Fast Poisson brackets and constraint algebras in canonical gravity

TL;DR

This work introduces Hamilcar, a Mathematica/xAct-based package that automates Poisson-bracket calculations and the reconstruction of constraint algebras in canonical gravity. It demonstrates the workflow on pure GR, a pure $R^2$ theory, and GR at two loops with order reduction, deriving the Dirac–Bergmann constraint structure, first-/second-class counts, and propagating degrees of freedom. The tool enables automated verification of the GR Dirac algebra (with $N_{\text{First}}=4$, $N_{\text{Second}}=0$, and $N_{\text{phy}}=2$) and exposes how higher-derivative theories introduce additional constraints and modes (e.g., $N_{\text{phy}}=3$ for $R^2$) or deformed algebras under order reduction. By providing automated manipulation through smearing, integration-by-parts, and dimensional identities, the package offers a practical route to test modified gravity theories and their quantum-corrected Hamiltonians, with immediate relevance to phenomenology.

Abstract

In the study of alternative or extended theories of gravity, Dirac's Hamiltonian constraint algorithm is invaluable for enumerating the propagating modes and gauge symmetries. For gravity, this canonical approach is frequently applied as a means for finding pathologies such as strongly coupled modes; more generally it facilitates the reconstruction of gauge symmetries and the quantization of gauge theories. For gravity, however, the algorithm can become notoriously arduous to implement. We present a simple computer algebra package for efficiently computing Poisson brackets and reconstructing constraint algebras. The tools are stress-tested against pure general relativity and modified gravity, including the order reduction of general relativity at two loops.

Figures (2)

• Figure 1: The canonical formulation of a field theory such as general relativity allows the propagation of initial data to subsequent foliations by means of first-order equations for phase-space variables ($\MetricFoliation{_{ij}}$ and $\ConjugateMomentumMetricFoliation{^{ij}}$), subject to additional constraints ($\SuperConstraint{}$ and $\SuperConstraint{_i}$) and a gauge choice to fix the values of undetermined Lagrange multipliers ($\Lapse$ and $\Shift{^i}$) on each foliation.
• Figure 2: Essential diagrams contributing to the two-loop divergence of pure GR in the background field approach, once the internal lines are carefully treated Goroff:1985sz.