Retarded and Advanced Green Functions

Ask a condensed-matter experimentalist what they measure: the imaginary part of a retarded Green function. ARPES maps $-\tfrac{1}{\pi}\mathrm{Im}\,G^{+}$ — the spectral function — band by band; scanning tunneling spectroscopy reads the same object at a surface atom. Quantum transport codes (the non-equilibrium Green function method behind nanodevice simulation) propagate $G^{+}$ and $G^{-}$ as their two working variables, their difference supplying the spectral weight exactly as the discontinuity formula dictates. And the causality-analyticity doctrine runs quality control across all of it: a response function violating Kramers–Kronig is unphysical by contour geometry, before any model is consulted.

Strike a bell at $t=0$ and listen. The physical answer — silence before, ringing after — is so obvious that it hides a mathematical scandal: the bell's wave equation is perfectly happy with the reversed movie, ringing that converges from the past and dies exactly at the strike. Both are point-source solutions. The equation transmits the kick; it does not decide the direction. You decide, by declaring which side of $t=0$ may respond, and each declaration names a Green function.

Quantum mechanics inherits the ambiguity. The operator $i\hbar\partial_t - \hat H$ is the "bell", the delta source the kick, and the two honest answers are the retarded Green function, which propagates the disturbance into the future, and the advanced one, which propagates it into the past. The retarded choice is the one experiments live in — prepare, then measure. The advanced one is the retarded function of the time-reversed world, and formal scattering theory needs both: states departing a collision and states arriving into one. The second picture is the one this article installs. In the energy domain the story compresses into geography on the complex $E$-plane. The eigenvalues of $\hat H$ are landmines on the real axis — $(E-\hat H)^{-1}$ blows up on each. The $\pm i\epsilon$ moves the mines infinitesimally off the road: down for retarded, up for advanced. The Fourier integral back to time is a contour whose closing direction is dictated by $t$ — and whether it encircles the mines or misses them is the statement that response follows kick.

The raw material is owned upstream: Green Functions in Quantum Mechanics built $\hat G^{+}(t) = -\tfrac{i}{\hbar}\theta(t)\hat U(t)$, proved via $\theta' = \delta$ that it is a Green function of the Schrödinger operator, and introduced the resolvent $\hat G(E) = (E-\hat H)^{-1}$ with its spectral representation. What it deliberately did not do is name the other member. Define both on equal footing:

where: $\hat U(t) = e^{-i\hat H t/\hbar}$ is the evolution operator and $\theta$ the unit step. $\hat G^{+}$ lives only at $t>0$ — it carries the kicked state forward. $\hat G^{-}$ lives only at $t<0$: it is the solution that arrives at the kick, the bookkeeping of a final condition rather than an initial one. Note both contain the same $\hat U(t)$ — dynamics is shared; only the window differs.

Fourier transforming the pair was half-done upstream: Green Functions in Quantum Mechanics showed that the transform of $\hat G^{+}(t)$ needs a convergence factor $e^{-\epsilon t/\hbar}$ to tame the step function's infinite tail, and that the factor lands as a shift of the energy:

where: $\epsilon \to 0^{+}$ at the end of every calculation; the $\pm$ records which time tail needed taming — future tail forces $+i\epsilon$, past forces $-i\epsilon$. Equation (2) is "retarded versus advanced" compressed into one sign. What remains — this article's direction — is the return trip: showing the inverse transform of $(E-\hat H+i\epsilon)^{-1}$ really vanishes for $t<0$, by an argument that makes the mechanism visible.

A doctrine card before moving on, because this is the article's exportable lesson. Causality is analyticity. $\hat G^{+}(E)$, continued into the complex plane, has all its singularities in the lower half-plane — so it is analytic everywhere in the upper half-plane, precisely because $\theta(t)$ killed the past. Any response function respecting cause-before-effect enjoys the same analyticity, and half-plane analyticity forces integral relations between real and imaginary parts on the axis — the Kramers–Kronig dispersion relations, stated here in one line and owned by the electromagnetism track.

On the real axis the two functions differ — and the difference is the entire spectral content of $\hat H$. The pointwise tool is the Sokhotski–Plemelj identity, splitting an on-shell denominator into principal value plus delta:

where: $\mathcal{P}$ denotes the principal value — the symmetric excision of the singular point — and the identity means both sides agree when integrated against any smooth test function. The real part is common to both prescriptions; only the delta flips sign. That asymmetry is about to become a theorem.

Two consequences organize the singularity landscape. Discrete spectrum: at an isolated eigenvalue the jump is a single delta — a simple pole at $E = E_n$ with residue $|n\rangle\langle n|$. Continuous spectrum: the poles merge into a branch cut along the allowed energies, across which $\hat G$ jumps by the continuous spectral density. The resolvent is a holographic record of the spectrum, readable from where it fails to be analytic — the geography the previous article surveyed; the discontinuity formula is its legend.

The two functions are each other's time-reverse. Dagger $\hat G^{+}(-t) = -\tfrac{i}{\hbar}\theta(-t)e^{+i\hat Ht/\hbar}$ (hermitian $\hat H$ flips the exponent, $-i$ conjugates to $+i$):

where: the energy-domain version follows because conjugating $(E-\hat H+i\epsilon)^{-1}$ flips only the $i\epsilon$. Advanced physics is retarded physics run backwards — anticipating the antiunitary machinery of Time Reversal in Quantum Dynamics, where this pairing becomes a symmetry statement about the dynamics itself. Averaging instead of subtracting gives the third member: $\hat G^{P} = \tfrac12(\hat G^{+}+\hat G^{-})$ — the pure principal value of (3), no delta. It carries standing waves: equal parts outgoing and incoming, no net flux.

The free particle makes the table concrete. In one dimension (the derivation route is owned by Green Functions in Quantum Mechanics, which built the outgoing member; the incoming twin follows by the conjugation (4) and is checked directly in this article's verify script):

where: the retarded member radiates away from the source — outgoing waves — while the advanced member converges onto it. In 3D the same structure reads $G_0^{\pm} = -\tfrac{m}{2\pi\hbar^2}\,e^{\pm ikR}/R$, $R = |\mathbf r - \mathbf r'|$ (stated; the scattering chapter evaluates the contour). Outgoing versus incoming is retarded versus advanced, in space rather than time.

Example 1 — resolvent spectroscopy of a two-level system (confidence builder). Take $\hat H = \begin{pmatrix} 2.0 & 0.5 \\ 0.5 & 3.0 \end{pmatrix}$ eV. Locate the poles of $\hat G^{+}(E)$, give their residues, and confirm the density of states integrates to the number of levels.

Step 1. Poles sit at eigenvalues. Diagonalizing: $E_{\pm} = 2.5 \mp \sqrt{0.25 + 0.25}$, i.e. $E_1 = \mathbf{1.7929}$ eV and $E_2 = \mathbf{3.2071}$ eV, split by exactly $\sqrt{2} = 1.4142$ eV. $\hat G^{+}(E)$ is a $2\times2$ matrix function with simple poles at both.

Step 2. Residues are projectors. Evaluating $\hat G^{+}$ at $E_1 + 10^{-7}$ eV and multiplying by $(E-E_1)$ reproduces the outer product $|1\rangle\langle 1|$ of the ground eigenvector — the residue is the state.

Step 3. Count states by the jump. With $\epsilon = 0.01$ eV, the density $-\tfrac{1}{\pi}\mathrm{Im}\,\mathrm{Tr}\,\hat G^{+}(E)$ is two width-$\epsilon$ Lorentzians on the eigenvalues; integrating over energy gives $\mathbf{2.00}$ — the level count. The discontinuity formula, audited on the smallest nontrivial example.

Example 2 — evaluating the free pair for a real electron (exam level). An electron has energy such that $k = 1.0\times10^{10}\,\mathrm{m}^{-1}$. Evaluate both $G_0^{\pm}$ at separation $|x-x'| = 1$ nm and interpret the difference.

Step 1. Fix the energy scale. $E = \hbar^2k^2/2m = \mathbf{3.81}$ eV — a typical conduction-band electron.

Step 2. Magnitude, common to both. $|G_0^{\pm}| = m/(\hbar^2 k) = \mathbf{1.31\ (eV\cdot nm)^{-1}}$, independent of separation — in one dimension a point source's influence does not dilute geometrically (contrast the 3D $1/R$).

Step 3. Phase, where they part ways. The accumulated phase is $k|x-x'| = 10^{10} \times 10^{-9} = \mathbf{10}$ rad. The retarded function carries $e^{+10i}$ — a wave that has traveled outward one nanometer from the source; the advanced carries $e^{-10i}$, a wave converging inward that will arrive at the source. Same magnitude, opposite senses of propagation: the $\pm i\epsilon$, read off an interferometer.

Step 4. Sanity-check the units. In 1D, $(E-\hat H)\hat G = \delta(x-x')$ gives $[\hat G] = 1/(\text{energy}\times\text{length})$ — matching $(eV\cdot nm)^{-1}$, and flagging any formula transplanted from 3D, whose kernel dilutes as $1/R$.

References. Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., §5.7–5.8 (Green functions and scattering) · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. II, complements on the resolvent · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 19 (scattering) · Economou, Green's Functions in Quantum Physics, 3rd ed., Chs. 1–2 · Griffiths, Introduction to Quantum Mechanics, 3rd ed., Ch. 11 (scattering Green function).

Provenance. A verifier should: (1) confirm the time-domain solutions satisfy $(i\hbar\partial_t-H)G^\pm=\delta(t-t')$ using $\partial_t\theta=\delta$ (Sakurai §5.7–5.8; Cohen-Tannoudji resolvent complements); (2) reproduce $G^\pm(E)=(E-H\pm i\epsilon)^{-1}$ by the convergence-factor Fourier transform; (3) re-derive the free $G_0^+(\mathbf r,\mathbf r';E)=-\tfrac{m}{2\pi\hbar^2}e^{ikR}/R$ by contour integration (Griffiths Ch.11, Sakurai); (4) verify Sokhotski–Plemelj and the discontinuity formula $G^+-G^-=-2\pi i\,\delta(E-H)$ (Economou Ch.1). All constants are standard textbook values; no nonstandard numbers introduced. For this edition, verification was machine-executed (sympy/numpy/scipy, 29 checks green): the finite-$\epsilon$ contour integrals reproducing $\mp\tfrac{i}{\hbar}\theta(\pm t)e^{-iE_0t}e^{\mp\epsilon t}$ by direct quadrature; the $\theta' = \delta$ bookkeeping for both members symbolically; $\hat G^{-}(t) = \hat G^{+}(-t)^{\dagger}$ on random hermitian matrices to $10^{-13}$; the exact finite-$\epsilon$ discontinuity identity and the Sokhotski–Plemelj split against a Gaussian test function; the 1D free pair's ODE, jump condition $2m/\hbar^2$, and mutual conjugacy; the worked numbers 1.7929/3.2071 eV (gap $\sqrt2$), DOS integrals 2.00 and 39.98/40, and the free-electron anchors 3.81 eV, 1.31 (eV·nm)⁻¹, 10 rad. Ownership: the resolvent, its spectral representation, the time→energy Fourier bridge, the DOS card, the 1D outgoing kernel's derivation, and the Dyson identity belong to Green Functions in Quantum Mechanics; the energy→time contour anatomy, the advanced twin, the discontinuity theorem, and the causal taxonomy are owned here; Lippmann–Schwinger and the Born series belong to the scattering chapter; Kramers–Kronig to the electromagnetism track; the Feynman prescription to the relativistic chapter. Stated as bridges only.

Builds on: The Time-Evolution Operator · Time-Dependent Schrodinger Equation · Spectral Decomposition of Time Evolution · Stationary States and Energy Eigenstates

Leads to: Time-Ordered Exponential and the Dyson Series · Lippmann–Schwinger Equation · Linear Response and Kubo Formula