Covariant Phase Space Formalism

• Covariant Phase Space (CPS) formalism is defined as an infinite-dimensional space of classical solutions to field equations, preserving full spacetime covariance.
• It utilizes a multisymplectic approach with a multidimensional Legendre transform to introduce conjugate multimomenta, enabling robust analysis of boundary terms and conserved charges.
• By incorporating the Peierls bracket and covariant symplectic structure, CPS forms a foundation for the quantization of gauge and gravitational theories while addressing phenomena like black hole entropy.

The covariant phase space (CPS) formalism is a framework for the Hamiltonian analysis of classical field theories that preserves full covariance with respect to spacetime symmetries. It addresses the core incompatibility between the classical Hamiltonian (canonical) approach—which requires a split into “space” and “time”—and the demands of relativistic or diffeomorphism-invariant dynamics, providing a geometric, coordinate-independent structure for the quantization and analysis of both matter and gauge fields, including gravitation.

1. Covariant Phase Space and the Multisymplectic Approach

The CPS formalism is rooted in the generalization of classical symplectic geometry to field theories, replacing the finite-dimensional phase space of mechanics with an infinite-dimensional space of solutions $\mathcal{E}$ to the field equations. Rather than isolating a preferred time direction, the multisymplectic approach treats all spacetime coordinates on an equal footing, starting with a Lagrangian density $\mathcal{L}[\phi] = \int L(x, y, dy)\, \beta$ on configuration bundle $Y$ over spacetime $X$ (Hélein, 2011).

A multidimensional Legendre transformation introduces conjugate multimomenta $p^\mu_i$ for the spacetime derivatives $\partial_\mu y^i$ and an energy variable $e$. The multisymplectic manifold $\mathcal{M}$ is locally described by $(x^\mu, y^i, e, p^\mu_i)$ with the canonical $(n+1)$-form

$$\omega = de \wedge \beta + dp^\mu_i \wedge dy^i \wedge \beta_\mu,$$

where $\beta_\mu$ are contractions of coordinate vectors $\partial/\partial x^\mu$ with the volume form $\beta$.

The Hamiltonian function is given by $\mathcal{J}(x, y, e, p) = e + H(x, y, p)$, and the field equations are defined by demanding that an $n$-dimensional Hamiltonian submanifold (the “Hamiltonian $n$-curve” $\Gamma$) satisfies

$$\omega(\xi, X) = d\mathcal{J}(\xi)\, \beta(X) \quad \forall\, \xi,$$

plus the independence condition $\beta|_\Gamma \neq 0$.

2. The Structure of Covariant Phase Space

The covariant phase space $\mathcal{E}$ is the space of all classical solutions to the equations of motion—i.e., all admissible Hamiltonian $n$-curves in $\mathcal{M}$. The central insight is that the multisymplectic structure of $\mathcal{M}$ induces a natural (pre)symplectic structure on $\mathcal{E}$, yielding an infinite-dimensional, covariant symplectic manifold.

For any $\Gamma \in \mathcal{E}$ and infinitesimal variation $\delta\Gamma$ represented by a tangent vector field $\xi$ along $\Gamma$, the canonical one-form is constructed as

$$\Theta^\Sigma_\Gamma(\delta\Gamma) = \int_{\Sigma \cap \Gamma} \xi \mathbin{\lrcorner}\, \theta,$$

where $\theta$ is an $n$-form such that $d\theta = \omega$, and $\Sigma$ is a spacelike Cauchy hypersurface (Hélein, 2011, Khavkine, 2014).

The symplectic two-form on phase space is obtained as the exterior derivative

$\Omega = \delta \Theta^\Sigma,$

which is independent of the choice of Cauchy slice (provided appropriate fall-off conditions), ensuring full spacetime covariance. Explicitly, in local coordinates,

$$\Omega_\Gamma(\delta_1\Gamma, \delta_2\Gamma) = \int_{\Sigma \cap \Gamma} \xi_1 \wedge \xi_2\, \omega.$$

3. Covariant Brackets, Observables, and Quantization

The traditional canonical Poisson bracket is not generally available in the covariant phase space, particularly for field variables corresponding to forms of differing degree. Instead, the Peierls bracket naturally arises as the covariant analog (Sharan, 2012, Khavkine, 2014). For observable functionals $A$ and $B$, their bracket measures the “backreaction” of deforming the action by $B$ on $A$ (and vice versa), focusing on the causal propagation of variations in field configurations.

Given a small perturbation of the Hamiltonian $H \to H + \epsilon B$, the induced vector field $X_B$ on the extended phase space generates the corresponding change in an observable $A$ via the Lie derivative $L_{X_B}A$. The antisymmetrized difference defines the Peierls bracket:

$[A,B] = \delta_B A - \delta_A B.$

For linear theories, this reduces to field commutators aligned with causal Green's functions (e.g., the Pauli–Jordan function in scalar field theory). The Peierls bracket coincides with the Poisson bracket constructed from the CPS symplectic form under sufficient regularity (contractibility of the field bundle, hyperbolizability of the equations) (Khavkine, 2014, Harlow et al., 2019).

In the context of quantization, the covariant symplectic structure and Peierls bracket jointly provide the geometric substrate for canonical quantization and construction of the commutator algebra of quantum observables in a manifestly covariant manner.

4. Boundary Terms, Relative Frameworks, and Ambiguities

The presence of boundaries—spatial, temporal, or null—requires a systematic treatment of total derivative terms and their contribution to the action, symplectic structure, and charges. Modern treatments employ the relative bicomplex framework (Margalef-Bentabol et al., 2020) or algorithmic prescription for incorporating boundary terms (Harlow et al., 2019):

• The variational principle is formulated for an action expressed as a pair $(L, \bar\ell)$, where $L$ is the bulk Lagrangian $n$-form and $\bar\ell$ is an $(n-1)$-form boundary Lagrangian.
• The variation yields (relative) Euler–Lagrange equations in the bulk and on the boundary.
• The presymplectic potential and current are constructed to absorb ambiguities from total derivatives, and the (pre)symplectic form is defined as

$$\Omega^{\imath}_S = \int_{(\Sigma,\partial\Sigma)} \underline{\mathbf{d}}\,\underline{\imath}^*(\Theta^L, \overline{\theta}^{(L,\overline{\ell})})$$

on solutions $\phi$ satisfying both bulk and boundary field equations.

Algorithmic boundary prescriptions produce the necessary additional terms in the Hamiltonian or Noether charges (e.g., the $C$ term in $(\Theta + \delta\ell)|_\Gamma = dC$ or the corner contributions in Lovelock gravity (Neri et al., 25 Apr 2024)), ensuring a well-posed variational principle and finite, conserved charges even for higher-derivative or higher-curvature theories.

Boundary ambiguities in the symplectic structure are resolved by constructing contour integrals around the region of interest, as shown via the path-integral definition of the symplectic structure, eliminating ambiguities due to addition of exact forms (Kirklin, 2019).

5. Historical Evolution and Principal Contributors

The development of the covariant phase space formalism synthesizes concepts traced through several historical strands (Hélein, 2011, Khavkine, 2014):

• Originating in the multisymplectic and De Donder–Weyl approaches (Jacobi, Clebsch, Volterra, Weyl, De Donder) for classical variational problems.
• The foundational realization by Poincaré, Cartan, and later Peierls that the space of solutions naturally admits a (pre)symplectic or Poisson structure independent of a privileged time coordinate.
• Modern rigorous formulations and practical computations were established by Crnković, Witten, Zuckerman, Lee, and Wald, focusing especially on gravity and gauge theories.
• Kijowski–Szczyrba provided a systematic construction of the covariant phase space via multisymplectic geometry, clarifying the connection to observables and Poisson brackets.
• Forger–Romero broadened the scope to explicitly include constrained and gauge-invariant systems and provided mathematical criteria (such as “hyperbolizability”) for well-defined symplectic structures.
• Vitagliano extended the geometry to higher-order theories and clarified the Hamilton–Jacobi structure in a covariant setting.

A diverse set of recent work has focused on resolving challenges for boundary contributions, extending the formalism to corners, fluctuating boundaries, and the treatment of entanglement and subregion factorization (Kirklin, 2019, Margalef-Bentabol et al., 2020, Adami et al., 3 Jul 2024).

6. Significance for Relativistic Dynamics, Gauge Theories, and Quantum Gravity

The CPS framework reconciles the need for Hamiltonian methods with the core principle of relativistic covariance. It provides:

• A canonical structure for field theory without breaking spacetime symmetries, essential for the quantization of gauge and gravitational systems.
• Geometric definitions of observables, conserved charges, and Noether currents, applicable to general backgrounds, including curved spacetimes.
• A precise apparatus for encoding edge modes, memory effects, and microphysical “soft hair” degrees of freedom in gauge and gravity theories, as seen in applications to black hole entropy, soft theorems, and the Kerr/CFT correspondence (Setare et al., 2021).
• Tools to unambiguously analyze thermodynamic laws, including the first law and Smarr formulas for black holes, through the inclusion of all relevant boundary and corner terms (Kim et al., 2023, Neri et al., 25 Apr 2024).

The flexibility and generality of this approach have led to wide applications in gauge theory, gravitation, noncommutative geometry, and the statistical mechanics of deformed or constrained phase spaces.

7. Core Mathematical Formulas

Key equations (cf. (Hélein, 2011)):

Quantity	Formula	Remarks
Multisymplectic $(n+1)$-form	$$\displaystyle \omega = de \wedge \beta + dp_i^\mu \wedge dy^i \wedge \beta_\mu$$	(1)
Covariant Hamilton’s eq. (geometry)	$$\displaystyle \omega(\xi, X) = d\mathcal{J}(\xi)\, \beta(X)$$	(2)
CPS 1-form on solutions	$$\displaystyle \Theta^\Sigma(\delta \Gamma) = \int_{\Sigma\cap\Gamma} \xi \mathbin{\lrcorner}\, \theta$$	(3)
Symplectic form on CPS	$$\displaystyle \Omega(\delta_1\Gamma, \delta_2\Gamma) = \int_{\Sigma\cap\Gamma} \xi_1 \wedge \xi_2\, \omega$$	(6)
Peierls bracket	$[A, B] = \delta_B A - \delta_A B$	Covariant

All these ingredients encode the structure by which the CPS formalism both generalizes and unifies symplectic, Hamiltonian, and Lagrangian approaches to classical and quantum field theories in a way that maintains full compatibility with the principles of relativity and gauge invariance.