[Paper Review] Superselection Rules from Measurement Theory

Shogo Tanimura|arXiv (Cornell University)|Dec 24, 2011

Quantum Mechanics and Applications29 references3 citations

TL;DR

This paper derives superselection rules from the symmetry properties of quantum measurement processes, showing that conserved quantities during measurement forbid the measurement of non-commuting observables via covariant indicators. It establishes that isolated conservation laws—such as charge or color symmetry—impose fundamental limits on measurability, while spontaneous symmetry breaking can circumvent these rules, offering a dynamical foundation for superselection rules in quantum theory.

ABSTRACT

In quantum theory, physically measurable quantities of a microscopic system are represented by self-adjoint operators. However, not all of the self-adjoint operators correspond to measurable quantities. The superselection rule is a criterion to distinguish measurable quantities. Any measurable quantity must obey the superselection rules. By contraposition, any quantity which does not obey the superselection rules cannot be measured. Although some of superselection rules were proved, the raizon d'être of the superselection rules has been still obscure. In this paper we deduce the superselection rules from an assumption on symmetry property of measurement process. We introduce the notion of covariant indicator, which is a macroscopic observable whose value indicates the value of a microscopic object observable. We prove that if an object system has a quantity that is conserved during the measurement process, other quantities that do not commute with the conserved quantity are non-measurable by the covariant indicator. Our derivation of superselection rules is compared with the uncertainty relation under the restriction by a conservation law. An implication of the color superselection rule for the color confinement is discussed. It is also argued that spontaneous symmetry breaking enables a measurement that the superselection rule prohibits.

Motivation & Objective

• To clarify the physical origin of superselection rules, which currently lack a fundamental justification despite their role in restricting measurable quantum observables.
• To establish a connection between measurement processes and superselection rules by analyzing the symmetry of interaction between microscopic systems and macroscopic measurement devices.
• To demonstrate that covariant indicators—measuring devices transforming consistently under symmetry—can only measure observables commuting with conserved quantities in the system.
• To explore how spontaneous symmetry breaking enables measurement of observables otherwise forbidden by superselection rules.
• To interpret superselection rules as emergent from isolation and symmetry in measurement, rather than as absolute principles.

Proposed method

• Introduces the concept of a 'covariant indicator'—a macroscopic observable that transforms in a way consistent with the symmetry of the microscopic system being measured.
• Defines a measurement process as one preserving an isolated conservation law, such as electric charge or color charge, ensuring the symmetry is maintained during interaction.
• Applies the Ozawa equality to relate measurement accuracy and disturbance, showing that non-commuting observables cannot be simultaneously measured under symmetry constraints.
• Uses the von Neumann measurement model to formalize the correlation between object system and apparatus, requiring covariance under symmetry group actions.
• Analyzes the role of symmetry in measurement by comparing conservation laws (e.g., angular momentum) with superselection rules, showing that only observables commuting with the conserved charge are measurable.
• Considers the implications of non-abelian gauge symmetries (e.g., color symmetry) and shows that the unobservability of colored particles arises from such symmetry constraints.

Experimental results

Research questions

• RQ1 What is the physical basis for superselection rules, and why do they restrict which self-adjoint operators can represent measurable quantities?
• RQ2 How does the symmetry of the measurement process—specifically, the invariance under a conservation law—affect the set of measurable observables?
• RQ3 Can a measurement process that preserves a symmetry still measure observables that do not commute with the conserved quantity?
• RQ4 Under what conditions can superselection rules be circumvented, and how does spontaneous symmetry breaking enable such measurements?
• RQ5 How do superselection rules emerge from the structure of quantum measurement, and what is their role in the emergence of classical behavior?

Key findings

• Superselection rules arise naturally from the requirement that measurement processes preserve an isolated conservation law, ensuring that the indicator transforms covariantly under the symmetry group.
• Any observable that does not commute with a conserved quantity cannot be measured by a covariant indicator, establishing a direct link between symmetry and measurability.
• The univalence superselection rule (for half-integer spin fields) and the electric charge superselection rule are derived from symmetry principles, not postulated a priori.
• Color confinement in quantum chromodynamics is interpreted as a consequence of the non-abelian color symmetry, where colored states are unobservable due to the same mechanism.
• Spontaneous symmetry breaking allows the construction of macroscopic systems (e.g., with external fields or interacting subsystems) that can measure observables otherwise forbidden by superselection rules.
• The class of measurable observables is not absolute but depends on the degree of isolation and symmetry of the measurement interaction, implying a hierarchy of micro-macro distinctions based on conserved quantities.