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Partial and Complete Observables for Hamiltonian Constrained Systems

B. Dittrich

TL;DR

This work develops a general framework to construct Dirac observables in Hamiltonian systems with first-class constraints by deploying partial and complete observables, extending Rovelli’s approach to arbitrary numbers of constraints. It derives a system of first-order PDEs that governs complete observables F_{[f,T_i]}( au_i,x) as functions of clock parameters and shows that, under weak abelianization, the clock flows become simpler and the complete observables remain gauge-invariant. The paper connects the finite-dimensional construction to Kuchař’s Bubble Time formalism and demonstrates how gauge fixing via clock variables leads to Dirac brackets and a gauge action on the observable algebra, with extensions to field theories through functional PDEs and field-model examples like Einstein–Rosen waves. It also investigates partially invariant observables and discusses potential paths for quantization and perturbative analyses, highlighting both the practicality and limitations of the approach in complex constrained systems.

Abstract

We will pick up the concepts of partial and complete observables introduced by Rovelli in order to construct Dirac observables in gauge systems. We will generalize these ideas to an arbitrary number of gauge degrees of freedom. Different methods to calculate such Dirac observables are developed. For background independent field theories we will show that partial and complete observables can be related to Kuchař's Bubble Time Formalism. Moreover one can define a non-trivial gauge action on the space of complete observables and also state the Poisson brackets of these functions. Additionally we will investigate, whether it is possible to calculate Dirac observables starting with partially invariant partial observables, for instance functions, which are invariant under the spatial diffeomorphism group.

Partial and Complete Observables for Hamiltonian Constrained Systems

TL;DR

This work develops a general framework to construct Dirac observables in Hamiltonian systems with first-class constraints by deploying partial and complete observables, extending Rovelli’s approach to arbitrary numbers of constraints. It derives a system of first-order PDEs that governs complete observables F_{[f,T_i]}( au_i,x) as functions of clock parameters and shows that, under weak abelianization, the clock flows become simpler and the complete observables remain gauge-invariant. The paper connects the finite-dimensional construction to Kuchař’s Bubble Time formalism and demonstrates how gauge fixing via clock variables leads to Dirac brackets and a gauge action on the observable algebra, with extensions to field theories through functional PDEs and field-model examples like Einstein–Rosen waves. It also investigates partially invariant observables and discusses potential paths for quantization and perturbative analyses, highlighting both the practicality and limitations of the approach in complex constrained systems.

Abstract

We will pick up the concepts of partial and complete observables introduced by Rovelli in order to construct Dirac observables in gauge systems. We will generalize these ideas to an arbitrary number of gauge degrees of freedom. Different methods to calculate such Dirac observables are developed. For background independent field theories we will show that partial and complete observables can be related to Kuchař's Bubble Time Formalism. Moreover one can define a non-trivial gauge action on the space of complete observables and also state the Poisson brackets of these functions. Additionally we will investigate, whether it is possible to calculate Dirac observables starting with partially invariant partial observables, for instance functions, which are invariant under the spatial diffeomorphism group.

Paper Structure

This paper contains 11 sections, 2 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. Complete and Partial Observables for Systems with one Constraint
  4. Complete and Partial Observables for Systems with Several Constraints
  5. A System of Partial Differential Equations for Complete Observables
  6. Weak Abelianization of the Constraint Algebra
  7. Bubble Time Formalism
  8. Gauge Fixing and Dirac Brackets
  9. Partially Invariant Partial Observables
  10. Field Theories
  11. Discussion

Key Result

Theorem 3.1

Let $f,T$ be two phase space functions and $x \in \mathcal{M}$ a phase space point, fulfilling the condition: $\alpha_C^t(f)(x)= \alpha_C^s(f)(x)$ for all values $s,t \in {\hbox{$\rm R$}}$ for which $\alpha_C^t(T)(x)= \alpha_C^s(T)(x)$. Then $F_{[f,T]}(\tau,x)$ is invariant under the flow generated

Theorems & Definitions (2)

  • Theorem 3.1
  • Theorem 4.1