Canonical Formalism of Gravity

• Canonical Formalism of Gravity is a Hamiltonian framework that recasts gravitational theories as constrained dynamical systems, clarifying gauge structures and degrees of freedom.
• It employs methods such as ADM decomposition, Ostrogradsky constructions, and systematic constraint classifications to analyze models from GR to bigravity and higher-order theories.
• The framework underpins canonical quantization and offers critical insights into modified gravity and discrete approaches, ensuring consistency in gauge and geometric formulations.

The canonical formalism of gravity is a Hamiltonian framework that recasts general and modified theories of gravity as constrained dynamical systems on phase space. This approach provides a foundation for canonical quantization, clarifies the gauge structure and degrees of freedom for various gravity models, and is fundamental both in continuum metric-based general relativity and in extensions such as bigravity, higher-derivative gravity, and discrete (simplicial) gravity. The canonical approach exposes the full set of first-class and second-class constraints, their algebra, and their geometric and physical content.

1. Foundational Structure: ADM Formalism and General Relativity

The canonical (Hamiltonian) formulation of general relativity—initiated by Arnowitt, Deser, and Misner (ADM)—starts from a spacetime manifold foliated into a family of spatial Cauchy hypersurfaces $S$ parametrized by time $t$. The spacetime metric $g_{\mu\nu}$ is decomposed as

$$ds^2 = -N^2 dt^2 + h_{ij}(dx^i + N^i dt)(dx^j + N^j dt),$$

where $h_{ij}$ is the induced metric on $S$, $N$ is the lapse, and $N^i$ is the shift vector. The canonical phase space consists of the pair $(h_{ij}(x), \pi^{ij}(x))$, where $\pi^{ij}$ is the momentum conjugate to $h_{ij}$, related to the extrinsic curvature $K_{ij}$.

The Hamiltonian is a linear combination of first-class constraints: $$H[N, N^i] = \int_S d^3x \left( N(x)\mathcal{H}(x) + N^i(x)\mathcal{H}_i(x) \right),$$ with the local constraint densities

$$\mathcal{H}_i = -2 \nabla_j \pi^j{}_i, \qquad \mathcal{H} = \frac{1}{\sqrt{h}}\Bigl(\pi^{ij}\pi_{ij}-\frac{1}{2}\pi^2\Bigr) - \sqrt{h}R^{(3)},$$

where $R^{(3)}$ is the Ricci scalar of $h_{ij}$. These encode spatial diffeomorphism symmetry and the Hamiltonian (Wheeler–DeWitt) constraint. All constraints are first-class and their Dirac algebra reflects the four-dimensional diffeomorphism invariance (Witten, 2022).

2. Constraints, Degrees of Freedom, and Algebraic Structures

In the canonical setting, constraints arise due to gauge invariances of the action (diffeomorphism and, for tetrad or Plebanski actions, internal gauge symmetries). These constraints are classified as:

• First-class constraints: Generate gauge transformations, reduce the number of physical degrees of freedom, and close under the Poisson bracket.
• Second-class constraints: Do not generate symmetries; they are imposed strongly via Dirac brackets, further reducing the phase space.

In general relativity, there are four first-class constraints per space point (three momentum and one Hamiltonian constraint), leading to two local configuration degrees of freedom per point—the graviton polarizations (Soloviev, 2020, Witten, 2022). The canonical algebra among smeared constraints reproduces the Dirac (hypersurface-deformation) algebra, encoding the embeddability of the spatial slices in a spacetime and ensuring full covariance (Deng et al., 4 Dec 2025).

Constraint Algebra in Bigravity

In ghost-free bigravity, the canonical analysis yields a more intricate set of constraints due to the doubled metric (or tetrad) content. Only the diagonal subgroup of diffeomorphisms remains as a gauge symmetry, and there is a set of ten second-class constraints removing the Boulware–Deser ghost and fixing relative tetrad symmetry. The remaining first-class constraints and their algebra closely mimic the structure of GR, with modifications from the bigravity potential (Soloviev, 2020).

3. Extension to Modified and Higher-Order Gravity

Higher-Order (Quadratic and $f(R)$) Gravity

Canonical formulations of higher-derivative gravity, such as $f(R)$ theories and quadratic gravity (both $R^2$ and Weyl-squared terms), require sophisticated Ostrogradsky-type constructions. To maintain general covariance and avoid promoting lapse and shift to dynamical variables, time derivatives are replaced with Lie derivatives along the hypersurface normal (Ezawa et al., 2013, Ohkuwa et al., 2014). Additional configuration variables (e.g., $\mathcal{L}_n h_{ij}$) and their conjugate momenta are introduced. The extended phase space carries new (mostly trace) degrees of freedom corresponding to higher-curvature modes (e.g., the scalaron in $f(R)$ gravity). The constraint algebra remains first-class and isomorphic to that of GR, and the number of local degrees of freedom increases according to the higher-order nature of the Lagrangian (Oda, 14 May 2025, Ohkuwa et al., 2014).

Canonical Formalism for Bigravity

In bigravity, the canonical phase space consists of two sets of tetrad variables and an auxiliary Lorentz boost, with Lagrange multipliers enforcing primary constraints. The Hamiltonian comprises contributions from both sectors plus interaction potential, and the full constraint analysis must address removal of the ghost via second-class constraints, preserving the first-class algebra for the diagonal gauge symmetries (Soloviev, 2020).

Deformed Bracket and Non-Standard Constructions

Generalized uncertainty principle (GUP)–motivated deformations of the canonical Poisson brackets require constructing a Hamiltonian (with modified structure functions) that closes under the new bracket and still leads to consistent covariant field equations. The canonical formalism can always be restored by identifying appropriate “shape functions” such that the algebra of constraints remains first-class and the correct metric is reproduced (Gingrich, 25 Nov 2025).

4. Hamiltonian Formalism for Embedding and Alternative Variables

Canonical gravity can be formulated using variables arising from embeddings of spacetime into higher-dimensional flat spaces. In embedding gravity, the canonical variables are the embedding functions $y^A(x^\mu)$ and their conjugate momenta. Primary constraints enforce that the momenta have no directions along the embeddings’ tangents, and additional constraints guarantee equivalence with general relativity (the “Einsteinian” constraints). The first-class constraint algebra replicates the Dirac algebra upon proper gauge fixing and solving of second-class constraints (0711.0576, Paston et al., 2015, 2207.13654).

Unimodular gravity yields further canonical refinements (e.g., Hamiltonian density not constrained to vanish, but to be constant), and the embedding provides natural “global clock” variables, potentially addressing the “problem of time” inherent in canonical quantization of GR (Gielen et al., 14 Nov 2024, Paston et al., 2015).

5. Canonical Structure in Discrete and Simplicial Gravity

Discrete gravity models, as in Regge calculus, can be endowed with a fully canonical formalism by treating the action of elementary simplicial “time steps” (Pachner moves) as discrete generating functions. The canonical variables are edge lengths and their conjugate discrete momenta (exterior angles). The dimension of the phase space is adjusted dynamically as moves introduce or remove edges, and pre- and post-constraints enforce the equivalence with the covariant equations. Although the precise Dirac algebra is replaced by its discrete analog, the continuum limit recovers the usual ADM constraint structure (Hoehn, 2011, Dittrich et al., 2011).

Approach	Canonical Variables	Constraints
Metric/ADM GR	$h_{ij},\,\pi^{ij}$	1 Hamiltonian, 3 diffeo
Bigravity (tetrad)	$e^a_i,\bar e^a_i, p_a, P^a$	7 first-class, 10 second
$f(R)/$higher-order	$h_{ij},Q_{ij},\pi^{ij},P^{ij}$	Extended, mostly first-class
Embedding	$y^A,\pi_A$	8 (or 7 with time gauge)
Simplicial/Regge	$l_e, p_e$ (edge, angle)	Pre/post-projective

6. Quantization and Gauge-Fixing

The canonical formalism provides a framework for canonical quantization and path-integral quantization via gauge-fixed measures. In the asymptotically AdS context, the Hilbert space of quantum gravity is constructed by gauge-fixing the conformal mode (constant-mean-curvature slicing), imposing BRST symmetry, and reducing to the cotangent bundle over equivalence classes of conformal metrics. The constraint equations are implemented as operator equations on wavefunctionals, effecting the Wheeler–DeWitt approach (Witten, 2022).

In higher-derivative and gauge-fixed settings, additional (BRST or secondary) symmetries are carefully incorporated, and reality conditions are crucial in complex formulations (e.g., Plebanski, Ashtekar variables in Lorentzian signature) (Oda, 14 May 2025, Gielen et al., 14 Nov 2024).

7. Physical Implications and Model Comparisons

The canonical formalism exposes the true local and global physical degrees of freedom, the nature of gauge variables, and the algebraic consistency (absence of anomalies) in a range of gravity theories. For example:

• In bigravity, only the diagonal diffeomorphism survives as a gauge symmetry, and the theory has 7 physical polarizations: 2 massless, 5 massive (Soloviev, 2020).
• In $f(R)$ or quadratic gravity, an extra scalar mode is present, and the non-unitarity of higher-derivative ghosts arises explicitly (Oda, 14 May 2025, Ohkuwa et al., 2014).
• For unimodular and embedding models, the cosmological constant becomes a global integration constant rather than a coupling, and “unimodular time” provides an internal evolution parameter (Gielen et al., 14 Nov 2024, Paston et al., 2015).
• In discrete gravity, the multi-fingered time evolution, pre- and post-constraints, and variable phase-space dimension correspond naturally to the piecewise-linear geometry of the underlying spacetime (Dittrich et al., 2011, Hoehn, 2011).

The canonical formalism, in all cases, provides an indispensable foundation for both classical analysis and any attempt at quantization of gravitational theories, making the structure, dynamics, and gauge content transparent and mathematically precise.