The time-identity tradeoff

TL;DR

This paper explores the transition between particle indistinguishability and distinguishability in quantum systems, introducing a timescale and temperature scale that govern when particles are effectively identical or different.

Contribution

It provides a general framework linking measurement timescales and temperature to particle distinguishability, addressing the continuum between identical and non-identical particles.

Findings

A timescale $ au_d$ determines when particles appear identical or different.

A temperature scale $T_d$ governs thermal equilibration relative to particle distinguishability.

As differences vanish, $ au_d$ and $T_d$ approach infinity and zero, respectively.

Abstract

Distinguishability plays a major role in quantum and statistical physics. When particles are identical their wave function must be either symmetric or antisymmetric under permutations and the number of microscopic states, which determines entropy, is counted up to permutations. When the particles are distinguishable, wavefunctions have no symmetry and each permutation is a different microstate. This binary and discontinuous classification raises a few questions: one may wonder what happens if particles are almost identical, or when the property that distinguishes between them is irrelevant to the physical interactions in a given system. Here I sketch a general answer to these questions. For any pair of non-identical particles there is a timescale, $\tau_d$, required for a measurement to resolve the differences between them. Below $\tau_d$, particles seem identical, above it - different, and the uncertainty principle provides a lower bound for $\tau_d$. Thermal systems admit a conjugate temperature scale, $T_d$. Above this temperature the system appears to equilibrate before it resolves the differences between particles, below this temperature the system identifies these differences before equilibration. As the physical differences between particles decline towards zero, $\tau_d \to \infty$ and $T_d \to 0$.