Session X15: Quantum Gravity: Mathematical Formalism

Can Tabletop Experiments Discover the Graviton?

We argue that if the Newtonian gravitational field of a body can mediate entanglement with another body, then it should also be possible for the body producing the Newtonian field to entangle directly with on-shell gravitons. Our arguments are made by revisiting a gedankenexperiment previously analyzed by Belenchia et al., which showed that a quantum superposition of a massive body requires both quantized gravitational radiation and local vacuum fluctuations of the spacetime metric in order to avoid contradictions with complementarity and causality. We provide a precise and rigorous description of the entanglement and decoherence effects occurring in this gedankenexperiment, thereby significantly improving upon the back-of-the-envelope estimates given in the analysis by Belenchia et al., and also showing that their conclusions are valid in much more general circumstances. As a by-product of our analysis, we show that under the protocols of the gedankenexperiment, there is no clear distinction between entanglement mediated by the Newtonian gravitational field of a body and entanglement mediated by on-shell gravitons emitted by the body. This suggests that Newtonian entanglement implies the existence of graviton entanglement and supports the view that the experimental discovery of Newtonian entanglement may be viewed as implying the existence of the graviton.   

Midisuperspace Foam

I describe a midisuperspace -- a model for quantum gravity in which most of the degrees of freedom are frozen out -- that is rich enough to allow a version of Wheeler's spacetime foam.   In this setting the Wheeler-DeWitt equation becomes tractable, and there is strong evidence that wave functions can be nearly time independent even in the presence of a large cosmological constant, with expansion in some regions balanced by contraction in others.    

Validity of the Semiclassical Approximation in 1+1 Electrodynamics: Solutions to the Linear Response Equation

For semiclassical electrodynamics to be an accurate approximation, quantum fluctuations must be small. The validity of the semiclassical approximation in 1+1 electrodynamics has been investigated using a criterion which involves the stability of solutions to the linear response equation, which is found by perturbing the semiclassical backreaction equation. If the solutions grow rapidly in time, then the criterion states that the semiclassical approximation breaks down. Numerical solutions to the linear response equation will be presented for the case of an electric field that is coupled to a homogeneous classical electric current and, through the semiclassical approximation, the current due to a quantized spin 1/2 field in a homogeneous state. The focus of this talk will be on the impact quantum fluctuations have on solutions to the linear response equation in the context of the above validity criterion.   

Time evolution of the entanglement entropy in isolated systems: Page curve and fluctuations

The average entanglement entropy of a random pure state was determined by Page in 1993. Here we consider an isolated quantum system and study the evolution of the entanglement entropy at early and late times, assuming that the system is initially in a non-entangled state. We conduct the analysis for: (i) random-matrix Hamiltonians, which can be used to model the full SYK Hamiltonian, and (ii) random quadratic fermionic Hamiltonians, which are also the SYK-2 Hamiltonians. We show that the entanglement entropy initially grows as $-\alpha t^2 \log(\alpha t^2)$ and determine the scale $\alpha$ in terms of the interaction strength. We also show that, for a random-matrix Hamiltonian, we recover the Page average at late times and estimate the size of the fluctuations around the average. Subsequently, we derive analogous results for random quadratic fermionic systems. Through this analysis, we derive expression for the time-scale at which the entanglement entropy reaches its equilibrium value. The result is of interest for the analysis of systems governed by SYK Hamiltonians and for investigations of information in black hole evaporation.   

Gauge-invariant subsystems and entanglement

One of the most basic notions in physics is the partitioning of a system into subsystems, and the study of correlations among its parts. In standard quantum mechanical systems, this partitioning is encoded in a tensor product structure of the Hilbert space of the system. However, for quantum field theories, this tensor product structure fails due to the universal divergence of the entanglement entropy, and so subsystems must be identified with commuting subalgebras of observables that encode a notion of locality. In canonical approaches to quantum gravity, this algebraic construction is complicated by the fact that gauge invariant Dirac observables are inherently non-local and can be difficult to construct in practice. For a quantum theory with gauge constraints, I will show that while the unconstrained (kinematical) Hilbert space may admit a tensor product structure, the physical (gauge-invariant) Hilbert space need not inherit this tensor product structure upon implementation of the constraints. A theorem will be proven providing necessary and sufficient conditions for the physical Hilbert space to inherit the kinematical subsystem structure. I will also leverage the framework of quantum reference frames to show that different reference frames induce generically different subsystem algebras of observables. The main thesis of this work is the identification of a gauge-invariant but frame-dependent notion of subsystems and entanglement in constrained systems. Since gravitation is an example of such a system, I will briefly comment on the issue of constructing physical gravitational subsystems.   

First Law for Kerr Taub-NUT AdS Black Holes

The first law of black hole mechanics, which relates the change of energy to the change of entropy and other conserved charges, has been the main motivation for probing the thermodynamic properties of black holes. In this talk, we will investigate the thermodynamics of Kerr Taub-NUT AdS black holes. Employing two diffreent methods (Komar integrals and Brown-York charges) we show that our expressions for the black hole charges satisfy a Smarr formula. Further, we will establish the first law for the Kerr Taub-NUT AdS black holes. We indicate the relevance of these results to the entropy of rotating AdS black holes as one quarter of the event horizon area.

A Geometric Analysis of Stabilizer States

The holographic entropy cone initiates a geometric description of multipartite entanglement by defining a state's entropy vector, constructed from the ordered tuple of subsystem entanglement entropies. Necessary satisfaction of appropriate entropy inequalities constrains the position of valid entropy vectors, those representing holographically realizable states, to the interior of a convex, polyhedral holographic entropy cone, a strict subspace of the ambient 2ⁿ-1 dimensional entropy vector space. The set of stabilizer states, those quantum states reachable from vacuum through combinations of the Hadamard, Phase, and CNOT operations, constitutes a strict superset of the holographic states and similarly admit a classical dual and complete characterization of entropy within the stabilizer entropy cone. While extensive analysis of these entropy cones has returned insight into the nested structure of entropy vector subspaces, much is unknown about the transition from one class of states to another. Notably, the set of stabilizer states that are not holographically realizable indicates the existence of some non-geometric process that ejects a system from the space of holographic states, into the space of stabilizer states. Continued examination of stabilizer states reveals discerning connections between distinct classes of quantum states, illustrating the foundations of entanglement in certain quantum systems. In this work, we present a new way to analyze and classify stabilizer states and stabilizer evolution within a graph-theoretic framework and exhibit the utility of this geometric analysis in studying entanglement structures arising in high-energy physics.   

Exploration of The Duality Between Generalized Geometry and Extraordinary Magnetoresistance

We outline the duality between the extraordinary magnetoresistance (EMR), observed in semiconductor-metal hybrids, and nonsymmetric gravity coupled to a diffusive U(1) gauge field. The corresponding gravity theory may be interpreted as the generalized complex geometry of the semidirect product of the symmetric metric and the antisymmetric Kalb-Ramond field: (gμν + βμν). We construct the four-dimensional covariant field theory and compute the resulting equations of motion. The equations encode the most general form of EMR within a well defined variational principle, for specific lower dimensional embedded geometric scenarios. Our formalism also reveals the emergence of additional diffusive pseudocurrents for a completely dynamic field theory of EMR. The proposed equations of motion now include terms that induce geometrical deformations in the device geometry in order to optimize the EMR. This bottom-up dual description between EMR and generalized geometry/gravity lends itself to a deeper insight into the EMR effect with the promise of potentially new physical phenomena and properties.   

Emergence of quantum chaos in non-equilibrium cosmology

The early universe is likely filled with a large number of interacting fields with unknown interactions. How can we quantitatively understand particle production (for example, during inflation and reheating after inflation) when such fields undergo a sufficient number of non-adiabatic, non-perturbative interactions? A recent proposal of a precise mapping between stochastic particle production events during inflation and re-heating in cosmology to conduction phenomena in disordered quasi one-dimensional wires provide a powerful statistical framework to resolve such seemingly intractable calculations. In my talk, I will describe our work where we use this precise correspondence to present a novel derivation for quantum corrections to the Fokker-Planck equation without dissipation responsible for studying the dynamical features of such stochastic particle creation events during the inflationary and re-heating epoch of the universe. Stochastic particle creation events can involve a large number of non-adiabatic events (scatterings). Such non-linearities give rise to signatures of quantum chaos. I will discuss our computation on the role of out-of-time ordered correlators to describe quantum chaos in the present non-equilibrium field theoretic setup and its consequences in stochastic inflation and reheating epoch of the universe using Spectral Form Factors from the principles of Gaussian Random Matrix Theory. Furthermore, my talk will focus on our proposed bound on the rate of growth of quantum chaos during stochastic particle production events in the early universe for non-adiabatic couplings with Gaussian matrix-models, namely polynomial matrix-models of Wigner-Dyson type.