A starting point for the present work was the statement recently discussed in the literature that two Hamiltonian formulations for the theory of gravity, the one proposed by Dirac and the other by Arnowitt - Deser - Misner, may not be related by a canonical transformation. In its turn, it raises a question about the equivalence of these two Hamiltonian formulations and their equivalence to the original formulation of General Relativity. We argue that, since the transformation from components of metric tensor to the ADM variables touches gauge degrees of freedom, which are non-canonical from the point of view of Dirac, the problem cannot be resolved in the limits of the Dirac approach. The proposed solution requires the extension of phase space by treating gauge degrees of freedom on an equal footing with other variables and introducing missing velocities into the Lagrangian by means of gauge conditions in differential form. We illustrate with a simple cosmological model the features of Hamiltonian dynamics in extended phase space. Then, we give a clear proof for the full gravitational theory that the ADM-like transformation is canonical in extended phase space in a wide enough class of possible parametrizations.
We discuss the relation between the canonical Hamilton-Jacobi theory and the De Donder-Weyl Hamilton-Jacobi theory known in the calculus of variations using the examples of a scalar field in curved space-time and general relativity. By restricting ourselves to the Gaussian coordinates we show how the canonical Hamilton-Jacobi equation of general relativity can be derived from the De Donder-Weyl Hamilton-Jacobi formulation of the Einstein equations.
We present the Hamiltonian formulation of the Ghost Free Mimetic Massive Gravity theory. The linearized theory is studied and the Hamiltonian equations of motion are analyzed. Poisson brackets are computed and closure is proved. To prove that this theory is ghost-free, the number of degrees of freedom is analyzed showing that we have only five degrees of freedom.
We derive a Hamiltonian for an extended spinning test body in a curved background spacetime, to quadratic order in the spin, in terms of three-dimensional position, momentum, and spin variables having canonical Poisson brackets. This requires a careful analysis of how changes of the spin supplementary condition are related to shifts of the body's representative worldline and transformations of the body's multipole moments, and we employ bitensor calculus for a precise framing of this analysis. We apply the result to the case of the Kerr spacetime and thereby compute an explicit canonical Hamil…
This paper is devoted to the analysis of the covariant canonical formalism of $F(R)$ gravity in Einstein frame. We also find canonical transformation between covariant canonical formulation of F(R) gravity in Jordan frame and Einstein frames and we also determine corresponding generating function.
We investigate the canonical formulation of the (bosonic) E$_{6(6)}$ exceptional field theory. The explicit non-integral (not manifestly gauge invariant) topological term of E$_{6(6)}$ exceptional field theory is constructed and we consider the canonical formulation of a model theory based on the topological two-form kinetic term. Furthermore we construct the canonical momenta and the canonical Hamiltonian of the full bosonic E$_{6(6)}$ exceptional field theory. Most of the canonical gauge transformations and some parts of the canonical constraint algebra are calculated. Moreover we discuss ho…
In this work, we present a consistent Hamiltonian analysis of cosmological perturbations for generalized non-canonical scalar fields. In order to do so, we introduce a new phase-space variable that is uniquely defined for different non-canonical scalar fields. We also show that this is the simplest and efficient way of expressing the Hamiltonian. We extend the Hamiltonian approach of [arXiv:1512.02539] to non-canonical scalar field and obtain a new definition of speed of sound in phase-space. In order to invert generalized phase-space Hamilton's equations to Euler-Lagrange equations of motion,…
The canonical Hamiltonian $H_C$ of the metric General Relativity is reduced to its natural form. The natural form of canonical Hamiltonian provides numerous advantages in actual applications to the metric GR, since the general theory of dynamical systems with such Hamiltonians is well developed. Furthermore, many analytical and numerically exact solutions for dynamical systems with natural Hamiltonians have been found and described in detail. In particular, based on this theory we can discuss an obvious analogy between gravitational field(s) and few-particle systems where particles are connect…
Jordan and Einstein frame are studied under the light of Hamiltonian formalism. Dirac's constraint theory for Hamiltonian systems is applied to Brans-Dicke theory in the Jordan Frame. In both Jordan and Einstein frame, Brans-Dicke theory has four secondary first class constraints and their constraint algebra is closed. We show, contrary to what is generally believed, the Weyl (conformal) transformation, between the two frames, is not a canonical transformation, in the sense of Hamiltonian formalism. This addresses quantum mechanical inequivalence as well. A canonical transformation is shown.
On canonical transformations between equivalent Hamiltonian formulations of General Relativity
两个广义相对论的哈密顿公式，由于皮拉尼，Schild和斯金纳（物理Rev. 87，452，1952）和狄拉克（Proc。Soc. A 246，333，1958）。这两个公式，尽管有不同的表达式的约束，允许一个推导四维的同构不变性。这两个配方之间的关系在所有阶段的狄拉克方法的约束哈密顿系统进行了分析。它示出，他们的相空间变量的完整的集合是由一个变换，满足普通的条件，规范已知的无约束哈密顿，此外，转换成另一个总的哈密顿量，从而保持形式不变性的广义哈密顿方程的约束系统。
Different routes towards the canonical formulation of a classical theory result in different canonically equivalent Hamiltonians, while their quantum counterparts are related through appropriate unitary transformation. However, for higher-order theory of gravity, although two Hamiltonians emerging from the same action differing by total derivative terms are related through canonical transformation, the difference transpires while attempting canonical quantization, which is predominant in non-minimally coupled higher-order theory of gravity. We follow Dirac's constraint analysis to formulate ph…
A detailed canonical treatment of a new action for a nonrelativistic particle coupled to background gravity, recently given by us, is performed both in the Lagrangian and Hamiltonian formulations. The equation of motion is shown to satisfy the geodesic equation in the Newton-Cartan background, thereby clearing certain confusions. The Hamiltonian analysis is done in the gauge independent as well as gauge fixed approaches, following Dirac's analysis of constraints. The physical (canonical) variables are identified and the path to canonical quantisation is outlined by explicitly deriving the Schr…
A covariant formulation for the Newton-Hooke particle is presented by following an algorithm developed by us \cite{BMM1, BMM2, BMM3}. It naturally leads to a coupling with the Newton-Cartan geometry. From this result we provide an understanding of gravitation in a Newtonian geometric background. Using Dirac's constrained analysis a canonical formulation for the Newton-Hooke covariant action is done in both gauge independent and gauge fixed approaches. While the former helps in identifying the various symmetries of the model, the latter is able to define the physical variables. From this analys…
Two different Hamiltonian formulations of the metric gravity are discussed and applied to describe a free gravitational field in the $d$ dimensional Riemann space-time. Theory of canonical transformations, which relate equivalent Hamiltonian formulations of the metric gravity, is investigated in details. In particular, we have formulated the conditions of canonicity for transformation between the two sets of dynamical variables used in our Hamiltonian formulations of the metric gravity. Such conditions include the ordinary condition of canonicity known in classical Hamilton mechanics, i.e., th…
The problem of obtaining canonical Hamiltonian structures from the equations of motion, without any knowledge of the action, is studied in the context of the spatially flat Friedmann-Robertson-Walker models. Modifications to Raychaudhuri equation are implemented independently as quadratic and cubic terms of energy density without introducing additional degrees of freedom. Depending on their sign, modifications make gravity repulsive above a curvature scale for matter satisfying strong energy condition, or more attractive than in the classical theory. Canonical structure of the modified theorie…
The Hawking effect is one of the most extensively studied topics in modern physics. Yet it remains relatively under-explored within the framework of canonical quantization. The key difficulty lies in the fact that the Hawking effect is principally understood using the relation between the ingoing modes which leave past null infinity and the outgoing modes which arrive at future null infinity. Naturally, these modes are described using advanced and retarded null coordinates instead of the usual Schwarzschild coordinates. However, null coordinates do not lead to a true Hamiltonian that describes…
We establish the Hamiltonian analysis and the canonical path integral for a local formulation of vacuum energy sequestering. In particular, by considering the state of the universe as a superposition of vacuum states corresponding to different values of the cosmological and gravitational constants, the path integral is extended to include integrations over the cosmological and gravitational constants. The result is an extension of the Ng-van Dam form of the path integral of unimodular gravity. It is argued to imply a relation between the fraction of the most likely values of the gravitational …
Starting from a local action for mimetic gravity that includes higher derivatives of a scalar field $ϕ$, we derive a gauge-fixed canonical action of the theory in the ADM canonical formalism in the time gauge $ϕ=t$. This reduced action reveals (i) a non-vanishing conserved physical Hamiltonian that is a sum of two terms, the expression for the Hamiltonian constraint of general relativity and a function of the expansion scalar, and (ii) a reduced symplectic structure that geometrically provides the Dirac brackets. As applications of our general analysis, we compute the physical Hamiltonians and…
Hamiltonian Expression of Curvature Tensors in the York Canonical Basis: II) The Weyl Tensor, Weyl Scalars, the Weyl Eigenvalues and the Problem of the Observables of the Gravitational Field
We work on the Lagrangian and the Hamiltonian formulations of the Palatini action. In the Lagrangian formulation, we find that we need to assume the metric compatibility and the torsion zero or to assume the tetrad compatibility to describe General Relativity. In the Hamiltonian formulation, we obtain the Einstein's equations only with assuming the tetrad compatibility. The Hamiltonian from assuming the metric compatibility and the torsion zero should be used to quantize General Relativity.
We systematically derive an action for a nonrelativistic spinning partile in flat background and discuss its canonical formulation in both Lagrangian and Hamiltonian approaches. This action is taken as the starting point for deriving the corresponding action in a curved background. It is achieved by following our recently developed technique of localising the flat space galilean symmetry \cite{BMM1, BMM3, BMM2}. The coupling of the spinning particle to a Newton-Cartan background is obtained naturally. The equation of motion is found to differ from the geodesic equation, in agreement with earli…
We obtain the Arnowitt-Deser-Misner formulation of general relativity in $n$ dimensions ($n \geq 3$) from its either $SO(n-1,1)$ [$SO(n)$] or $SO(n-1)$ Palatini Hamiltonian formulations and vice versa [we recall that $SO(n-1,1)$ [$SO(n)$] requires no gauge fixing whereas $SO(n-1)$ involves the time gauge]. Similarly, the Hamiltonian formulation of general relativity in terms of Ashtekar-Barbero variables can also be directly obtained from the Arnowitt-Deser-Misner Hamiltonian formulation and vice versa, which is an alternative approach to the way followed by Barbero. We give the relevant maps …
This chapter of the proceedings for the Ninth Meeting on CPT and Lorentz Symmetry is dedicated to the Hamiltonian formulation of the minimal gravitational Standard-Model Extension. Some theoretical questions associated with the latter shall be reviewed. First, we recall the properties of the Hamiltonian, which was computed elsewhere, and discuss how it is linked to the modified Einstein equations. Second, we describe how the covariant and Hamiltonian formulations are shown to be consistent with each other.
