Holographic Influence Functionals and Open-Sector Dynamics

Integrating out a holographic bath produces an influence functional, not merely a friction coefficient. Its retarded kernel controls causal dissipation; its symmetrized kernel controls fluctuations; local counterterms remove ultraviolet contact divergences; and a Markovian Langevin equation follows only when the bath memory time is short compared with the probe evolution time.

Required background. Schwinger–Keldysh Contours and Real-Time Bulk Geometries supplies the doubled bulk solution, and System–Environment Splits and Influence Functionals supplies the open-QFT construction.

Helpful background. Noise, Dissipation, and Fluctuation Relations fixes the kernel relations. The gravitational extensions are Validity, Decoherence, and What Stochastic Gravity Does Not Capture, In-In Effective Actions and Causal Mean Backreaction, The Stress-Tensor Noise Kernel, Influence Functionals, Dissipation, and Noise, Fluctuation–Dissipation Relations in Stationary States, and Einstein–Langevin Dynamics.

Integrating out a thermal operator

Couple a probe coordinate $q$ to a Hermitian bath operator,

$$S_{\mathrm{int}}=g\int dt\,q(t)\mathcal O(t).$$

To quadratic order in $g$, integrating out the bath on the closed-time path gives

$$S_{\mathrm{IF}}^{(2)} =-g^2\int dt\,dt'\, q_a(t)G_R(t-t')q_r(t') +\frac{i g^2}{2}\int dt\,dt'\, q_a(t)G_{\mathrm{sym}}(t-t')q_a(t').$$

Here $q_r=(q_1+q_2)/2$, $q_a=q_1-q_2$,

$$G_R(t)=-i\theta(t)\langle[\mathcal O(t),\mathcal O(0)]\rangle, \qquad G_{\mathrm{sym}}(t)=\frac12\langle\{\mathcal O(t),\mathcal O(0)\}\rangle .$$

The first kernel is causal; positivity of the reduced density matrix requires the noise quadratic form to be nonnegative. In a thermal state,

$$G_{\mathrm{sym}}(\omega) =-\coth\!\left(\frac{\beta\omega}{2}\right) \operatorname{Im}G_R(\omega).$$

The holographic calculation obtains both kernels from one Schwinger–Keldysh bulk saddle. A trailing string endpoint provides a canonical example: horizon absorption supplies the retarded force kernel and horizon fluctuations supply the noise, as shown by Son and Teaney 2009.

From memory to a Langevin equation

The probe equation is

$$M_{\mathrm{ren}}\ddot q(t)+V'_{\mathrm{ren}}(q) +g^2\int_{-\infty}^{t}dt'\,G_R(t-t')q(t') =\xi(t),$$

with $\langle\xi(t)\xi(t')\rangle=g^2G_{\mathrm{sym}}(t-t')$. If the small-frequency expansion is analytic,

$$g^2G_R(\omega)=\delta K-i\gamma\omega+O(\omega^2\tau_{\mathrm{mem}}^2),$$

the local equation contains a renormalized spring constant and friction $\gamma$. For $\omega\ll T$, KMS gives white noise,

$$\langle\xi(t)\xi(t')\rangle\simeq2T\gamma\,\delta(t-t').$$

This is the first application: compute the infalling bulk response, extract $\operatorname{Im}G_R/\omega$ at small $\omega$, compute the symmetrized solution on the same contour, and verify the fluctuation–dissipation relation before taking the local limit. The Brownian-motion realization and its time scales were analyzed by de Boer et al. 2009.

Adversarial memory and renormalization checks

Suppose $G_R(t)$ has a power-law tail. Replacing it by $\gamma\delta'(t)$ discards contributions from times comparable to the probe evolution and can move poles into the wrong half-plane. The Markovian claim fails even though the exact nonlocal influence functional remains causal.

At coincident times, $G_{\mathrm{sym}}$ generally contains local ultraviolet divergences. Using it before holographic renormalization can make the noise variance regulator-dependent or apparently negative. Counterterms may change local real pieces; they may not change the absorptive spectral density.

Evidence ceiling and handoff

The kernels describe the declared probe, bath state, coupling order, and time window. They do not prove autonomous unitary evolution of the open sector, and a horizon-derived Markovian model does not determine exact finite-$N$ bath recurrences. Thermal and Nonequilibrium QFT owns open-system theory, curved-spacetime QFT owns stochastic backreaction, and Chapter 11 applies these kernels to transport models.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• de Boer, Jan; Hubeny, Veronika E.; Rangamani, Mukund; and Shigemori, Masaki. “Brownian Motion in AdS/CFT.” Journal of High Energy Physics 2009, 094 (2009). doi:10.1088/1126-6708/2009/07/094.
• Son, Dam T., and Derek Teaney. “Thermal Noise and Stochastic Strings in AdS/CFT.” Journal of High Energy Physics 2009, 021 (2009). doi:10.1088/1126-6708/2009/07/021.