Green Function Table

This table collects common Green-function conventions used in quantum dynamics. The formulas here concern inverse kernels with specified boundary conditions. In many-body and QFT language, Green functions can also be correlators; those do not all invert the same local operator. The same local operator can have retarded, advanced, outgoing, incoming, time-ordered, Euclidean, or finite-temperature Green functions.

The conventions below match the Dynamics pages:

$$R(z) = (z-H)^{-1},$$

and, for time-independent $H$,

$$G^R(t) = -\frac{i}{\hbar}\theta(t)e^{-iHt/\hbar}, \qquad G^A(t) = \frac{i}{\hbar}\theta(-t)e^{-iHt/\hbar}.$$

The energy transform convention is

$$G(E) = \int_{-\infty}^{\infty}dt\, e^{iEt/\hbar}G(t).$$

With different Fourier conventions, factors of $2\pi$, $\hbar$, and signs of $i0$ may move. Always translate conventions before comparing formulas from different sources.

Resolvents and Spectral Weight

Object	Formula	Prescription	Main use
Resolvent	$R(z)=(z-H)^{-1}$	$z\notin\sigma(H)$	spectral inverse
Retarded resolvent	$G^R(E)=(E-H+i0)^{-1}$	upper half-plane	causal response
Advanced resolvent	$G^A(E)=(E-H-i0)^{-1}$	lower half-plane	complementary boundary value
Spectral function	$A(E)=i(G^R-G^A)$	real-axis discontinuity	spectral weight
Density of states	$\rho(E)=-(1/\pi)\operatorname{Im}\operatorname{Tr}G^R(E)$	trace and retarded sign	state counting
Coordinate Green function	$G(x,x';z)=\langle x\rvert(z-H)^{-1}\lvert x'\rangle$	domain and boundary data	inverse kernel

Resolvent

The resolvent is

$$R(z) = (z-H)^{-1},$$

where $z$ lies away from the spectrum of $H$. It satisfies

$$(z-H)R(z)=I.$$

For a discrete nondegenerate spectrum,

$$R(z) = \sum_n \frac{\lvert n\rangle\langle n\rvert}{z-E_n}.$$

The poles locate eigenvalues, and the residues give projectors onto eigenspaces. For continuous spectra, the real-axis structure is usually read through boundary values.

Retarded and Advanced Boundary Values

The retarded and advanced energy-domain Green functions are

$$G^R(E) = \lim_{\eta\to0^+} \frac{1}{E-H+i\eta},$$

and

$$G^A(E) = \lim_{\eta\to0^+} \frac{1}{E-H-i\eta}.$$

The sign of $i0$ records the time-support condition:

Object	Time support	Energy boundary value
retarded	zero for $t\lt0$	$E+i0$
advanced	zero for $t\gt0$	$E-i0$

For self-adjoint $H$,

$$G^A(E) = \bigl(G^R(E)\bigr)^\dagger$$

as a boundary-value statement when the relevant limits exist.

Spectral Function and Delta Operator

The distribution identity

$$\frac{1}{x+i0} = \operatorname{PV}\frac{1}{x} - i\pi\delta(x)$$

implies

$$G^R(E)-G^A(E) = -2\pi i\,\delta(E-H).$$

Thus the spectral function convention

$$A(E) = i\bigl(G^R(E)-G^A(E)\bigr)$$

gives

$$A(E) = 2\pi\,\delta(E-H).$$

Some many-body texts define spectral functions with additional signs, factors, matrix elements, or $1/(2\pi)$ normalizations. The safest habit is to state both $G^R-G^A$ and the chosen definition of $A$.

Density of States

For a finite system or a specified regulated trace, the total density of states is

$$\rho(E) = \operatorname{Tr}\delta(E-H).$$

Using the retarded resolvent,

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

The local density of states in coordinate representation is

$$\rho(x,E) = -\frac{1}{\pi} \operatorname{Im} G^R(x,x;E),$$

with the trace or integral over $x$ recovering the total density when defined. Infinite-volume totals usually diverge; use density per volume or local spectral measures. Coincident Green kernels can require an ultraviolet regulator. For an operator-valued resolvent, $\operatorname{Im}G=(G-G^\dagger)/(2i)$, not entrywise imaginary parts in an arbitrary basis.

Coordinate-Space Inverse Kernel

The coordinate-space energy Green function, derived in Energy Green Function, is

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

It solves

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

with boundary conditions determined by $z$ and the Hamiltonian domain. For example, the same differential expression on the line, a half-line, a box, or a ring gives different Green functions.

This is the energy-domain inverse-kernel analogue of the propagator table’s warning: boundary conditions are part of the object.

Time-Domain Response

For a driven equation

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

a retarded solution is

$$\psi_R(t) = \int dt'\, G^R(t-t')\eta(t').$$

Since $G^R(t-t')=0$ for $t\lt t'$, the response at $t$ depends only on earlier source values. This is the causal-response meaning of the retarded prescription.

The advanced Green function solves the same local equation but obeys the opposite support condition. It is a useful analytic boundary value, not the usual initial-value response.

Free-Particle Energy Green Functions

For a one-dimensional free particle,

$$H = \frac{p^2}{2m}, \qquad E = \frac{\hbar^2k^2}{2m}, \qquad k\gt0,$$

the outgoing boundary value is

$$G^+(x,x';E) = -\frac{im}{\hbar^2k} e^{ik|x-x'|}.$$

For this real free Hamiltonian the incoming kernel is the complex conjugate of the outgoing kernel, changing both the phase and the imaginary prefactor:

$$G^-(x,x';E) = \frac{im}{\hbar^2k} e^{-ik|x-x'|}.$$

In three dimensions, with $R=|\mathbf r-\mathbf r'|$,

$$G^+(\mathbf r,\mathbf r';E) = -\frac{m}{2\pi\hbar^2} \frac{e^{ikR}}{R}.$$

These are Green functions of $E-H\pm i0$, not time-evolution kernels. They solve Helmholtz-type inverse problems with outgoing or incoming boundary conditions.

Relation to Propagator Kernels

The time-evolution kernel is

$$K(x_f,t_f;x_i,t_i) = \langle x_f|e^{-iH(t_f-t_i)/\hbar}|x_i\rangle.$$

The resolvent is related to time evolution by a one-sided transform. For $\operatorname{Im}z\gt0$,

$$R(z) = \frac{1}{i\hbar} \int_0^\infty dt\, e^{izt/\hbar} e^{-iHt/\hbar}.$$

Thus propagator kernels and Green functions are related, but they are not interchangeable. The propagator evolves initial data in time; the Green function inverts an operator with a boundary prescription.

Common Mistakes

• Saying “the Green function” without specifying the operator and boundary prescription.
• Confusing $K(t)$ with $(E-H+i0)^{-1}$.
• Dropping the $i0$ sign when taking imaginary parts.
• Mixing retarded, advanced, outgoing, incoming, and time-ordered prescriptions.
• Forgetting Fourier-transform conventions when comparing signs and factors of $i$.
• Treating local density of states as a total density without tracing or integrating.
• Using free-space Green functions in bounded domains without imposing boundary conditions.

References

• E. N. Economou, Green’s Functions in Quantum Physics, 3rd ed., Springer, 2006.
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover, 2003.
• G. D. Mahan, Many-Particle Physics, 3rd ed., Kluwer Academic/Plenum, 2000.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, revised and enlarged ed., Academic Press, 1980.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.

Exercises

1. Derive the spectral function identity from the retarded and advanced resolvents.

Solution

Using

$$\frac{1}{x+i0} - \frac{1}{x-i0} = -2\pi i\,\delta(x),$$

and applying the identity through the spectral theorem with $x=E-H$, one gets

$$G^R(E)-G^A(E) = -2\pi i\,\delta(E-H).$$

Therefore, with $A(E)=i(G^R-G^A)$,

$$A(E) = 2\pi\,\delta(E-H).$$

2. Show that the density-of-states formula follows from the spectral function.

Solution

The retarded imaginary-part identity is

$$-\frac{1}{\pi}\operatorname{Im}G^R(E) = \delta(E-H).$$

Taking the trace gives

$$-\frac{1}{\pi} \operatorname{Im} \operatorname{Tr}G^R(E) = \operatorname{Tr}\delta(E-H) = \rho(E).$$

3. Verify the one-dimensional outgoing free Green function away from $x=x'$.

Solution

For $x\ne x'$,

$$G^+(x,x';E) = -\frac{im}{\hbar^2k} e^{ik|x-x'|}$$

is proportional to a solution of

$$\left( \frac{d^2}{dx^2}+k^2 \right)G=0.$$

Since

$$E-H = \frac{\hbar^2}{2m} \left( \frac{d^2}{dx^2}+k^2 \right),$$

the equation $(E-H)G=0$ holds away from the source. The derivative jump is $\partial_xG|_{x'+}-\partial_xG|_{x'-}=2m/\hbar^2$, so multiplying it by $\hbar^2/(2m)$ gives the unit delta source, and the phase $e^{ik|x-x'|}$ gives outgoing waves on both sides.

4. Why can two Green functions solve the same differential equation but represent different physics?

Solution

The local differential equation fixes only the inverse relation away from boundary data. The Green function also includes a boundary or support prescription. Retarded and advanced Green functions solve the same local equation but have opposite time support. Outgoing and incoming Green functions solve the same Helmholtz-type equation but impose different behavior at infinity. The prescription states the physical problem.