Retarded and Advanced Green Functions

A retarded Green function gives the response after a source acts. An advanced Green function gives the complementary boundary prescription in which the response is supported before the source. The local differential equation may be the same; the boundary condition changes the Green function. In nonrelativistic dynamics, temporal retardation does not by itself imply finite spatial propagation; see Causality, Support, and Interpretation in Nonrelativistic QM.

For a Schrödinger-type operator

$$L = i\hbar\frac{\partial}{\partial t} - H,$$

a time-domain Green function satisfies

$$L_tG(t,t') = \delta(t-t')I.$$

The retarded and advanced choices differ by support:

$$G^R(t,t')=0 \quad \text{for }t\lt t',$$

while

$$G^A(t,t')=0 \quad \text{for }t\gt t'.$$

The support condition is not a minor add-on. It is the part of the definition that states the physical question being answered.

Retarded Response

For a time-independent Hamiltonian,

$$U(t,t') = e^{-iH(t-t')/\hbar}$$

is the unitary evolution operator from $t'$ to $t$. With a common normalization, the retarded Green operator is

$$G^R(t,t') = -\frac{i}{\hbar} \theta(t-t')U(t,t').$$

It is zero before the source time $t'$. To verify the Green-function equation, use

$$i\hbar\frac{\partial}{\partial t}U(t,t') = HU(t,t')$$

and

$$\frac{d}{dt}\theta(t-t') = \delta(t-t').$$

The derivative of the step function supplies the delta function, while the derivative of $U$ cancels the explicit Hamiltonian term. Thus

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

If a driven equation is written as

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

then the retarded solution is schematically

$$\psi_R(t) = \int_{-\infty}^{\infty}dt'\, G^R(t,t')\eta(t').$$

Because $G^R(t,t')$ vanishes for $t\lt t'$, only earlier source values contribute to the response at time $t$.

Advanced Response

The advanced Green operator uses the opposite support condition:

$$G^A(t,t') = \frac{i}{\hbar} \theta(t'-t)U(t,t').$$

It also satisfies

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

The sign differs from the retarded expression because

$$\frac{d}{dt}\theta(t'-t) = -\delta(t-t').$$

Advanced Green functions are not the usual causal response functions for initial-value problems. They are nevertheless important as complementary boundary values, as ingredients in spectral identities, and as part of the analytic structure connecting retarded, advanced, time-ordered, and Euclidean objects.

Boundary Conditions, Not New Equations

Retarded and advanced Green functions solve the same local equation. The difference is global:

Object	Support condition	Typical use
$G^R(t,t')$	zero for $t\lt t'$	causal response to a source
$G^A(t,t')$	zero for $t\gt t'$	complementary boundary value, formal identities
$G^F(t,t')$	time-ordered prescription	vacuum perturbation theory preview

This is why the phrase “the Green function of $L$” is incomplete. One must also specify the boundary prescription. The same differential expression can have retarded, advanced, outgoing, incoming, Feynman, Euclidean, or finite-temperature Green functions.

Fourier Transform and the i0 Prescription

Assume time-translation invariance, so $G(t,t')=G(t-t')$. Use the energy transform convention

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

For the retarded Green function, the step function restricts the integral to $t\gt0$. The convergence prescription is

$$e^{i(E-H)t/\hbar} \quad\longrightarrow\quad e^{i(E-H)t/\hbar}e^{-\epsilon t/\hbar}, \qquad \epsilon\gt0.$$

This gives the boundary value

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

Similarly,

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

Thus the retarded prescription approaches the real energy axis from the upper half-plane, while the advanced prescription approaches from the lower half-plane:

$$G^R(E)=G(E+i0), \qquad G^A(E)=G(E-i0).$$

The sign of $i0$ is not decoration. It is the Fourier-domain record of the time-support condition. For coordinate-space source equations and outgoing free kernels, see Energy Green Function.

Spectral Form

If the Hamiltonian has discrete eigenstates,

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

then

$$G^R(E) = \sum_n \frac{|n\rangle\langle n|} {E-E_n+i0},$$

and

$$G^A(E) = \sum_n \frac{|n\rangle\langle n|} {E-E_n-i0}.$$

The distribution identity

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

implies

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

Equivalently,

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

as an operator-valued distribution, interpreted through matrix elements or traces. This is the seed of the density-of-states relation developed in Spectral Representation of Green Functions.

Worked Example: One Energy Eigenstate

For a one-dimensional Hilbert space with Hamiltonian $H=E_0$, the retarded Green function is

$$G^R(t) = -\frac{i}{\hbar} \theta(t) e^{-iE_0t/\hbar}.$$

It satisfies

$$\left( i\hbar\frac{d}{dt} - E_0 \right)G^R(t) = \delta(t).$$

In energy space,

$$G^R(E) = \frac{1}{E-E_0+i0}.$$

The pole prescription determines which inverse Fourier transform is allowed. With the retarded convention, the result vanishes for negative time. The advanced object has the opposite boundary value:

$$G^A(E) = \frac{1}{E-E_0-i0}.$$

This example contains the whole idea: the local inverse is fixed by the pole, and the time boundary condition is fixed by how the pole is bypassed.

Comparison with Feynman Propagators

Retarded and Feynman Green functions are often confused because both are called propagators in some contexts. They answer different questions.

A retarded correlator in many-body or field-theory notation commonly has the form

$$G^R_{AB}(t) = -\frac{i}{\hbar} \theta(t) \langle[A(t),B(0)]\rangle,$$

for bosonic operators. It measures causal linear response: a disturbance coupled to $B$ affects measurements of $A$ only at later times.

Green Functions in Many-Body QM owns the distinct normal fermionic propagator, whose retarded bracket is an anticommutator and whose intermediate states lie in $N\pm1$ sectors. That object is not an ordinary Kubo susceptibility.

Retarded and Advanced Response owns the many-body observable commutator pair, its adjoint identities, response spectral density, dispersion relations, and stability diagnostics. The present page retains the inverse-kernel boundary prescription.

A Feynman or time-ordered correlator has a different ordering prescription, schematically

$$G^F_{AB}(t) = -\frac{i}{\hbar} \langle\mathcal T A(t)B(0)\rangle.$$

It is the natural object in vacuum perturbation theory. It is not the same as the retarded response function, and its analytic prescription is not obtained by replacing every denominator with the same $+i0$ rule.

The QFT bridge page From Propagators in QM to Propagators in QFT explains this distinction from the field-theory side.

Common Mistakes

• Dropping the $i0$ prescription after writing $E-H\pm i0$.
• Calling $G^R$, $G^A$, and $G^F$ interchangeable because they solve related equations.
• Treating the advanced Green function as a causal response function for ordinary initial-value dynamics.
• Forgetting that the sign convention depends on the Fourier transform convention and on the normalization of $G$.
• Inferring that a nonzero Feynman propagator at spacelike separation would by itself imply acausal signaling.

Cross-Links

• What Is a Green Function?
• Resolvent Operator
• Energy Green Function
• Green Functions and Response Preview
• Spectral Representation of Green Functions
• Green Functions and Density of States
• Propagator Kernel
• Causality, Support, and Interpretation in Nonrelativistic QM
• Formula Sheet
• Green Function Table
• From Propagators in QM to Propagators in QFT
• Green Functions
• Correlation Functions
• Green Functions in Many-Body QM
• Retarded and Advanced Response

References

• G. D. Mahan, Many-Particle Physics, 3rd ed., Kluwer Academic/Plenum, 2000.
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover, 2003.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.
• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014.

Exercises

1. Verify that $G^R(t,t')=-(i/\hbar)\theta(t-t')U(t,t')$ satisfies the Green-function equation for a time-independent Hamiltonian.

Solution

Differentiate the product of the step function and the evolution operator:

$$\frac{\partial G^R}{\partial t} = -\frac{i}{\hbar} \delta(t-t')U(t,t') - \frac{i}{\hbar} \theta(t-t') \frac{\partial U(t,t')}{\partial t}.$$

The first term gives $\delta(t-t')I$ after multiplication by $i\hbar$, because $U(t',t')=I$. The second term cancels $HG^R$ because $i\hbar\,\partial_tU=HU$. Therefore

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

2. Show why the retarded Fourier transform gives $G^R(E)=(E-H+i0)^{-1}$.

Solution

For $t\gt0$,

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

Insert a convergence factor $e^{-\epsilon t/\hbar}$:

$$G^R(E) = -\frac{i}{\hbar} \int_0^\infty dt\, e^{i(E-H)t/\hbar} e^{-\epsilon t/\hbar}.$$

The integral gives

$$G^R(E) = \frac{1}{E-H+i\epsilon}.$$

Taking $\epsilon\to0^+$ yields the retarded boundary value.

3. Use the distribution identity for $1/(x\pm i0)$ to derive $G^R(E)-G^A(E)=-2\pi i\,\delta(E-H)$.

Solution

The scalar identities are

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

and

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

Subtracting gives

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

Applying this through the spectral theorem with $x=E-H$ gives

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

4. Explain why a time-ordered Green function is not the same object as a retarded Green function.

Solution

A retarded Green function is supported only after the source time and is tied to causal linear response. A time-ordered Green function orders operators according to time and is tied to vacuum or perturbative boundary prescriptions. In operator language, retarded functions involve commutators or anticommutators with a step function, while Feynman functions involve time-ordered products. They may be related by analytic continuation or spectral formulas in special settings, but they are not interchangeable definitions.