[Paper Review] Locality and the classical limit of quantum systems

T. Banks|ArXiv.org|Jul 6, 2009

Quantum many-body systems8 references3 citations

TL;DR

The paper argues that locality in interactions suppresses quantum coherence in macroscopic systems, making quantum superpositions of classical states exponentially unlikely due to environmental decoherence from internal degrees of freedom. It shows that overlaps between wave functions for different classical trajectories decay exponentially with system size, rendering observable quantum behavior in macroscopic superpositions effectively impossible within cosmological timescales.

ABSTRACT

I argue that conventional estimates of the criterion for classical behavior of a macroscopic body are incorrect in most circumstances,because they do not take into account the locality of interactions, which characterizes the behavior of all systems described approximately by local quantum field theory. The deviations from classical behavior of a macroscopic body, except for those that can be described as classical uncertainties in the initial values of macroscopic variables,are {\it exponentially} small as a function of the volume of the macro-system in microscopic units. Conventional estimates are correct only when the internal degrees of freedom of the macrosystem are in their ground state, and the classical motion of collective coordinates is adiabatic. Otherwise, the system acts as its own environment and washes out quantum phase correlations between different classical states of its collective coordinates. I suggest that it is likely that we can only achieve meso-scopic superpositions, for systems which have topological variables, and for which we can couple to those variables without exciting phonons.

Motivation & Objective

• To challenge conventional estimates that suggest power-law corrections to classical behavior in macroscopic systems.
• To address the psychological unease surrounding the classical limit and Schrödinger's cat paradox by re-examining decoherence mechanisms.
• To argue that macroscopic systems naturally suppress quantum phase correlations between different classical states due to their internal structure.
• To explain why experiments attempting to observe mesoscopic superpositions are unlikely to succeed without extreme control over internal degrees of freedom.
• To clarify the role of locality and collective coordinates in the emergence of classical behavior from quantum mechanics.

Proposed method

• Analyzes the quantum mechanics of collective coordinates in large systems, focusing on how external forces couple to individual constituents via local interactions.
• Uses the framework of local quantum field theory to model how different classical motions of collective coordinates induce time-dependent Hamiltonians for internal degrees of freedom.
• Applies variational approximations (e.g., Hartree-Fock, Jastrow) to represent the many-body wave function as a product of localized cluster wave functions.
• Calculates the overlap between wave functions for different classical trajectories as (1 - ε)^N, where ε is a small perturbation and N is the number of microscopic correlation volumes.
• Estimates the time scale for observing phase correlations between superselection sectors as ~10^{cN}, where c is order-1 and N is the system size in microscopic units.
• Applies statistical mechanics and quantum field theory to argue that the density of states supports a statistical description of the system’s behavior.

Experimental results

Research questions

• RQ1 Why do conventional estimates of quantum corrections to classical behavior fail for macroscopic systems?
• RQ2 How does locality in interactions affect the stability of quantum superpositions in macroscopic bodies?
• RQ3 What determines the rate at which quantum phase coherence is lost between different classical trajectories of a macroscopic system?
• RQ4 Under what conditions might mesoscopic superpositions of collective variables be observable?
• RQ5 Why is the classical limit of quantum mechanics not simply governed by mass or size, but by the number of localizable constituents?

Key findings

• Deviations from classical behavior in macroscopic systems are exponentially small in the number of microscopic correlation volumes, not power-law as conventionally assumed.
• The overlap between wave functions for different classical trajectories decays as (1 - ε)^N, leading to exponential suppression of quantum coherence.
• The time required to observe phase correlations between distinct classical states scales as 10^{cN}, which exceeds the age of the universe for N ~ 10^3 correlation volumes.
• Only systems with topological variables—where coupling can be done without exciting phonons—may allow for robust mesoscopic superpositions.
• Conventional estimates of quantum corrections are valid only when internal degrees of freedom are in their ground state and collective motion is adiabatic.
• The predictions of quantum mechanics for macroscopic systems are indistinguishable from classical statistical mechanics with initial conditions derived from quantum microsystems.