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arXiv:1912.05914 · Journal of Mathematical Physics

Mathematics of the classical and the quantum

Alexey A. Kryukov

Published Aug 1, 2020• 6 citations
arXivHTMLPDF

Abstract

Newtonian and Schrödinger dynamics can be formulated in a physically meaningful way within the same Hilbert space framework. This fact was recently used to discover an unexpected relation between classical and quantum motions that goes beyond the results provided by the Ehrenfest theorem. The Newtonian dynamics was shown to be the Schrödinger dynamics of states constrained to a submanifold of the space of states, identified with the classical phase space of the system. Quantum observables are identified with vector fields on the space of states. The commutators of observables are expressed through the curvature of the space. The resulting embedding of the Newtonian and Schrödinger dynamics into a unified geometric framework is rigid in the sense that the Schrödinger dynamics is a unique extension of the Newtonian one. Under the embedding, the normal distribution of measurement results associated with a classical measurement implies the Born rule for the probability of transition of quantum states. The mathematics of the discovered relationship between the classical and the quantum is reviewed and investigated here in detail and applied to the process of measurement of spin and position observables.

DOI
10.1063/1.5135015

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Newtonian and Schrödinger dynamics can be formulated in a physically meaningful way within the same Hilbert space framework. This fact was recently used to discover an unexpected relation between classical and quantum motions that goes beyond the results provided by the Ehrenfest theorem. The Newtonian dynamics was shown to be the Schrödinger dynamics of states constrained to a submanifold of the space of states, identified with the classical phase space of the system. Quantum observables are identified with vector fields on the space of states. The commutators of observables are expressed through the curvature of the space. The resulting embedding of the Newtonian and Schrödinger dynamics into a unified geometric framework is rigid in the sense that the Schrödinger dynamics is a unique extension of the Newtonian one. Under the embedding, the normal distribution of measurement results associated with a classical measurement implies the Born rule for the probability of transition of quantum states. The mathematics of the discovered relationship between the classical and the quantum is reviewed and investigated here in detail and applied to the process of measurement of spin and position observables.
Newtonian and Schrodinger dynamics can be formulated in a physically meaningful way within the same Hilbert space framework. This fact was recently used to discover an unexpected relation between classical and quantum motions that goes beyond the results provided by the Ehrenfest theorem. The Newtonian dynamics was shown to be the Schrodinger dynamics of states constrained to a submanifold of the space of states, identified with the classical phase space of the system. Quantum observables are identified with vector fields on the space of states. The commutators of observables are expressed through the curvature of the space. The resulting embedding of the Newtonian and Schrodinger dynamics into a unified geometric framework is rigid in the sense that the Schrodinger dynamics is a unique extension of the Newtonian one. Furthermore, under the embedding, the normal distribution of measurement results associated with a classical measurement implies the Born rule for the probability of transition of quantum states. In this paper, the implications of the obtained theory to the process of measurement in quantum theory are analyzed. The double-slit, EPR and Schrodinger cat type experiments are reviewed anew. It is shown that, despite reproducing the usual results of quantum theory, the framework is not simply a reformulation of the theory. New experiments to discover the predicted effects are proposed.
Newtonian and Schrödinger dynamics can be formulated in a physically meaningful way within the same Hilbert space framework. This fact was recently used to discover an unexpected relation between classical and quantum motions that goes beyond the results provided by the Ehrenfest theorem. A formula relating the normal probability distribution and the Born rule was also found. Here the dynamical mechanism responsible for the latter formula is proposed and applied to measurements of macroscopic and microscopic systems. A relationship between the classical Brownian motion and the diffusion of state on the space of states is discovered. The role of measuring devices in quantum theory is investigated in the new framework. It is shown that the so-called collapse of the wave function is not measurement specific and does not require a “concentration” near the eigenstates of the measured observable. Instead, it is explained by the common diffusion of a state over the space of states under interaction with the apparatus and the environment. This in turn provides us with a basic reason for the definite position of macroscopic bodies in space.
We develop a dynamical framework for quantum measurement based on stochastic but unitary evolution in projective state space. Random Hamiltonians drawn from the Gaussian Unitary Ensemble generate stochastic unitary dynamics of the quantum state, while equivalence classes reflecting finite detector resolution define classical observables as well as classical configuration-space and phase-space submanifolds. When the evolution is constrained to the phase-space submanifold, free Schrödinger dynamics reduces to Newtonian motion, while stochastic motion constrained to the classical configuration-space submanifold yields ordinary Brownian motion in classical space. Transition probabilities under the stochastic dynamics satisfy the Born rule, whereas the constrained classical evolution produces the normal probability distributions characteristic of classical measurements. We show that, in this setting, measurement, state reduction, and the quantum-classical transition emerge from unitary dynamics alone, without invoking nonunitary collapse or additional postulates. Entanglement and EPR correlations arise geometrically from the evolution of joint states in composite state space, preserving locality in spacetime. The framework provides a unified dynamical account of measurement and classicality compatible with the structure of quantum mechanics.
Abstract Newtonian and Scrödinger dynamics can be formulated in a physically meaningful way within the same Hilbert space framework. The resulting unexpected relation between classical and quantum motions goes beyond the results provided by the Ehrenfest theorem. The normal probability distribution and the Born rule turn out to be related. Here a dynamical mechanism responsible for the latter formula is proposed and applied to measurements of macroscopic and microscopic systems. A new meaning of the wave function collapse is proposed. The results are used to explore the classical behavior of macroscopic bodies in quantum theory.
The paper investigates the physical content of a recently proposed mathematical framework that unifies the standard formalisms of classical mechanics, relativity and quantum theory. In the framework states of a classical particle are identified with Dirac delta functions. The classical space is "made" of these functions and becomes a submanifold in a Hilbert space of states of the particle. The resulting embedding of the classical space into the space of states is highly non-trivial and accounts for numerous deep relations between classical and quantum physics and relativity. One of the most striking results is the proof that the normal probability distribution of position of a macroscopic particle (equivalently, position of the corresponding delta state within the classical space submanifold) yields the Born rule for transitions between arbitrary quantum states.