Most recently 't Hooft has postulated (G. 't Hooft, Class. Quantum Grav.16, 3263 (1999)) that quantum states at the "atomic scale" can be understood as equivalence classes of primordial states governed by a dissipative deterministic theory underlying quantum theory at the "Planck scale". Defining invariant subspaces clearly for primordial states according to a given evolution, we mathematically reformulate 't Hooft's theory as a quotient space construction with the time-reversible evolution operator induced naturally. With this observation and some analysis, 't Hooft's theory is generalized beyond his case where the evolution at the "Planck scale" is a power of a one-time-step evolution or the time is discrete.
Most recently 't Hooft has postulated (G. 't Hooft, Class. Quantum Grav.16, 3263 (1999)) that quantum states at the "atomic scale" can be understood as equivalence classes of primordial states governed by a dissipative deterministic theory underlying quantum theory at the "Planck scale". Defining invariant subspaces clearly for primordial states according to a given evolution, we mathematically reformulate 't Hooft's theory as a quotient space construction with the time-reversible evolution operator induced naturally. With this observation and some analysis, 't Hooft's theory is generalized beyond his case where the evolution at the "Planck scale" is a power of a one-time-step evolution or the time is discrete.
Motivated by the conceptual problems concerning the quantisation of gravity, the Dutch theoretical physicist G. 't Hooft (1999 Nobel prize in physics) put forward the notion that quantum mechanics must be the emergent theory of some underlying, deterministic theory.This proposal usually goes by the name quantum mechanics as an emergent phenomenon.This research line, initiated by 't Hooft in the late 1990's, has been the subject of intense research over the last 15 years, by 't Hooft himself as well as by many other researchers.In this PhD thesis we present our own approach to quantum mechanics as an emergent phenomenon.According to the emergence paradigm for quantum mechanics, information-loss effects in the underlying deterministic theory lead to the arrangement of states of the latter into equivalence classes, that one identifies as quantum states of the emergent quantum mechanics.In brief, quantisation is dissipation, according to 't Hooft.Moreover it has been argued in the literature that, in the presence of weak gravitational fields, quantum effects must be indistinguishable from thermal effects.Since the latter are typically dissipative in nature, the presence of a weak gravitational field should provide a framework in which quantum effects can be explained as due to thermal, dissipative fluctuations.Furthermore, since gravitational effects can be locally gauged away (thanks to the equivalence principle), there should exist some kind of equivalence principle for quantum effects, i.e., some kind of relativity principle for the notion of quantumness as opposed to the notion of classicality.In this PhD thesis we elaborate on this idea.Once a reference frame is fixed, however, quantum effects cannot be gauged away, and the statement quantisation is dissipation lends itself to a thermodynamical treatment.In this thesis we also present one mechanism whereby quantum mechanics is seen to emerge, thus explicitly realising 't Hooft's proposal.This mechanism is based on a dictionary between semiclassical quantum mechanics, on the one hand, and the classical theory of irreversible thermodynamics in the linear regime, on the other.This thermodynamical formalism, developed by Nobel prize winners Onsager and Prigogine, can be easily mapped into that of semiclassical quantum mechanics. Chapter 3Emergent QM as a thermal ensemble This chapter is based on the article Emergent Quantum Mechanics as a Thermal Ensemble, by P.
According to ’t Hooft [Class. Quantum. Grav. 16, 3263 (1999)], quantum gravity can be postulated as a dissipative deterministic system, where quantum states at the “atomic scale” can be understood as equivalence classes of primordial states governed by a dissipative deterministic dynamics law at the “Planck scale.” In this paper, it is shown that for a quantum system to have an underlying deterministic dissipative dynamics, the time variable should be discrete if the continuity of its temporal evolution is required. Besides, the underlying deterministic theory also imposes restrictions on the energy spectrum of the quantum system. It is also found that quantum symmetry at the “atomic scale” can be induced from ’t Hooft’s coarse graining classification of primordial states at the “Planck scale.”
We explore whether quantum field theory can be understood as the statistical mechanics of a time-reversal-invariant stochastic generalization of Hamiltonian dynamics. The motivation for this project, started with this paper, is to assign sharp values to all observables and thereby avoid the quantum measurement problem. In classical mechanics, motion is deterministic and corresponds to an evolution of the phase space probability density according to Liouville's equation that is governed by first derivatives of the Hamiltonian in phase space. We derive a generalization of the Liouville equation with natural constraints -- namely, reduction to classical Hamiltonian dynamics as the stochasticity parameter ℏ↦0, Fokker-Planck form for the probability density evolution, local Hamiltonian dependence, time-reversal invariance, energy conservation, and minimality -- which turns out to be a Fokker-Planck equation with a generalized diffusion matrix that is symmetric, traceless, and constructed from the Hessian of the Hamiltonian. We then show that the Schrödinger equation in the coherent-state phase-space formulation of certain bosonic QFTs has precisely this form, with the Husimi function playing the role of the phase space probability density. The question to what extent this equation can be interpreted in terms of objective stochastic field theories is discussed in a companion paper.
This monograph provides a comprehensive and rigorous commentary on the paper “Effective Dynamics of a Relational Operator K with GUT-Scale and Planck-Suppressed Corrections.” It is designed as an independent companion volume that expands the mathematical, physical, and conceptual foundations underlying the relational master equation introduced in the main article. The work develops a unified operator-theoretic framework that integrates: Self-adjoint GUT-scale dynamics (reversible quantum evolution) Maximal dissipative operators (irreversible decoherence and information loss) Planck-suppressed higher-dimensional operators (non-linear EFT corrections from quantum gravity) Using tools from functional analysis, semigroup theory, perturbation theory, and relative boundedness, the framework establishes global well-posedness of the nonlinear master equation governing the relational operator (K(t)) on the trace-class space (T_1(H)). The commentary explains these mathematical structures through the “Six Pillars” of operator theory—Banach spaces, closed operators, semigroups, maximal dissipativity, Kato–Rellich perturbation theory, and Banach fixed-point methods. On the physics side, the monograph connects the operator-theoretic results to concrete observables in: High-energy scattering (GUT-scale deviations from higher-dimensional operators) Macroscopic dissipation (environment-induced decoherence from the dissipative generator (D)) Primordial non-Gaussianity (Planck-suppressed signatures in the early universe) The commentary also situates the relational operator (K) within broader conceptual frameworks such as Relational Quantum Mechanics, quantum information, and effective field theory, providing intuitive explanations, diagrams, and a glossary of 50 key terms for peer reviewers. This document is intended for researchers in mathematical physics, quantum foundations, operator theory, high-energy physics, and cosmology. It serves both as an accessible introduction and a rigorous technical reference for the relational master equation and its implications