We consider the quantum-to-classical transition for macroscopic systems coupled to their environments. By applying Born's Rule, we are led to a particular set of quantum trajectories, or an unravelling, that describes the state of the system from the frame of reference of the subsystem. The unravelling involves a branch dependent Schmidt decomposition of the total state vector. The state in the subsystem frame, the conditioned state, is described by a Poisson process that involves a non-linear deterministic effective Schrodinger equation interspersed with quantum jumps into orthogonal states. We then consider a system whose classical analogue is a generic chaotic system. Although the state spreads out exponentially over phase space, the state in the frame of the subsystem localizes onto a narrow wave packet that follows the classical trajectory due to Ehrenfest's Theorem. Quantum jumps occur with a rate that is the order of the effective Lyapunov exponent of the classical chaotic system and imply that the wave packet undergoes random kicks described by the classical Langevin equation of Brownian motion. The implication of the analysis is that this theory can explain in detail how classical mechanics arises from quantum mechanics by using only unitary evolution and Born's Rule applied to a subsystem.
We consider the quantum-to-classical transition for macroscopic systems coupled to their environments. By applying Born's Rule, we are led to a particular set of quantum trajectories, or an unravelling, that describes the state of the system from the frame of reference of the subsystem. The unravelling involves a branch dependent Schmidt decomposition of the total state vector. The state in the subsystem frame, the conditioned state, is described by a Poisson process that involves a non-linear deterministic effective Schrodinger equation interspersed with quantum jumps into orthogonal states. We then consider a system whose classical analogue is a generic chaotic system. Although the state spreads out exponentially over phase space, the state in the frame of the subsystem localizes onto a narrow wave packet that follows the classical trajectory due to Ehrenfest's Theorem. Quantum jumps occur with a rate that is the order of the effective Lyapunov exponent of the classical chaotic system and imply that the wave packet undergoes random kicks described by the classical Langevin equation of Brownian motion. The implication of the analysis is that this theory can explain in detail how classical mechanics arises from quantum mechanics by using only unitary evolution and Born's Rule applied to a subsystem.
It is shown that Schrodinger's equation and Born's rule are sufficient to ensure that the states of macroscopic collective coordinate subsystems are microscopically localized in phase space and that the localized state follows the classical trajectory with random quantum noise that is indistinguishable from the pseudo-random noise of classical Brownian motion. This happens because in realistic systems the localization rate determined by the coupling to the environment is greater than the Lyapunov exponent that governs chaotic spreading in phase space. For realistic systems, the trajectories of the collective coordinate subsystem are at the same time an "unravelling" and a set of "consistent/decoherent histories". Different subsystems have their own stochastic dynamics that generally knit together to form a global dynamics, although in certain contrived thought experiments, most notably Wigner's friend, in the contrary, there is observer complementarity.
Two approaches to quantum-classical correspondence are distinguished according to the classical dynamical theory with which quantum theory is compared. The first of these, Ehrenfest correspondence, defines a dynamical regime in which the quantum expectation values follow approximately a classical trajectory. The second of these, Liouville correspondence, applies when the quantum probability distributions remain well approximated by a density in the classical phase space. The former applies only for narrow states, whereas the latter may remain valid even for quantum states that have spread to the system size. A spin model is adopted for this correspondence study because the quantum state is discrete and finite- dimensional, and thus no artificial truncation of the Hilbert space is required. The quantum time-evolution is given by a discrete unitary mapping. The corresponding classical model is volume-preserving (non-dissipative) and the time-evolution is given by a symplectic map. In classically chaotic regimes, the widths of initially narrow quantum states grow, on average, exponentially with time, until saturation at the system size. This initial spreading rate is well approximated by the classical Lyapunov exponent when the accessible classical phase space is predominantly chaotic. Because of the exponential growth rate of the quantum variance, the Ehrenfest regime is delimited by a break-time that grows logarithmically with increasing quantum numbers. The small differences between quantum expectation values and corresponding Liouville averages of dynamical variables also grow exponentially, initially, if the classical behaviour is chaotic. This exponential rate is independent of h and consistently larger than the classical Lyapunov exponent by at least a factor of two. Interestingly, this exponential growth rate does not continue until these differences approach magnitudes of order the system size, but crosses over to power-law growth (or simply saturates), when the differences have reached magnitudes no larger than order ħ. Because of the exponential growth of these quantum-classical differences, the Liouville regime is also delimited by a break-time that grows logarithmically with increasing quantum numbers. However, due to the early saturation of the exponential growth of the differences, the Liouville log break-time applies only in a restricted domain. After spreading to the system size, the quantum states relax to an approximately time-independent state. This is characterised by equilibrium distributions that are subject to rapidly oscillating fluctuations about the coarse-grained classical steady-state. If the accessible classical phase space is predominantly chaotic, then the equilibrium quantum probability distributions are nearly microcanonical. Otherwise the steady-state quantum probability distributions accurately reflect the details of the KAM surfaces in the classical phase space. The equilibrium regime quantum fluctuations, which are quantum-classical differences, are shown to approach zero as a negative power of the quantum numbers. Since the differences between quantum expectation values and individual classical trajectories grow exponentially in time to a magnitude that scales with the system size, the Ehrenfest regime is delimited by an unrestricted log break-time. The log break-time may be short compared to experimental time-scales for some chaotic macroscopic bodies, and, as a result, an observable breakdown of Ehrenfest correspondence may arise for some macroscopic systems. Although the Liouville regime is also delimited by a log break-time, an observable breakdown of correspondence between quantum mechanics and Liouville mechanics for macroscopic bodies is not likely since their differences, expressed relative to the dimensions of the observable, approach zero in the limit of large quantum numbers.
In the frames of classical mechanics the generalized Langevin equation is derived for an arbitrary mechanical subsystem coupled to the harmonic bath of a solid. A time-acting temperature operator is introduced for the quantum Klein-Kramers and Smoluchowski equations, accounting for the effect of the quantum thermal bath oscillators. The model of Brownian emitters is theoretically studied and the relevant evolutionary equations for the probability density are derived. The Schrodinger equation is explained via collisions of the target point particles with the quantum force carriers, transmitting the fundamental interactions between the point particles. Thus, electrons and other point particles are no waves and the wavy chapter of quantum mechanics originated for the force carriers. A stochastic Lorentz-Langevin equation is proposed to describe the underlaying Brownian-like motion of the point particles in quantum mechanics. Considering the Brownian dynamics in the frames of the Bohmian mechanics, the density functional Bohm-Langevin equation is proposed, and the relevant Smoluchowski-Bohm equation is derived. A nonlinear master equation is proposed by proper quantization of the classical Klein-Kramers equation. Its equilibrium solution in the exact canonical Gibbs density operator, while the well-known Caldeira-Leggett equation is simply a linearization at high temperature. In the case of a free quantum Brownian particles, a new law for the spreading of the wave packet it discovered, which represents the quantum generalization of the classical Einstein law of Brownian motion. A new projector operator is proposed for the collapse of the wave function of a quantum particle moving in a classical environment. Its application results in dissipative Schrodinger equations, as well as in a new form of dissipative Liouville equation in classical mechanics.
The Newtonian motion of a macroscopic particle is derived from the linear Schrödinger equation with a Hamiltonian consisting of the free-particle term and a random Hamiltonian drawn from the Gaussian Unitary Ensemble. The random term models interaction with the environment. We show that the parameters governing the resulting state-space random walk, together with the treatment of experimentally indistinguishable states as equivalence classes, explain the contrasting behavior of microscopic and macroscopic systems. The analysis extends previous work deriving the Born rule for microscopic particles when the free-particle term is negligible.
The equations of motion for phase-space moments and correlations are derived systematically for quantum and classical dynamics, and are solved numerically for chaotic and regular motions of the H\'enon-Heiles model. For very narrow probability distributions, Ehrenfest's theorem implies that the centroid of the quantum state will approximately follow a classical trajectory. But the error in Ehrenfest's theorem does not scale with $\ensuremath{\Elzxh},$ and is found to be governed essentially by classical quantities. The difference between the centroids of the quantum and classical probability distributions, and the difference between the variances of those distributions, scale as ${\ensuremath{\Elzxh}}^{2},$ and so are the true measures of quantum effects. For chaotic motions, these differences between quantum and classical motions grow exponentially, with a larger exponent than does the variance of the distributions. For regular motions, the variance of the distributions grows as t2, whereas the differences between the quantum and classical motions grow as t3.
It was recently shown that the Born rule for transition probabilities of a microscopic particle can be derived from the linear Schr{\"o}dinger equation with a Hamiltonian represented by random matrices from the Gaussian Unitary Ensemble, taken to be independent at different time points. Under such an evolution, the particle's state undergoes an isotropic random walk on the projective state space, and the relative frequencies of reaching different regions obey the Born rule. In this work, we extend these results to demonstrate how the same framework accounts for the emergence of Newtonian dynamics for macroscopic bodies on a submanifold of state space, thereby providing a unified geometric account of the quantum-classical transition.