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Approximately Macroscopically Unique States and Quantum Mechanics

Huaxin Lin, Hang Wang

TL;DR

The paper addresses Mumford's Approximately Macroscopically Unique (AMU) states for macroscopic observables represented by (potentially unbounded) self-adjoint operators with small commutators. It develops an almost commutative one-point compactification and bounded-transform techniques to translate unbounded tuples into a C*-algebraic setting, enabling synthetic-spectrum methods to guarantee the existence of AMU states. The main contributions include extending AMU existence to unbounded operators in position–momentum systems and angular momentum, and showing that AMU states do not imply a classical commuting limit as $|\hbar|\to0$, thereby clarifying the quantum-classical boundary. The results provide a rigorous framework linking synthetic spectrum, bounded transforms, and AMU existence, with implications for understanding classical limits and nearly-commuting observables in quantum mechanics.

Abstract

We show that Mumford's Approximately Macroscopically Unique (AMU) states exist for quantum systems consisting of unbounded self-adjoint operators when the commutators are small. In particular, AMU states always exist in position and momentum systems when the Planck constant $|\hbar|$ is sufficiently small. However, we show that these standard quantum mechanical systems are far away from classical mechanical (commutative) systems even when $|\hbar|\to 0.$

Approximately Macroscopically Unique States and Quantum Mechanics

TL;DR

The paper addresses Mumford's Approximately Macroscopically Unique (AMU) states for macroscopic observables represented by (potentially unbounded) self-adjoint operators with small commutators. It develops an almost commutative one-point compactification and bounded-transform techniques to translate unbounded tuples into a C*-algebraic setting, enabling synthetic-spectrum methods to guarantee the existence of AMU states. The main contributions include extending AMU existence to unbounded operators in position–momentum systems and angular momentum, and showing that AMU states do not imply a classical commuting limit as , thereby clarifying the quantum-classical boundary. The results provide a rigorous framework linking synthetic spectrum, bounded transforms, and AMU existence, with implications for understanding classical limits and nearly-commuting observables in quantum mechanics.

Abstract

We show that Mumford's Approximately Macroscopically Unique (AMU) states exist for quantum systems consisting of unbounded self-adjoint operators when the commutators are small. In particular, AMU states always exist in position and momentum systems when the Planck constant is sufficiently small. However, we show that these standard quantum mechanical systems are far away from classical mechanical (commutative) systems even when
Paper Structure (7 sections, 33 theorems, 373 equations)

This paper contains 7 sections, 33 theorems, 373 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Unbounded $n$-tuples of self-adjoint operators and almost $n$-spheres
  4. Synthetic spectrum
  5. AMU States
  6. Synthetic Spectrum revisited
  7. Momentum and position operators

Key Result

Theorem 1.1

Let $n\in\mathbb{N}$, $\epsilon>0$ and $M>1.$ There exists $0<\eta<\epsilon/4$ (depending only on $\epsilon$ and $M$) and $\delta({n},\eta, M, \epsilon)>0$ satisfying the following: Let $\{h_1,h_2,...,h_n\}$ be an $n$-tuple of (unbounded) self-adjoint operators densely defined on an infinite dimensi Then, for any $\lambda=(\lambda_1, \lambda_2,...,\lambda_n)\in s{\rm Sp}_M^{\eta}((h_1, h_2,...,h_n

Theorems & Definitions (89)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 79 more