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Existence of Approximately Macroscopically Unique States

Huaxin Lin

Abstract

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the C*-algebra of bounded operators on $H.$ Suppose that $T_1,T_2,..., T_n$ are self-adjoint operators in $B(H).$ We show that, if commutators $[T_i, T_j]$ are sufficiently small in norm, then ``Approximately Macroscopically Unique" states always exist for any values in a synthetic spectrum of the $n$-tuple of self-adjoint operators. This is achieved under the circumstance for which the $n$-tuple may not be approximated by commuting ones. This answers a question proposed by David Mumford for measurements in quantum theory. If commutators are not small in norm but small modulo compact operators, then ``Approximate Macroscopic Uniqueness" states also exist.

Existence of Approximately Macroscopically Unique States

Abstract

Let be an infinite dimensional separable Hilbert space and the C*-algebra of bounded operators on Suppose that are self-adjoint operators in We show that, if commutators are sufficiently small in norm, then ``Approximately Macroscopically Unique" states always exist for any values in a synthetic spectrum of the -tuple of self-adjoint operators. This is achieved under the circumstance for which the -tuple may not be approximated by commuting ones. This answers a question proposed by David Mumford for measurements in quantum theory. If commutators are not small in norm but small modulo compact operators, then ``Approximate Macroscopic Uniqueness" states also exist.
Paper Structure (4 sections, 9 theorems, 76 equations)

This paper contains 4 sections, 9 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Synthetic spectra
  3. The case of small commutators
  4. The case of compact commutators

Key Result

Theorem 1.4

Let $n\in \mathbb{N}$ and $\epsilon>0.$ There exists $\delta(n, \epsilon)>0$ satisfying the following: Suppose that $H$ is an infinite dimensional separable Hilbert space and $T_1, T_2,...,T_n\in B(H)$ are self-adjoint operators with $\|T_j\|\le 1$ ($1\le j\le n$) such that Then, for any $\lambda=(\lambda_1, \lambda_2,...,\lambda_n)\in s{\rm Sp}^{\epsilon/4}((T_1,T_2,...,T_n)),$ there exists $v\i

Theorems & Definitions (26)

  • Definition 1.1
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 16 more