Fluctuations of Quantum Fields and Thermodynamics of Spacetime

My research in the past 40 years can summarily be represented in a 2D phase space with QuantumClassical on the horizontal axis, and micro-Macro on the vertical. You can tell from the title of my

talk that the intention is to connect the micro features of quantum matter to the macrostructures of spacetime: matter via quantum field theory, spacetime via general relativity, and micro/macro interface via nonequilibrium statistical mechanics [1]. This stretch may sound wild

on surface, but not so surprising if you knew that in my view, general relativity is in the nature of a hydrodynamic theory [arXiv:gr-qc/9607070, arXiv:gr-qc/9511077], valid only in the long wavelength, low energy domain. GR is a beautiful theory, yet only an effective theory, emergent from quantum gravity -- theories describing the microscopic structures of spacetime at the Planck length (10^{-35} m), not unlike thermo- and hydro- to molecular-dynamics. To me, hydrodynamics and thermodynamics not only serve as a set of useful tools, but provide the correct

perspective to ask meaningful questions about the nature and behavior of spacetime as we know it. My recent work with H. T. Cho and J. T. Hsiang attempts to link up the stress energy fluctuations of quantum fields in spacetimes (with nontrivial topology or curvature) with their thermodynamic properties. The former is represented by the noise kernel, the stress energy tensor correlator of quantum matter fields, while the latter by the heat capacity and the (adiabatic and isothermal) compressibility. Noise kernel is the centerpiece of stochastic gravity [2], a theory for the dynamics of curved spacetimes based on the Einstein-Langevin equation [arXiv:0802.0658] which incorporates fully and self-consistently the backreaction effects from the mean values and the

fluctuations of quantum field stress tensors. Examining the noise kernel from a thermodynamic perspective can add a new dimension to our understanding of its physical properties. E.g., heat

capacity gives a measure of the fluctuations of the energy density to the mean, acting as a criterion for the validity of the canonical distribution. An intriguing fact coming from the past 3 decades of work [3] is that the fluctuations of energy density to the mean is close to unity for quantum

fields in many different spacetimes. From a thermodynamic perspective we conjecture that this feature, even for quantum fields in ordinary Minkowski spacetime, is an indication that the balance between spacetime and quantum matter fields has some built-in thermodynamic instability, that their co-existence meets with a saturation criterion in the “capacity of spacetime” to “hold" the quantum field, a theme we view as worthy of deeper thoughts. * Based on 1) H. T. Cho, J. T. Hsiang and B. L. Hu, Quantum Capacity and Vacuum Compressibility of

Spacetimes: Thermal Fields, Universe 8, 291 (2022). 2) Yucun Xie, J. T. Hsiang and B. L. Hu,

Dynamical Vacuum Compressibility of Space, Phys. Rev. D109, 065027 (2024)

[1] E. Calzetta and B. L. Hu, Nonequilibrium Quantum Field Theory (Cambridge University Press,

Cambridge, 2008).

[2] B. L. Hu and E. Verdaguer, Semiclassical and Stochastic Gravity-- Quantum Field Effects on Curved

Spacetimes (Cambridge University Press, Cambridge, 2020). Liv. Rev. Rel. 11 (2008) 3

[3] C. I. Kuo and L. H. Ford, Phys. Rev. D47, 4510 (1993). N. G. Phillips and B. L. Hu, Phys. Rev.

D55, 6123; D62 (2000) 084017. H. T. Cho and B. L. Hu, Phys. Rev. D84, 044032 (2011).