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Coarse descriptions, microcanonical ensembles and the Born rule

Blind Review

Published Dec 7, 2023• 0 citations
Publisher

Abstract

It is conventionally thought that a large macroscopic object (universe) converges to a microcanonical ensemble. We ask how the microcanonical ensemble assumption can be consistent with the initial system amplitude information preserved (usually in the diagonal entries of the system density matrix) in a quantum measurement process. Patching these two assumptions together, as well as interpreting the initial system state and the microcanonical ensemble as reflecting epistemic uncertainty and providing a coarse description, results in the Born rule.

DOI
10.31219/osf.io/8xr94

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It is conventionally thought that a large macroscopic object (universe) converges to a microcanonical ensemble. We ask how the microcanonical ensemble assumption can be consistent with the initial system amplitude information preserved (usually in the diagonal entries of the system density matrix) in a quantum measurement process. Patching these two assumptions together, as well as interpreting the initial system state and the microcanonical ensemble as reflecting epistemic uncertainty and providing a coarse description, results in the Born rule.
An approach to quantum mechanics is developed which makes the Heisenberg cut between the deterministic microscopic quantum world and the partly deterministic, partly stochastic macroscopic world explicit. The microscopic system evolves according to the Schrodinger equation with stochastic behaviour arising when the system is probed by a set of coarse grained macroscopic observables whose resolution scale defines the Heisenberg cut. The resulting stochastic process can account for the different facets of the classical limit: Newton's laws (ergodicity broken); statistical mechanics of thermal ensembles (ergodic); and solve the measurement problem (partial ergodicity breaking). In particular, the usual rules of the Copenhagen interpretation, like the Born rule, emerge, along with completely local descriptions of EPR type experiments. The formalism also re-introduces a dynamical picture of equilibration and thermalization in quantum statistical mechanics and provides insight into how classical statistical mechanics can arise in the classical limit and in a way that alleviates various conceptual problems.
The measurement problem is the issue of explaining how the objective classical world emerges from a quantum one. Here we take a different approach. We assume that there is an objective classical system, and then ask that the standard rules of probability theory apply to it when it interacts with a quantum system. Under mild assumptions, we recover the unitary dynamics, collapse and Born rule postulates from quantum theory. Nonetheless, there is no decoherence, because the quantum state remains pure conditioned on the classical trajectory. This results in one world, rather than many-worlds. Our main technical tool is to exploit a change of measure on the space of classical paths, the functional form of which is shown to characterise the quantum dynamics and Born rules of a class of quantum-like theories.
The Born rule assigns probabilities to measurement outcomes under the assumption that a quantum system retains a well-defined identity within a stable coarse-grained description. Experimental studies of open quantum systems, error correction, monitored dynamics, and decoherence report abrupt “avalanche”-like transitions in which probabilistic descriptions of subsystems rapidly fail, even though global unitary evolution is preserved. This work provides a structural interpretation of such transitions by identifying an identity tolerance boundary, denoted Ω, beyond which the thermodynamic identity of a quantum system ceases to be well defined. When cumulative irreversible entanglement or decoherence exceeds this boundary, Born-rule probabilities no longer apply—not due to a failure of quantum mechanics, but because the probabilistic description presupposes a persisting system identity that has been lost. The framework does not introduce new physical laws, collapse mechanisms, or modifications to quantum theory. Rather, it clarifies the domain of applicability of probabilistic descriptions by explicitly separating fine-grained unitary evolution from the persistence of macroscopic or subsystem identity. This perspective unifies observations across decoherence avalanches, error-correction thresholds, and measurement-induced transitions without proposing operational procedures or control mechanisms.
For a macroscopic, isolated quantum system in an unknown pure state, the expectation value of any given observable is shown to hardly deviate from the ensemble average with extremely high probability under generic equilibrium and nonequilibrium conditions. Special care is devoted to the uncontrollable microscopic details of the system state. For a subsystem weakly coupled to a large heat bath, the canonical ensemble is recovered under much more general and realistic assumptions than those implicit in the usual microcanonical description of the composite system at equilibrium.
Starting from the quantum mechanics for NNN particles, we show that we can directly derive the microcanonical ensemble average of the physical quantity AAA by using only the long time average and the equal probability assumption for the equal energy states. The system is considered to be embedded in the outer world and we describe them in terms of the density matrix method.
Abstract I provide a simple derivation of the Born rule as giving a classical probability, that is, the ratio of the measure of favorable states of the system to the measure of its total possible states. In classical systems, the probability is due to the fact that the same macrostate can be realized in different ways as a microstate. Despite the radical differences between quantum and classical systems, I show that the same can be applied to quantum systems, and the result is the Born rule. This works only if the basis is continuous (an eigenbasis of observables with continuous spectra), but all known physically realistic measurements involve a continuous basis (the position basis). The continuous basis is not unique, and for subsystems it depends on the observable. But for the entire universe, there are continuous bases that give the Born rule for all measurements, because all measurements reduce to distinguishing macroscopic pointer states, and macroscopic observations commute. This allows for the possibility of a unique ontic basis for the entire universe. In the wavefunctional formulation, the basis can be chosen to consist of classical field configurations, and the coefficients Ψ[ϕ]\Psi [\phi ]Ψ[ϕ] Ψ [ ϕ ] can be made real by absorbing them into a global U(1) gauge. For the many-worlds interpretation, this result gives the Born rule from micro-branch counting.