Causal Response in Semiclassical Equations

A nonlocal quantum effective action need not produce acausal metric response. The causal equation comes from an in-in, or closed-time-path, expectation value with an initial density matrix and retarded boundary prescription. An in-out action instead computes transition amplitudes; varying it generally produces a Feynman kernel that can be complex and is not the equation for a real expectation-value metric.

Required background. Quantum Information supplies density matrices, reduced states, and connected correlators.

Helpful background. Quantum Fields in Curved Spacetime supplies state-dependent stress tensors; Einstein Equations supplies gauge constraints; Black-Hole Geometry and Mechanics supplies causal boundaries; Semiclassical Gravity supplies mean-field response; and Causality and Global Structure supplies $J^\pm$ and domains of dependence.

The closed-time-path equation

Choose an initial Cauchy surface $\Sigma_0$ and density operator $\rho_0$. Duplicate the metric, $g^+_{ab}$ on a forward time branch and $g^-_{ab}$ on a backward branch, and define

$$e^{i\Gamma_{\rm CTP}[g^+,g^-]} =\operatorname{Tr}\!\left( U_{g^+}\rho_0 U_{g^-}^{\dagger} \right)e^{iS_{\rm grav}[g^+]-iS_{\rm grav}[g^-]}.$$

The two histories agree at the return surface of the contour. Vary first and then take the physical limit $g^+=g^-=g$:

$$\left. \frac{2}{\sqrt{-g^+(x)}} \frac{\delta\Gamma_{\rm CTP}}{\delta g^{+ab}(x)} \right|_{g^+=g^-}=0.$$

This yields a real semiclassical equation with $\langle T_{ab}\rangle_{\rho_0}$ rather than an in-out matrix element. Hu and Verdaguer develop this influence-action construction and its relation to stochastic gravity Hu and Verdaguer 2008, §§ 4.1–4.3, PDF.

The contour boundary conditions are part of the physics. Both histories begin at the same $\Sigma_0$ with matrix elements weighted by $\rho_0$, their field configurations are sewn together at the return surface, and variations vanish there before the physical limit is taken. Moving the return surface later than every observation cannot change an earlier expectation value. This composition property is the contour-level reason future sources cancel between the two branches; merely doubling symbols without the initial density matrix and sewing condition would not establish causality.

Retarded polarization and support

Perturb a self-consistent solution by $g_{ab}\mapsto g_{ab}+h_{ab}$. Following the volume-wide convention, define the complete retarded functional kernel by

$$\delta\langle T_{ab}(x)\rangle =\int_{\Sigma_0}^{x}d^4x'\sqrt{-g(x')}\, \Pi^{\rm ret}_{ab}{}^{cd}(x,x')h_{cd}(x') +\delta T^{\rm init}_{ab}(x).$$

For the covariant perturbation $h_{cd}=\delta g_{cd}$, the matter-action variation is $\delta S_{\rm m}=\tfrac12\int dV\,\hat T^{cd}h_{cd}$. Holding the initial state fixed and using $\hbar=1$, the nonlocal part of this kernel is

$$\Pi^{\rm ret,nonlocal}_{ab}{}^{cd}(x,x') =\frac{i}{2}\,\theta_\Sigma(x,x') \left\langle \left[\hat T_{ab}(x),\hat T^{cd}(x')\right] \right\rangle_{\rho_0}$$

where $\theta_\Sigma(x,x')$ vanishes unless $x'$ lies to the past of $x$ relative to the initial-value foliation. The sign and factor follow from the covariant metric coupling; changing to an inverse-metric perturbation changes that convention. The complete kernel also contains contact terms from the explicit metric dependence of the renormalized stress operator. Hu and Verdaguer’s response definitions give this normalization after combining their local and nonlocal terms Hu and Verdaguer 2008, Eqs. (4.6)–(4.9) and (4.17), PDF.

Microcausality makes the commutator vanish at spacelike separation, so the integral has support only in $J^-(x)\cap J^+(\Sigma_0)$. The initial-state term records an independent perturbation of $\rho_0$. It vanishes when the initial density matrix is held fixed in the chosen identification of the matter algebras; if the preparation changes, its contribution must be retained.

Substitution gives the linear-response equation

$$\delta G_{ab}+\Lambda\,h_{ab} +\delta H^{\rm local}_{ab} =8\pi G\,\delta\langle T_{ab}\rangle.$$

Here the finite curvature counterterms assigned to $H^{\rm local}_{ab}$ remain on the gravitational side; they must not also be counted in $\Pi^{\rm ret}$. Their variations may contain derivatives at $x$, but do not introduce dependence on future sources. A symbol such as $\ln(-\Box/\mu^2)$ becomes causal only after its retarded spectral or contour prescription is supplied.

Ward identities, gauge, and renormalization

Diffeomorphism invariance implies a response Ward identity. On a background satisfying the mean equation,

$$\nabla^a\delta\langle T_{ab}\rangle +(\delta\nabla^a)\langle T_{ab}\rangle=0.$$

The gravitational side obeys the matching linearized Bianchi identity. These relations are an independent check on contact terms and signs. A calculation that retains only a nonlocal two-point function while omitting its local variation can violate the constraint and falsely appear acausal.

The same cosmological, Newton, and curvature-squared counterterms that renormalize the mean equation renormalize linear response; no state-dependent counterterm may be invented for a particular perturbation. Anderson, Molina-París, and Mottola show this closure explicitly and formulate stability in terms of bounded gauge-invariant perturbations Anderson, Molina-París, and Mottola 2003, §§ II–IV, PDF.

Approximation regime and checks

The equation is controlled at the declared loop or large-$N$ order, for Hadamard initial data, frequencies and curvatures below the EFT cutoff, and perturbations small enough for linearization. Order reduction or a specified prescription must remove cutoff-scale runaway roots without deleting physical low-frequency response. A deterministic mean geometry additionally requires small fluctuations of the relevant smeared metric observables, including the noise kernel propagated through the retarded response. A stress-variance-to-mean ratio alone does not test that condition.

In a stationary flat-space vacuum, Fourier transformation gives a polarization analytic in the upper half frequency plane; the retarded inverse transform vanishes for negative time separation. Taking $G\to0$ removes metric backreaction while retaining ordinary causal QFT response. By contrast, replacing $\Pi^{\rm ret}$ with the time-ordered correlator produces nonzero support on both time orderings and may acquire an imaginary part: that is a transition-amplitude object, not the desired mean-field evolution.

Exercise

Let a scalar response be

$$\delta O(t)=\int_{t_0}^{\infty}dt'\, \theta(t-t')e^{-\gamma(t-t')}J(t'), \qquad \gamma\gt 0.$$

A source pulse is supported only at $t'=t_1$, with $t_1\gt t_0$. Evaluate the response for $t\lt t_1$ and $t\gt t_1$, and identify the feature that enforces causality. With Fourier convention $\widetilde K(\omega)=\int dt\,e^{i\omega t}K(t)$, locate the pole of the response kernel.

Solution

For $J(t')=J_0\delta(t'-t_1)$,

$$\delta O(t)=J_0\theta(t-t_1)e^{-\gamma(t-t_1)}.$$

It vanishes for $t\lt t_1$ and decays after the pulse for $t\gt t_1$. The step function, which is the time-domain form of the retarded boundary prescription, excludes future support. Its transform is $\widetilde K(\omega)=(\gamma-i\omega)^{-1}$, with its pole at $\omega=-i\gamma$ in the lower half-plane, consistent with a stable retarded response. Replacing the step function with a symmetric kernel would spoil this initial-value interpretation even if the same differential operator appeared.

References

• Anderson, P. R., Molina-París, C., and Mottola, E. (2003). “Linear Response, Validity of Semiclassical Gravity, and the Stability of Flat Space.” Physical Review D 67, 024026. DOI. Open PDF.
• Hu, B. L., and Verdaguer, E. (2008). “Stochastic Gravity: Theory and Applications.” Living Reviews in Relativity 11, 3. DOI. Open PDF.