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arXiv:2005.12903 · International Journal of Geometric Methods in Modern Physics

On the origin of the weak equivalence principle in a theory of emergent quantum mechanics

Ricardo Gallego Torromé

Published Jul 18, 2020• 8 citations
arXivHTMLPDF

Abstract

We argue that in a framework for emergent quantum mechanics, the weak equivalence principle is a consequence of concentration of measure in large-dimensional spaces of [Formula: see text]-Lipshitz functions. Furthermore, as a consequence of the emergent framework and the properties that we assume for the fundamental dynamics, it is argued that gravity must be a classical, emergent interaction.

DOI
10.1142/s0219887820501571

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We argue that in a framework for emergent quantum mechanics, the weak equivalence principle is a consequence of concentration of measure in large-dimensional spaces of [Formula: see text]-Lipshitz functions. Furthermore, as a consequence of the emergent framework and the properties that we assume for the fundamental dynamics, it is argued that gravity must be a classical, emergent interaction.
Quantum systems are viewed as emergent systems from the fundamental degrees of freedom. The laws and rules of quantum mechanics are understood as an effective description, valid for the emergent systems and specially useful to handle probabilistic predictions of observables. After introducing the geometric theory of Hamilton-Randers spaces and reformulating it using Hilbert space theory, a Hilbert space structure is constructed from the Hilbert space formulation of the underlying Hamilton-Randers model and associated with the space of wave functions of quantum mechanical systems. We can prove the emergence of the Born rule from ergodic considerations. A geometric mechanism for a natural spontaneous collapse of the quantum states based on the concentration of measure phenomena as it appears in metric geometry is discussed.We show the existence of stable vacua states for the quantized matter Hamiltonian. Another consequence of the concentration of measure is the emergence of a weak equivalence principle for one of the dynamics of the fundamental degrees of freedom. We suggest that the reduction of the quantum state is driven by a gravitational type interaction. Such interaction appears only in the dynamical domain when localization of quantum observables happens, it must be a classical interaction. We discuss the double slit experiment in the context of the framework proposed, the interference phenomena associated with a quantum system in an external gravitational potential, a mechanism explaining non-quantum locality and also provide an argument in favour of an emergent interpretation of every macroscopic time parameter. Entanglement is partially described in the context of Hamilton-Randers theory and how naturally Bell's inequalities should be violated.
Quantum systems are viewed as emergent systems from the fundamental degrees of freedom. The laws and rules of quantum mechanics are understood as an effective description, valid for the emergent systems and specially useful to handle probabilistic predictions of observables. After introducing the geometric theory of Hamilton-Randers spaces and reformulating it using Hilbert space theory, a Hilbert space structure is constructed from the Hilbert space formulation of the underlying Hamilton-Randers model and associated with the space of wave functions of quantum mechanical systems. We can prove the emergence of the Born rule from ergodic considerations. A geometric mechanism for a natural spontaneous collapse of the quantum states based on the concentration of measure phenomena as it appears in metric geometry is discussed.We show the existence of stable vacua states for the quantized matter Hamiltonian. Another consequence of the concentration of measure is the emergence of a weak equivalence principle for one of the dynamics of the fundamental degrees of freedom. We suggest that the reduction of the quantum state is driven by a gravitational type interaction. Such interaction appears only in the dynamical domain when localization of quantum observables happens, it must be a classical interaction. We discuss the double slit experiment in the context of the framework proposed, the interference phenomena associated with a quantum system in an external gravitational potential, a mechanism explaining non-quantum locality and also provide an argument in favour of an emergent interpretation of every macroscopic time parameter. Entanglement is partially described in the context of Hamilton-Randers theory and how naturally Bell's inequalities should be violated.
Assuming the validity of the equivalence principle in the quantum regime, we argue that one of the assumptions of the usual definition of quantum mechanics, namely separation between the “classical” detector and the “quantum” system, must be relaxed. We argue, therefore, that if both the equivalence principle and quantum mechanics continue to survive experimental tests, then this favors “informational” interpretations of quantum mechanics (where formalism is built around relations between observables, defined as information that can be accessed by an observer of the system. In particular “collapse” is understood as a change of relative information as the detector interacts with the system) over “ontic ones” (assuming the physical reality of states and wave functions, which are assumed to be more than informational objects. In particular collapse is understood as a physical process). In particular, we show that relational-type interpretations can readily accommodate the equivalence principle via a minor modification of the assumptions used to justify the formalism. We qualitatively speculate what a full generally covariant quantum dynamics could look like, and comment on experimental investigations.
Hamilton-Randers dynamical systems, a particular class of Hamiltonian dynamical systems, are considered as mathematical models for deterministic emergent quantum mechanics. We show that in such framework, local diffeomorphism invariance, reversibility of the effective quantum dynamics and an universal maximal bound for proper acceleration emerge in that class of classical, deterministic and local models. Starting from the elements of Hamilton-Randers spaces, a phenomenological Hilbert space is constructed and associated with the space of wave functions of quantum mechanics. A geometric description for a spontaneous reduction of the quantum states, based on the concentration of measure phenomena as it appears in asymptotic Banach theory and probability theory, is described in general terms. It is also shown how the same concentration of measure phenomena plays a remarkable role in showing conditions for the existence of stable vacua states for the matter Hamiltonian. Furthermore, it is discussed the emergence of a weak equivalence principle from very fundamental principles of mathematical analysis and the basic assumptions of Hamilton-Randers theory. This fact, together with the existence in the theory of a maximal speed and the property of diffeomorphism invariance of the interaction driving the reduction of the quantum state, suggest that the reduction of the quantum state is driven by a gravitational type interaction. Moreover, since such interaction appears only in the dynamical domain when localization happens, it must be associated with a classical interaction. We make the hypothesis that such identification is universal and that indeed gravity is a domain of the dynamics of Hamilton-Randers systems. Hence Hamilton-Randers theory provides an unification scheme where quantum mechanics and classical gravity are both emergent.
Since the advent of quantum mechanics we have mainly been concerned with its predictions from the perspective of an external observer. This is in strong contrast to the theory of general relativity, where the physics is governed by the intrinsic properties of space-time. At the same time, the precise relation between space-time and quantum mechanics is still one of the greatest problems of theoretical physics. This immediately raises the question on the completeness of our understanding of quantum mechanics. Here we address the problem by making an intrinsic analysis of observables in generic quantum systems. We show that there is an extreme fine tuning problem for the emergence of physics from the Hilbert space dynamics. However, for any initial condition and Hamiltonian, there exists a special set of observables. We show that these observables are intimately linked to the natural configuration space in which an area law for the entanglement is inevitable. We argue that this implies emergent gravity.