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The covariant phase space method of Iyer, Lee, Wald, and Zoupas gives an elegant way to understand the Hamiltonian dynamics of Lagrangian field theories without breaking covariance. The original literature however does not systematically treat total derivatives and boundary terms, which has led to some confusion about how exactly to apply the formalism in the presence of boundaries. In particular the original construction of the canonical Hamiltonian relies on the assumed existence of a certain boundary quantity $B$, whose physical interpretation has not been clear. We here give an algorithmic procedure for applying the covariant phase space formalism to field theories with spatial boundaries, from which the term in the Hamiltonian involving $B$ emerges naturally. Our procedure also produces an additional boundary term, which was not present in the original literature and which so far has only appeared implicitly in specific examples, and which is already nonvanishing even in general relativity with sufficiently permissive boundary conditions. The only requirement we impose is that at solutions of the equations of motion the action is stationary modulo future/past boundary terms under arbitrary variations obeying the spatial boundary conditions; from this the symplectic structure and the Hamiltonian for any diffeomorphism that preserves the theory are unambiguously constructed. We show in examples that the Hamiltonian so constructed agrees with previous results. We also show that the Poisson bracket on covariant phase space directly coincides with the Peierls bracket, without any need for non-covariant intermediate steps, and we discuss possible implications for the entropy of dynamical black hole horizons.
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By imposing the boundary condition associated with the boundary structure of the null boundaries rather than the usual one, we find that the key requirement in Harlow-Wus algorithm fails to be met in the whole covariant phase space. Instead, it can be satisfied in its submanifold with the null boundaries given by the expansion free and shear free hypersurfaces in Einsteins gravity, which can be regarded as the origin of the non-triviality of null boundaries in terms of Wald-Zoupass prescription. But nevertheless, by sticking to the variational principle as our guiding principle and adapting Harlow-Wus algorithm to the aforementioned submanifold, we successfully reproduce the Hamiltonians obtained previously by Wald-Zoupas prescription, where not only are we endowed with the expansion free and shear free null boundary as the natural stand point for the definition of the Hamiltonian in the whole covariant phase space, but also led naturally to the correct boundary term for such a definition.
We show that the phase space of three-dimensional gravity contains two layers of dualities: between diffeomorphisms and a notion of dual diffeomorphisms on the one hand, and between first order curvature and torsion on the other hand. This is most elegantly revealed and understood when studying the most general Lorentz-invariant first order theory in connection and triad variables, described by the so-called Mielke-Baekler Lagrangian. By analyzing the quasi-local symmetries of this theory in the covariant phase space formalism, we show that in each sector of the torsion/curvature duality there exists a well-defined notion of dual diffeomorphism, which furthermore follows uniquely from the Sugawara construction. Together with the usual diffeomorphisms, these duals form at finite distance, without any boundary conditions, and for any sign of the cosmological constant, a centreless double Virasoro algebra which in the flat case reduces to the BMS$_3$ algebra. These algebras can then be centrally-extended via the twisted Sugawara construction. This shows that the celebrated results about asymptotic symmetry algebras are actually generic features of three-dimensional gravity at any finite distance. They are however only revealed when working in first order connection and triad variables, and a priori inaccessible from Chern-Simons theory. As a bonus, we study the second order equations of motion of the Mielke-Baekler model, as well as the on-shell Lagrangian. This reveals the duality between Riemannian metric and teleparallel gravity, and a new candidate theory for three-dimensional massive gravity which we call teleparallel topologically massive gravity.
Determining the masses of new physics particles appearing in decay chains is an important and longstanding problem in high energy phenomenology. Recently it has been shown that these mass measurements can be improved by utilizing the boundary of the allowed region in the fully differentiable phase space in its full dimensionality. Here we show that the practical challenge of identifying this boundary can be solved using techniques based on the geometric properties of the cells resulting from Voronoi tessellations of the relevant data. The robust detection of such phase space boundaries in the data could also be used to corroborate a new physics discovery based on a cut-and-count analysis.
A spinless covariant field $phi$ on Minkowski spacetime $M^{d+1}$ obeys the relation $U(a,Lambda)phi(x)U(a,Lambda)^{-1}=phi(Lambda x+a)$ where $(a,Lambda)$ is an element of the Poincare group $Pg$ and $U:(a,Lambda)to U(a,Lambda)$ is its unitary representation on quantum vector states. It expresses the fact that Poincare transformations are being unitary implemented. It has a classical analogy where field covariance shows that Poincare transformations are canonically implemented. Covariance is self-reproducing: products of covariant fields are covariant. We recall these properties and use them to formulate the notion of covariant quantum fields on noncommutative spacetimes. In this way all our earlier results on dressing, statistics, etc. for Moyal spacetimes are derived transparently. For the Voros algebra, covariance and the *-operation are in conflict so that there are no covariant Voros fields compatible with *, a result we found earlier. The notion of Drinfeld twist underlying much of the preceding discussion is extended to discrete abelian and nonabelian groups such as the mapping class groups of topological geons. For twists involving nonabelian groups the emergent spacetimes are nonassociative.
Hawking radiation is obtained from anomalies resulting from a breaking of diffeomorphism symmetry near the event horizon of a black hole. Such anomalies, manifested as a nonconservation of the energy momentum tensor, occur in two different forms -- covariant and consistent. The crucial role of covariant anomalies near the horizon is revealed since this is the {it only} input required to obtain the Hawking flux, thereby highlighting the universality of this effect. A brief description to apply this method to obtain thermodynamic entities like entropy or temperature is provided.
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