[Paper Review] Constraints and gauge transformations: Dirac's theorem is not always valid

Julian Barbour, Brendan Foster|ArXiv.org|Aug 8, 2008

Noncommutative and Quantum Gravity Theories6 references64 citations

TL;DR

This paper challenges the foundational assumption in canonical quantum gravity that the Hamiltonian constraint generates only gauge transformations, not physical change. By analyzing reparametrization-invariant systems like Jacobi's principle, the authors show Dirac's theorem—on which this belief rests—fails when absolute time is not assumed, proving the Hamiltonian constraint can generate genuine physical evolution.

ABSTRACT

A standard tenet of canonical quantum gravity is that evolution generated by a Hamiltonian constraint is just a gauge transformation on the phase space and therefore does not change the physical state. The basis for this belief is a theorem of Dirac that identifies primary first-class constraints as generators of physically irrelevant motions. We point out that certain assumptions on which Dirac based his argument do not hold for reparametrization invariant systems, and show that the primary Hamiltonian constraint of these systems does generate physical motion. We show explicitly how the argument fails for systems described by Jacobi's principle, which has a structure closely resembling that of general relativity. We defer discussion of general relativity and the implications for quantum gravity to a later paper.

Motivation & Objective

• To challenge the widespread belief in canonical quantum gravity that the Hamiltonian constraint generates only gauge transformations.
• To identify the flaw in Dirac’s original argument, which assumes an absolute time parameter for Hamiltonian evolution.
• To demonstrate that in reparametrization-invariant systems—such as those governed by Jacobi’s principle—the Hamiltonian constraint generates physical, not just gauge, changes.
• To argue that constraints must be evaluated on a case-by-case basis, based on their physical origin, rather than assuming all first-class constraints are gauge generators.
• To lay the groundwork for rethinking the role of the Hamiltonian constraint in general relativity and quantum gravity

Proposed method

• Analyzes the structure of reparametrization-invariant (RI) systems, particularly Jacobi’s principle, which shares the mathematical form of general relativity’s action.
• Identifies that Dirac’s proof of primary first-class constraints as gauge generators relies on the assumption of an absolute time parameter, which does not hold in RI systems.
• Completes the missing explicit calculation in Dirac’s argument for RI systems, showing that the canonical evolution generated by the Hamiltonian constraint leads to physically distinct states.
• Contrasts the behavior of the Hamiltonian constraint in RI systems with standard gauge theories (e.g., electrodynamics), where the Gauss constraint truly generates gauge transformations.
• Uses the fact that in RI systems, the Hamiltonian vanishes identically (H = 0), yet the evolution is not trivial, indicating physical content.
• Demonstrates that the appearance of arbitrary functions in solutions is not due to gauge freedom per se, but due to the underlying physical constraint of initial data being a point and direction in configuration space

Experimental results

Research questions

• RQ1 Does Dirac’s theorem on primary first-class constraints as gauge generators hold for reparametrization-invariant systems like those in general relativity?
• RQ2 What is the physical role of the Hamiltonian constraint in systems where time is not absolute, such as in Jacobi’s principle?
• RQ3 Why do constraints in reparametrization-invariant systems generate physical changes despite being first-class and primary?
• RQ4 How does the absence of absolute time invalidate Dirac’s original proof that such constraints generate only gauge transformations?
• RQ5 Can the Hamiltonian constraint in general relativity be both a gauge generator and a generator of physical evolution?

Key findings

• Dirac’s proof that primary first-class constraints generate gauge transformations fails in reparametrization-invariant systems due to the absence of absolute time.
• The Hamiltonian constraint in Jacobi’s principle—structurally analogous to general relativity—generates physical evolution, not just gauge transformations.
• The appearance of arbitrary functions in solutions to constrained systems is not inherently a sign of gauge freedom but arises from the physical specification of initial position and direction.
• Not all first-class constraints generate unphysical motions; their physical significance must be assessed based on the underlying dynamics, not just their algebraic classification.
• The Hamiltonian constraint in general relativity may be partly a generator of physical change, challenging the standard orthodoxy in quantum gravity.
• Observables in quantum gravity do not need to be 'perennials' (i.e., operators commuting with the Hamiltonian constraint), especially in simpler reparametrization-invariant systems.