Retarded, Advanced, and Spectral Correlators

Retarded and advanced correlators are the future- and past-supported parts of a commutator. Its Fourier transform is the spectral function, and causal support turns into half-plane analyticity and a dispersion relation. For the site’s source convention $S\mapsto S+\int J\mathcal O$, the retarded correlator is also the linear response of $\langle\mathcal O\rangle$ to $J$. This page develops those statements for a centered Hermitian scalar operator in a translation-invariant vacuum and then closes every sign and normalization check on the free massive scalar. Thermal weights, KMS relations, transport limits, and nonequilibrium response remain downstream.

Required background. Lorentzian Boundary Conditions and the iε Prescription supplies the boundary-value, pole, support, and raw-correlator normalization conventions used here. Spectral Decomposition of Two-Point Functions supplies the vacuum state sum, spectrum condition, and positive mass spectral measure.

Helpful background. Hyperbolic Equations and Causal Propagators supplies the support-selected inverse and finite-propagation language.

The vacuum commutator carries the causal data

Let $\mathcal O(x)=\mathcal O(x)^\dagger$ be a centered bosonic scalar operator in a selected Poincaré-invariant vacuum $|\Omega\rangle$. Translation invariance makes the two-point objects functions of $x-y$; set $y=0$ and define

$$C_{\mathcal O}(x) \equiv \langle\Omega|[\mathcal O(x),\mathcal O(0)]|\Omega\rangle.$$

The retarded and advanced correlators in the site convention are

$$\begin{aligned} G_{\mathrm R,\mathcal O}(x) &=i\,\theta(x^0)C_{\mathcal O}(x),\\ G_{\mathrm A,\mathcal O}(x) &=-i\,\theta(-x^0)C_{\mathcal O}(x). \end{aligned}$$

For singular or composite operators, these products with step functions are distributional causal splittings: they are understood after smearing and, at coincidence, after choosing compatible retarded and advanced extensions. The two extensions can share local contact-term ambiguity, while their difference below is fixed by the commutator.

Consequently,

$$G_{\mathrm R,\mathcal O}(x) -G_{\mathrm A,\mathcal O}(x) =iC_{\mathcal O}(x).$$

The step functions alone give future or past time support. If $\mathcal O$ is local and microcausality applies in the chosen physical sector, $C_{\mathcal O}(x)$ also vanishes at spacelike separation, sharpening this to

$$\operatorname{supp}G_{\mathrm R,\mathcal O} \subseteq\overline V_+, \qquad \operatorname{supp}G_{\mathrm A,\mathcal O} \subseteq\overline V_-.$$

For a general interacting or composite $\mathcal O$, these are response correlators, not automatically inverses of a fixed differential operator. The free elementary scalar is special: its normalization makes $G_{\mathrm R}$ and $G_{\mathrm A}$ the delta-normalized Klein–Gordon inverses.

The source sign fixes the response sign

Perturb the action by a real compactly supported source,

$$S[J]=S+\int\mathrm d^d y\,J(y)\mathcal O(y).$$

The Hamiltonian perturbation is $H_{\mathrm{int}}(t)=-\int\mathrm d^{d-1}\mathbf y\,J(t,\mathbf y)\mathcal O(t,\mathbf y)$. Expanding real-time unitary evolution to first order about the vacuum gives

$$\delta\langle\mathcal O(x)\rangle = \int\mathrm d^d y\, G_{\mathrm R,\mathcal O}(x-y)J(y) +O(J^2),$$

or equivalently

$$\left. \frac{\delta\langle\mathcal O(x)\rangle_J} {\delta J(y)} \right|_{J=0} = i\,\theta(x^0-y^0) \langle[\mathcal O(x),\mathcal O(y)]\rangle.$$

This formula assumes that the measured operator has no additional explicit $J$-dependence. Background variations of currents, stress tensors, or derivative couplings can add seagull or contact terms. The formula also concerns expectation-value evolution from the selected initial vacuum; an in–out vacuum amplitude does not generate it by ordinary source differentiation. The next pages make that real-time state distinction systematic.

Altland and Simons derive the retarded/advanced definitions using $H_{\mathrm{int}}=+F\mathcal O$ Altland and Simons 2023, § 7.3, pp. 394–395. Their response therefore carries $-i\theta$; identifying $F=-J$ translates it to the site’s displayed $+i\theta$.

The spectral function is the Fourier-space commutator

Use the inherited Fourier convention and define the commutator spectral function

$$\varrho_{\mathcal O}(p) \equiv \int\mathrm d^d x\, e^{ip\cdot x}C_{\mathcal O}(x).$$

For the diagonal correlator of a Hermitian operator, $\varrho_{\mathcal O}(p)$ is a real distribution and

$$\varrho_{\mathcal O}(-p) =-\varrho_{\mathcal O}(p).$$

The vacuum state sum makes the sign structure explicit: positive-energy states contribute nonnegative weights at $p^0>0$, while the reversed ordering supplies their negative-frequency partners. Thus $\varrho_{\mathcal O}$ is not nonnegative on the whole energy axis. Positivity means that its positive-frequency part is a positive measure under the physical-Hilbert-space, Hermiticity, and diagonal-correlator hypotheses.

This energy–momentum spectral function is related to, but is not the same object as, the nonnegative Källén–Lehmann mass measure $\rho_{\mathcal O}(\mathrm d\mu^2)$. Under the scalar vacuum hypotheses,

$$\varrho_{\mathcal O}(p) = 2\pi\,\operatorname{sgn}(p^0) \int_{[0,\infty)} \rho_{\mathcal O}(\mathrm d\mu^2) \delta(p^2-\mu^2).$$

The formula is distributional; massless zero modes and atoms at the spectral edge require the corresponding smearing and limiting qualifications. Gauge-variant fields in an indefinite auxiliary space, ghosts, non-Hermitian pairs, and off-diagonal correlators do not inherit this scalar positivity statement automatically. The positive scalar measure and its assumptions are derived in Schwartz 2014, § 24.2.1, pp. 467–469.

Causal support becomes half-plane analyticity

At fixed spatial momentum, the retarded time support gives the Fourier–Laplace transform

$$\widetilde G_{\mathrm R,\mathcal O}(z,\mathbf p) = \int_0^\infty\mathrm dt\, e^{izt}\,iC_{\mathcal O}(t,\mathbf p), \qquad \operatorname{Im}z>0,$$

while $\widetilde G_{\mathrm A,\mathcal O}$ is analytic for $\operatorname{Im}z<0$. These statements require the usual tempered or polynomial-growth control and are understood after smearing when the time correlator is only a distribution. For a Hermitian diagonal correlator, the real-axis boundary values obey

$$\widetilde G_{\mathrm A,\mathcal O}(\omega,\mathbf p) = \widetilde G_{\mathrm R,\mathcal O}(\omega,\mathbf p)^*.$$

When the large-$|z|$ behavior permits an unsubtracted representation,

$$\widetilde G_{\mathrm R,\mathcal O}(z,\mathbf p) = \int_{-\infty}^{\infty} \frac{\mathrm d\omega'}{2\pi} \frac{\varrho_{\mathcal O}(\omega',\mathbf p)} {\omega'-z}, \qquad \operatorname{Im}z>0.$$

Approaching the real axis from above and below gives

$$\begin{aligned} \widetilde G_{\mathrm R,\mathcal O}(\omega,\mathbf p) -\widetilde G_{\mathrm A,\mathcal O}(\omega,\mathbf p) &=i\varrho_{\mathcal O}(\omega,\mathbf p),\\ \varrho_{\mathcal O}(\omega,\mathbf p) &=2\,\operatorname{Im} \widetilde G_{\mathrm R,\mathcal O}(\omega,\mathbf p),\\ \operatorname{Re}\widetilde G_{\mathrm R,\mathcal O}(\omega,\mathbf p) &= \operatorname{PV}\!\int_{-\infty}^{\infty} \frac{\mathrm d\omega'}{2\pi} \frac{\varrho_{\mathcal O}(\omega',\mathbf p)} {\omega'-\omega}. \end{aligned}$$

The sign $\varrho=+2\operatorname{Im}G_{\mathrm R}$ follows from the site’s $+i\theta$ response convention. Sources that define $G_{\mathrm R}=-i\theta\langle[\mathcal O,\mathcal O]\rangle$ instead obtain the familiar opposite sign.

The half-plane analyticity is developed in Altland and Simons 2023, § 7.3.1, pp. 396–397, and their spectral reconstruction and Kramers–Kronig formulas appear in § 7.3.2, pp. 403–404. Their spectral sign is translated by the response-sign crosswalk above.

If the response grows too rapidly, the spectral function fixes only the discontinuity. With $N$ subtractions at a point $z_*$ in an analytic domain,

$$\begin{aligned} \widetilde G_{\mathrm R}(z) &= \sum_{k=0}^{N-1} \frac{(z-z_*)^k}{k!} \,\partial_z^k\widetilde G_{\mathrm R}(z_*)\\ &\quad+ (z-z_*)^N \int_{-\infty}^{\infty} \frac{\mathrm d\omega'}{2\pi} \frac{\varrho(\omega')} {(\omega'-z)(\omega'-z_*)^N}. \end{aligned}$$

Spatial momentum labels were suppressed in this display. The subtraction data are not determined by $\varrho$; local contact counterterms contribute polynomial pieces with no spectral discontinuity. Analyticity therefore yields a dispersion relation only together with the stated growth, subtraction, and boundary assumptions.

Weinberg gives the corresponding $N$-subtracted two-point dispersion relation and polynomial ambiguity in Weinberg 1995, § 10.7, p. 460, footnote **. The displayed fixed-spatial-momentum form uses the site’s Fourier and response conventions.

The free scalar closes the dispersion calculation

For a canonically normalized free scalar with $m>0$ and

$$E_{\mathbf p}=\sqrt{\mathbf p^2+m^2},$$

the momentum-mode commutator is

$$C_0(t,\mathbf p) = -\frac{i}{E_{\mathbf p}} \sin(E_{\mathbf p}t).$$

Its spectral function is

$$\varrho_0(\omega,\mathbf p) = \frac{\pi}{E_{\mathbf p}} \left[ \delta(\omega-E_{\mathbf p}) -\delta(\omega+E_{\mathbf p}) \right].$$

The two delta functions make the retarded dispersion integral elementary:

$$\begin{aligned} \int_{-\infty}^{\infty} \frac{\mathrm d\omega'}{2\pi} \frac{\varrho_0(\omega',\mathbf p)}{\omega'-z} &= \frac1{2E_{\mathbf p}} \left( \frac1{E_{\mathbf p}-z} -\frac1{-E_{\mathbf p}-z} \right)\\ &= -\frac1{z^2-E_{\mathbf p}^2}. \end{aligned}$$

Taking the upper and lower boundary values gives exactly

$$\begin{aligned} \widetilde G_{\mathrm R,0}(\omega,\mathbf p) &= -\frac1{(\omega+i0)^2-E_{\mathbf p}^2},\\ \widetilde G_{\mathrm A,0}(\omega,\mathbf p) &= -\frac1{(\omega-i0)^2-E_{\mathbf p}^2}. \end{aligned}$$

The inverse transforms are

$$G_{\mathrm R,0}(t,\mathbf p) = \theta(t)\frac{\sin(E_{\mathbf p}t)}{E_{\mathbf p}}, \qquad G_{\mathrm A,0}(t,\mathbf p) = -\theta(-t)\frac{\sin(E_{\mathbf p}t)}{E_{\mathbf p}}.$$

They obey the independent contact and support checks

$$(\partial_t^2+E_{\mathbf p}^2)G_{\mathrm R/A,0}(t,\mathbf p) =\delta(t),$$

with retarded or advanced support. The canonical equal-time commutator also yields the moment checks

$$\int\frac{\mathrm d\omega}{2\pi}\,\varrho_0(\omega,\mathbf p)=0, \qquad \int\frac{\mathrm d\omega}{2\pi}\, \omega\varrho_0(\omega,\mathbf p)=1.$$

The second sum rule depends on canonical normalization and is not a universal identity for a rescaled or composite operator.

Positive mass weight produces a retarded spectral representation

Whenever the unsubtracted Källén–Lehmann integral exists, the same calculation gives

$$\widetilde G_{\mathrm R,\mathcal O}(z,\mathbf p) = -\int_{[0,\infty)} \frac{\rho_{\mathcal O}(\mathrm d\mu^2)} {z^2-\mathbf p^2-\mu^2}, \qquad \operatorname{Im}z>0.$$

An isolated atom in $\rho_{\mathcal O}(\mathrm d\mu^2)$ contributes a stable real-frequency pole pair; continuous mass support contributes the corresponding real-frequency continuum and cut. Neither feature changes the retarded half-plane assignment. For composite operators the integral can require subtractions, and local contact terms must be supplied separately. A spectral function therefore determines the nonanalytic response data, not every convention-dependent local term.

In the selected vacuum, define

$$W^+_{\mathcal O}(x) =\langle\Omega|\mathcal O(x)\mathcal O(0)|\Omega\rangle, \qquad W^-_{\mathcal O}(x) =\langle\Omega|\mathcal O(0)\mathcal O(x)|\Omega\rangle.$$

The spectrum condition separates their Fourier transforms away from zero-energy subtleties:

$$\widetilde W^+_{\mathcal O}(p) =\theta(p^0)\varrho_{\mathcal O}(p), \qquad \widetilde W^-_{\mathcal O}(p) =-\theta(-p^0)\varrho_{\mathcal O}(p).$$

This is vacuum information, not a generic stationary-state identity. At finite temperature the state-dependent Wightman weights are related by KMS, and out of equilibrium they require additional statistical data.

What the spectral function does not establish

• It does not turn a retarded correlator into a Feynman correlator. The commutator discontinuity is shared information, but time ordering and Feynman boundary values are different selection data.
• It does not prove locality. Future time support follows from the step function; causal-cone support additionally uses microcausality or a hyperbolic propagation theorem.
• It does not remove subtraction data. Real polynomial/contact pieces have no discontinuity and are invisible to $\varrho$.
• It does not make Euclidean inversion stable. Exact analytic continuation under theorem hypotheses and numerical reconstruction from finite noisy Euclidean data are different problems.
• It does not supply thermal or transport physics. KMS weights, fluctuation–dissipation relations, zero-frequency and zero-momentum limit order, conserved overlaps, and transport peaks require a specified thermal or nonequilibrium state.

Common pitfalls

Calling $\varrho$ a positive function. It is odd in energy. Under the physical scalar hypotheses, only its positive-frequency measure is nonnegative.

Copying a spectral-sign formula across source conventions. With the site’s $+J\mathcal O$ action source, $G_{\mathrm R}=+i\theta C$ and $\varrho=+2\operatorname{Im}G_{\mathrm R}$. A source using $-i\theta C$ has the opposite spectral sign.

Treating every retarded correlator as a Green inverse. The free elementary scalar passes that check; a general operator response need not obey a local differential equation with a delta source.

Reading a bump as a stable particle. A spectral enhancement can come from a threshold, interference, finite resolution, or a continued-sheet resonance. Stable-particle language requires the isolated spectral atom developed in the spectra chapter.

Check your understanding

1. Recover the response sign

Starting from $H_{\mathrm{int}}=-\int J\mathcal O$, expand the expectation value to first order. Why does the kernel contain $+i\theta[\mathcal O,\mathcal O]$?

Answer

The source appears in the evolution operator with $+i\int J\mathcal O$. Combining the first-order terms from $U$ and $U^\dagger$ gives $i\langle[\mathcal O(x),\mathcal O(y)]\rangle$, and the integration range imposes $y^0<x^0$.

2. Read the free poles from the spectral function

Insert the two free delta functions into the dispersion integral and identify the retarded pole locations.

Answer

The integral gives $-1/(z^2-E_{\mathbf p}^2)$. The upper boundary $z=\omega+i0$ puts both poles below the real $\omega$ axis, yielding the future-supported retarded kernel.

3. Diagnose a missing term

Two retarded functions have the same spectral function but differ by a real polynomial in frequency and momentum. Is that a contradiction?

Answer

No. A polynomial has no cut discontinuity, so it is invisible to $\varrho$. Subtraction conditions, local counterterms, Ward identities, or normalization data are needed to decide that difference.

4. Separate support from locality

What can be concluded from $G_{\mathrm R}=i\theta(t)C$ before microcausality is assumed?

Answer

Only that $G_{\mathrm R}$ vanishes for negative time separation. Vanishing at spacelike separation, and hence support inside the full future light cone, requires locality of the relevant operators or a corresponding causal-propagation result.

Where to continue

• In–Out versus In–In Expectation Values explains why the expectation-value response used here requires initial-state rather than transition-amplitude semantics.
• Thermal Propagators and Spectral Representations adds KMS weights and reconstructs the thermal correlator dictionary from a state-dependent spectral density.
• Sources, Linear Response, and Kubo Formulae develops conserved-current response, contact terms, transport limit order, and Kubo relations.
• The Källén–Lehmann Representation supplies the full positive-mass-measure derivation used in the vacuum spectral representation.

References

• Altland, Alexander, and Ben Simons. Condensed Matter Field Theory. 3rd ed. Cambridge University Press, 2023. DOI.

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014. DOI.

• Weinberg, Steven. The Quantum Theory of Fields. Volume I: Foundations. Cambridge University Press, 1995. DOI.