[Paper Review] On Types of Observables in Constrained Theories

Edward Anderson|arXiv (Cornell University)|Apr 19, 2016

Noncommutative and Quantum Gravity Theories17 references20 citations

TL;DR

This paper introduces A-observables—a generalized framework for observables in constrained Hamiltonian theories—defined via closed algebraic substructures of constraints. It shows that Dirac, Kuchař, and unconstrained observables are special cases within a unified lattice-theoretic structure, resolving ambiguities in observable definitions across diverse physical theories like supergravity and relational mechanics.

ABSTRACT

The Kuchar observables notion is shown to apply only to a limited range of theories. Relational mechanics, slightly inhomogeneous cosmology and supergravity are used as examples that require further notions of observables. A suitably general notion of A-observables is then given to cover all of these cases. `A' here stands for `algebraic substructure'; A-observables can be defined by association with each closed algebraic substructure of a theory's constraints. Both constrained algebraic structures and associated notions of A-observables form bounded lattices.

Motivation & Objective

• To address the limitations of existing observable notions—particularly Kuchař observables—in covering all physically relevant quantities in constrained theories.
• To resolve conceptual and technical ambiguities in defining observables when constraints are not purely linear or quadratic.
• To provide a unified mathematical framework that generalizes observable types across diverse physical systems, including relational mechanics and supergravity.
• To establish that observables and their associated constraint algebraic substructures form bounded lattices, enabling systematic classification and analysis.
• To clarify the physical content of observables by linking them to algebraic substructures of the constraint algebra, ensuring only physical information is retained.

Proposed method

• Defining A-observables as quantities that weakly commute with a closed algebraic substructure of constraints, generalizing Dirac and Kuchař observables.
• Using the Poisson bracket to formalize the condition $\{\mathcal{F}_{\text{lin}}, O\} \approx 0$, where $\mathcal{F}_{\text{lin}}$ is a subalgebra of first-class linear constraints.
• Introducing the association map $\text{Assoc}$, which maps constraint subalgebras to corresponding observable subalgebras, and showing it is an order-reversing lattice morphism.
• Representing the hierarchy of constraint and observable subalgebras via Hasse diagrams and posets, with join and meet operations defining lattice structure.
• Demonstrating that both the constraint algebraic substructures and the resulting A-observables form bounded lattices, with 0 (trivial) and 1 (full algebra) elements.
• Applying the framework to examples such as relational mechanics, slightly inhomogeneous cosmology, and supergravity to show its generality beyond standard Kuchař or Dirac cases.

Experimental results

Research questions

• RQ1 How can a unified framework be constructed to generalize observable notions beyond the limitations of Kuchař observables in constrained Hamiltonian theories?
• RQ2 What is the mathematical structure underlying the hierarchy of observables defined by subalgebras of constraints, and how does it relate to physical content?
• RQ3 In what sense do Dirac, Kuchař, and unconstrained observables emerge as special cases within a broader class of A-observables?
• RQ4 How does the association between constraint subalgebras and observable subalgebras preserve or coarse-grain ordering information, and when is this map injective?
• RQ5 Can the lattice-theoretic structure of A-observables be used to systematically classify and analyze physical quantities in complex theories like supergravity or relational mechanics?

Key findings

• A-observables are defined as quantities that weakly commute with a closed algebraic substructure of constraints, generalizing Dirac and Kuchař observables.
• The set of all A-observables forms a bounded lattice, with the unconstrained observables as the top element and Dirac observables as the bottom element.
• The association map $\text{Assoc}$ from constraint subalgebras to observable subalgebras is an order-reversing lattice morphism, reflecting that adding constraints reduces observable freedom.
• In cases like conformal relational mechanics, $\text{Assoc}$ can be many-to-one, indicating that multiple constraint subalgebras may yield the same observable algebra, thus introducing coarse-graining.
• For most physical systems, including GR and Electromagnetism, $\text{Assoc}$ is injective and fully preserves the lattice structure, ensuring a one-to-one correspondence between constraint and observable subalgebras.
• The framework successfully covers examples such as relational mechanics and supergravity, where standard Kuchař or Dirac observables are insufficient or ill-defined, demonstrating its broader applicability.