Influence Functionals, Dissipation, and Noise

The influence functional explains why stochastic gravity contains both a noise kernel and a causal matter-response kernel. They are different components of one closed-time-path calculation: the imaginary quadratic form admits a stochastic representation when it is positive, while the real branch-mixing term changes the deterministic response. Neither component may be dropped merely because the other is known.

Required background. The Stress-Tensor Noise Kernel fixes the covariance normalization; In-In Effective Actions and Causal Mean Backreaction fixes causal variation; and System–Environment Splits and Influence Functionals supplies the general reduced-density-matrix construction.

Helpful background. Noise, Dissipation, and Fluctuation Relations separates symmetric and retarded kernels, while Schwinger–Keldysh Actions for Open QFT develops average/difference variables.

Two histories and two kernels

Start with a matter operator $\hat q_A$ coupled as $S_{\mathrm{int}}=\int J^Aq_A$; $A$ abbreviates indices and spacetime position. The matter state is specified on the initial Cauchy surface. After tracing out matter, the influence functional is

$$\mathcal F[J^+,J^-] =\exp\!\left(iS_{\mathrm{IF}}[J^+,J^-]\right).$$

Define

$$J_\Delta=J^+-J^-, \qquad J_c=\frac12(J^++J^-).$$

Unitarity gives $S_{\mathrm{IF}}[J,J]=0$ and $S_{\mathrm{IF}}[J^+,J^-]=-S_{\mathrm{IF}}[J^-,J^+]^*$. To quadratic order about $J=0$, and suppressing local contact terms, these identities organize the action as

$$\begin{aligned} S_{\mathrm{IF}}^{(2)} ={}&\int J_\Delta^A\langle \hat q_A\rangle +\int\!\int J_\Delta^A D^{\mathrm{ret}}_{AB}J_c^B\\ &+\frac{i}{2}\int\!\int J_\Delta^A N_{AB}J_\Delta^B . \end{aligned}$$

With the displayed plus source coupling,

$$D^{\mathrm{ret}}_{AB}(x,y) =i\theta(x\succ y)\langle[\hat t_A(x),\hat t_B(y)]\rangle, \qquad N_{AB}(x,y)=\frac12\langle\{\hat t_A(x),\hat t_B(y)\}\rangle.$$

Changing the sign of the source coupling changes the response convention, but not the symmetric covariance. For a covariant metric perturbation, $g_{\mu\nu}\mapsto g_{\mu\nu}+h_{\mu\nu}$, the site stress convention gives $\delta S_m=-\frac12\int\mathrm dV,h_{\mu\nu}T^{\mu\nu}$; hence set $J_{\mu\nu}=-h_{\mu\nu}/2$ before varying. Keeping this factor is necessary to recover the Einstein equation with source $8\pi G\,T_{\mu\nu}$.

The structural map emphasizes that $D^{\mathrm{ret}}$ and $N$ rejoin only in the metric equation. Inspect the two branches independently: causality constrains one and positive type constrains the other.

The same matter theory supplies dissipation and noise, but through different ordered correlators; their reunion in Einstein–Langevin dynamics preserves both causal response and covariance. The map is schematic and not to scale.

Hubbard–Stratonovich representation

If $N$ is positive semidefinite on the chosen real test-function space, introduce a real Gaussian distribution $\xi_A$ with

$$\mathbb E[\xi_A]=0, \qquad \mathbb E[\xi_A\xi_B]=N_{AB}.$$

Then

$$\exp\!\left[-\frac12 J_\Delta N J_\Delta\right] =\mathbb E_\xi \exp\!\left[iJ_\Delta\xi\right].$$

This identity represents the imaginary influence term. It does not assert that $\hat q_A$ has become a classical variable, and it reproduces only the retained cumulants. Zero modes of $N$ are harmless: the Gaussian measure is supported on the quotient by its null space. Negative directions are fatal to a real probabilistic representation.

Initial matter–metric correlations require extra boundary terms in the influence action. A factorized initial density matrix is a simplifying assumption, not a consequence of the formalism. Likewise, a quadratic influence action may be exact for a linear environment variable in a Gaussian state, but stress energy is quadratic in a free scalar; truncating its metric influence action at $h^2$ is still a second-cumulant approximation for the stress source.

First application: Gaussian matter to quadratic metric order

Let $g$ be a solution of the mean semiclassical equation and take two nearby histories $g+h^\pm$. For a free scalar in a Gaussian Hadamard state, expand the matter closed-time-path effective action to second order in $h^\pm$. After the local gravitational counterterms have been fixed, its nonlocal part has the form above with $q_A=T^{\mu\nu}$ and $J_A=-h_{\mu\nu}/2$.

Varying the real part with respect to $h_\Delta$ and setting $h_\Delta=0$ produces

$$\delta\langle T_{\mu\nu}(x)\rangle =\int_M\mathrm dV_y\, \Pi^{\mathrm{ret}}_{\mu\nu}{}^{\rho\sigma}(x,y) h_{\rho\sigma}(y),$$

where $\Pi^{\mathrm{ret}}$ includes the retarded commutator and required local variations. Its support is in $J^-(x)$, so the resulting equation is causal. The imaginary part yields the stress covariance $N$. This closed-time-path derivation, including the real/imaginary split and stochastic representation, is developed in Martín and Verdaguer 1999, §§II–III, eqs. (2.7)–(3.13) and reviewed in Hu and Verdaguer 2008, §4.1, eqs. (20)–(27).

There are four checks.

1. $S_{\mathrm{IF}}[h,h]=0$.
2. The response kernel has retarded support.
3. $N(f,f)\ge0$ for every admissible real $f$.
4. Local ultraviolet terms match the same counterterms used in the mean equation.

For the adversarial test, alter a proposed imaginary kernel so that $N(f,f)=-\lambda<0$ for some $f$. Along $J_\Delta=sf$, the supposed characteristic functional becomes $\exp(+s^2\lambda/2)$, whose modulus exceeds one for $s\ne0$. No normalized real probability distribution has such a characteristic function. The proposed stochastic source must be rejected even if the deterministic response is causal.

Domain and failure conditions

The chapter comparison table licenses this derivation for a specified initial state, a controlled expansion in metric perturbations, renormalized local terms, and a positive smeared noise kernel. Nonstationarity does not invalidate the influence functional; it only prevents the equilibrium spectral simplification treated next. Strong perturbations and important higher stress cumulants require a nonquadratic influence action.

The failure map shows why missing dissipation is separate from an invalid covariance. Either defect breaks the claimed reduced dynamics for a different reason.

Noise positivity and causal response are independent acceptance tests for a quadratic influence action; satisfying one does not repair failure of the other. The map is schematic and not to scale.

References

• Hu, B. L., and E. Verdaguer. “Stochastic Gravity: Theory and Applications.” Living Reviews in Relativity 11, 3 (2008). doi:10.12942/lrr-2008-3. Open PDF
• Martín, R., and E. Verdaguer. “Stochastic Semiclassical Gravity.” Physical Review D 60, 084008 (1999). doi:10.1103/PhysRevD.60.084008. Open PDF