\pm(E) = \lim_{\epsilon\to0^+} \frac{1}{E-H\pm i\epsilon}.

The sign of $i\epsilon$ selects a boundary condition. In scattering, it distinguishes outgoing and incoming wave behavior. In time-domain language, related prescriptions distinguish retarded and advanced response. In many-body and QFT contexts, other prescriptions define time-ordered, Euclidean, or thermal Green functions.

The notation $i0$ is therefore not decoration. It carries physical and analytic information.

Why Green Functions Reveal Spectra

For a discrete spectrum,

G(z) = \sum_n \frac{|n\rangle\langle n|}{z-E_n}.

Near $z=E_k$ ,

G(z) \sim \frac{|k\rangle\langle k|}{z-E_k} +\text{regular terms}.

The pole location gives the energy, and the residue gives the projector onto the corresponding state. With degeneracies, the residue projects onto the degenerate eigenspace. With continuous spectra, the analogous information is spread over branch cuts and boundary values.

This is why Green functions appear in bound-state problems, scattering theory, density of states, response functions, and perturbation theory.

For two spectral parameters away from the spectrum, the resolvent identity is

G(z)-G(w)=(w-z)G(z)G(w).

It follows directly from the inverse-operator identity and is a useful sign check in perturbation theory.

For the boundary value $G^+(E)=(E-H+i0)^{-1}$ , the density of states is formally

\rho(E)=-\frac{1}{\pi}\operatorname{Im} \operatorname{Tr}G^+(E).

The trace may require a finite-volume, per-volume, or subtracted interpretation. The sign changes if the Fourier or resolvent convention is changed.

As a concrete outgoing-wave check, for a free particle with $E=\hbar^2k^2/(2m)$ and $k\gt0$ ,

G^+_{1\mathrm D}(x,x';E) =-\frac{im}{\hbar^2k}e^{ik\lvert x-x'\rvert},

whereas in three dimensions

G^+_{3\mathrm D}(\mathbf r,\mathbf r';E) =-\frac{m}{2\pi\hbar^2} \frac{e^{ik\lvert\mathbf r-\mathbf r'\rvert}} {\lvert\mathbf r-\mathbf r'\rvert}.

The different dimensions and prefactors reflect different delta-function normalizations. In numerical work, replacing $i0$ by $i\eta$ broadens sharp spectral features; convergence must be checked as both discretization and $\eta$ are reduced.

Difference from the Propagator Kernel

The time-domain propagator kernel answers:

Given an initial wavefunction, what is the amplitude to arrive at a later coordinate?

The Green-function question is more often:

Given a source, boundary condition, or spectral parameter, what inverse kernel solves the linear problem?

The two questions are related but not identical. A propagator kernel is tied to time evolution. A Green function is tied to an inverse problem plus a prescription. In nonrelativistic quantum mechanics, keeping that distinction prevents confusion. In QFT, the word “propagator” often refers to a Green function or correlation function, so conventions become even more important.

Why Green Functions Matter for QFT

Green functions are one of the main bridges from single-particle quantum mechanics to field theory. The bridge works because several ideas survive the transition:

poles and cuts encode spectral information;
boundary prescriptions define physical response or ordering;
source-response language becomes functional differentiation with sources;
perturbation theory organizes corrections to Green functions;
correlation functions replace simple wavefunction amplitudes as primary observables.

The bridge page Green Functions compares the quantum-mechanics and QFT meanings. The important warning is that QFT Green functions are not merely one-particle wavefunction propagators with different notation.

Green Functions in Many-Body QM develops the intermediate many-body meaning: creation and annihilation operators connect $N$ to exact $N\pm1$ sectors, so poles and continua carry addition and removal overlaps of the full system.

Common Mistakes

Saying “the Green function” without specifying the operator and boundary prescription.
Confusing the time-evolution kernel $K$ with the resolvent $(z-H)^{-1}$ .
Dropping the $i0$ prescription near the continuous spectrum.
Treating poles, branch cuts, and residues as optional complex-analysis details.
Comparing Green functions from different books without checking Fourier conventions and factors of $i$ .
Assuming QFT propagators are always ordinary transition amplitudes for a single particle.

Cross-Links

References

G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2013.
R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover, 2003.
A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press, 2010.

Exercises

Let $H$ have orthonormal eigenvectors $|n\rangle$ with eigenvalues $E_n$ . Verify that $G(z)=\sum_n |n\rangle\langle n|/(z-E_n)$ satisfies $(z-H)G(z)=I$ when $z$ is not an eigenvalue.

Solution

Apply $z-H$ :

(z-H)G(z) = \sum_n \frac{(z-H)|n\rangle\langle n|}{z-E_n}.

Since $H|n\rangle=E_n|n\rangle$ ,

(z-H)|n\rangle=(z-E_n)|n\rangle.

Therefore

(z-H)G(z) = \sum_n |n\rangle\langle n| = I.

In one sentence, explain why $G^+(E)$ and $G^-(E)$ are different even though both contain $E-H$ .

Solution

They approach the singular real energy axis from opposite sides of the complex plane, which selects different boundary conditions such as outgoing versus incoming scattering behavior.

A time-evolution kernel evolves an initial wavefunction by integration over $x_i$ . What extra information is needed before a Green function is defined?

Solution

One must specify the operator being inverted, the variables and measure, the domain or boundary conditions, and any prescription such as retarded, advanced, outgoing, incoming, or time ordered.

For a two-level Hamiltonian $H=E_1|1\rangle\langle1|+E_2|2\rangle\langle2|$ , locate the poles of $G(z)$ .

Solution

The resolvent is

G(z) = \frac{|1\rangle\langle1|}{z-E_1} + \frac{|2\rangle\langle2|}{z-E_2}.

It has simple poles at $z=E_1$ and $z=E_2$ , with residues $|1\rangle\langle1|$ and $|2\rangle\langle2|$ .

Prove the resolvent identity $G(z)-G(w)=(w-z)G(z)G(w)$ .

Solution

Use $A^{-1}-B^{-1}=A^{-1}(B-A)B^{-1}$ with $A=z-H$ and $B=w-H$ :

G(z)-G(w) =G(z)[(w-H)-(z-H)]G(w) =(w-z)G(z)G(w).