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arXiv:hep-th/0007006 · Journal of Mathematical Physics

Consequences of ’t Hooft’s equivalence class theory and symmetry by coarse graining

X. F. Liu, C. P. Sun

Published Aug 1, 2001• 4 citations
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Abstract

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.”

DOI
10.1063/1.1380250

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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.”
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.
We discuss an explicit realization of the dissipative dynamics anticipated in the proof of 't Hooft's existence theorem, which states that ' For any quantum system there exists at least one deterministic model that reproduces all its dynamics after prequantization' . – There is an energy-parity symmetry hidden in the Liouville equation, which mimics the Kaplan-Sundrum protective symmetry for the cosmological constant. This symmetry may be broken by the coarse-graining inherent in physics at scales much larger than the Planck length. We correspondingly modify classical ensemble theory by incorporating dissipative fluctuations (information loss) – which are caused by discrete spacetime continually 'measuring' matter. In this way, aspects of quantum mechanics, such as the von Neumann equation, including a Lindblad term, arise dynamically and expectations of observables agree with the Born rule. However, the resulting quantum coherence is accompanied by an intrinsic decoherence and continuous localization mechanism. Our proposal leads towards a theory that is linear and local at the quantum mechanical level, but the relation to the underlying classical degrees of freedom is nonlocal.
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.
We start from classical general relativity coupled to matter fields. Each configuration variable and its conjugate momentum, as also spacetime points are raised to the status of matrices [equivalently operators]. These matrices obey a deterministic Lagrangian dynamics at the Planck scale. By coarse-graining this matrix dynamics over time intervals much larger than Planck time, one derives quantum theory as a low energy emergent approximation. If a sufficiently large number of degrees of freedom get entangled, spontaneous localisation takes place, leading to the emergence of classical spacetime geometry and a classical universe. In our theory, dark energy is shown to be a large-scale quantum gravitational phenomenon. Quantum indeterminism is not fundamental, but results from our not probing physics at the Planck scale.
Constrained Hamiltonian dynamics is exploited to provide the mathematical framework of a coarse-grained description of the quantum system of two interacting qubits and of nonlinear interacting oscillators. The coarse-graining is treated as an equivalence relation on the set of quantum states resulting in the emergence of the classical phase-space. The equivalence relation imposes constraints on the Hamiltonian dynamics of the quantum system. It is seen that the evolution of the coarse-grained system preserves constant and minimal quantum fluctuations of the fundamental observables. This leads to the emergence of typical classical properties, like the relation between symmetry and integrability, and in the case of oscillators in the macro-limit implies the emergence of the classical system.
Constrained Hamiltonian dynamics is exploited to provide the mathematical framework of a coarse-grained description of the quantum system of nonlinear interacting oscillators. The coarse graining is treated as an equivalence relation on the set of quantum states resulting in the emergence of classical phase space. The equivalence relation imposes constraints on the Hamiltonian dynamics of the quantum system. It is seen that the evolution of the coarse-grained system preserves constant and minimal quantum fluctuations of the fundamental observables. This leads to the emergence of the corresponding classical system on a sufficiently large scale.