Hamilton–Jacobi Wave Theory in Manifestly-Covariant Classical and Quantum Gravity

Abstract

The axiomatic geometric structure which lays at the basis of Covariant Classical and Quantum Gravity Theory is investigated. This refers specifically to fundamental aspects of the manifestly-covariant Hamiltonian representation of General Relativity which has recently been developed in the framework of a synchronous deDonder–Weyl variational formulation (2015–2019). In such a setting, the canonical variables defining the canonical state acquire different tensorial orders, with the momentum conjugate to the field variable g μ ν being realized by the third-order 4-tensor Π μ ν α . It is shown that this generates a corresponding Hamilton–Jacobi theory in which the Hamilton principal function is a 4-tensor S α . However, in order to express the Hamilton equations as evolution equations and apply standard quantization methods, the canonical variables must have the same tensorial dimension. This can be achieved by projection of the canonical momentum field along prescribed tensorial directions associated with geodesic trajectories defined with respect to the background space-time for either classical test particles or raylights. It is proved that this permits to recover a Hamilton principal function in the appropriate form of 4-scalar type. The corresponding Hamilton–Jacobi wave theory is studied and implications for the manifestly-covariant quantum gravity theory are discussed. This concerns in particular the possibility of achieving at quantum level physical solutions describing massive or massless quanta of the gravitational field.

The Wave-Front Equation of Gravitational Signals in Classical General Relativity

“…Such types of contributions are necessarily ruled out in asynchronous principles, where by definition the variational metric tensor shares the same properties of the extremal one, and in particular it is allowed to raise/lower indices, so that identically, its covariant derivatives expressed in terms of Christoffel symbols are vanishing. In contrast, the kinetic term in the synchronous principle reveals itself to be crucial since: (a) it permits us to reach a representation of the Lagrangian for the gravitational field which has a structure analogous to that of other classical continuum fields, pointing out the role of the customary Ricci contribution to be a potential term; (b) it affords the derivation of corresponding classical Hamiltonian and Hamilton-Jacobi theories for the Einstein field equations, according to the developments reported in references [4,5]; (c) it is essential for the construction of a manifestly-covariant quantum gravity theory with canonical quantization method [50,51].…”

“…Indeed, both still represent theoretical challenges after more than one hundred years since the mathematical foundations of the theory originally were laid down (in 1915) by Einstein in terms of the famous Einstein-Hilbert variational principle [2]. This occurs because, as explained below, the mathematical setting of the theory concerning both its continuous Hamilton equations and the proper form of the corresponding Hamilton-Jacobi formulation has remained undetermined until recently [3][4][5].…”

“…Notably, as in conventional Hamiltonian theories, a related Hamilton-Jacobi theory is induced on the configuration space U g spanned by the variational metric tensor g. This permits us to determine an appropriate Hamilton-Jacobi equation which is fully equivalent to same canonical equations. Second, it is shown that the Hamilton-Jacobi equation determined in this way retains its customary meaning as the equation of wave-front dynamics [5].…”

“…The goal is to point out some crucial mathematical aspects of the problem, which are actually related to the innermost nature of the Einstein equations; i.e., their Hamiltonian representation and corresponding wave propagation phenomena. For this purpose the theoretical framework adopted here is based on the synchronous variational theory of covariant classical gravity (CCG) recently developed in series of papers (see reference [35] and also references [4,5]), which permits us to formulate a continuum Hamiltonian representation of the Einstein field equations.…”

“…Given these premises, we therefore consider here the problem of the physical conjecture that, in the context of classical GR, gravitational signals or general perturbations of the metric field tensor propagate with a well-definite invariant speed, and in particular, that they can be characterized by a suitable front surface (namely, a surface associated with the front-propagation of gravitational signals) which is endowed with a maximum speed of propagation coinciding with the speed of light c. Rather than investigating per se solutions of the Einstein field equations in their customary representation as a set of 2nd-order nonlinear PDEs for the components of the metric tensor, our aim was to carry out the study in the context of the manifestly-covariant Hamiltonian theory of GR previously developed in references [4,5]. This was realized in terms of a two-step approach.…”