Reparametrization-invariant Hamiltonian formalism in the general theory of relativity

A method is proposed that is appropriate for resolving the Hamiltonian constraint and which leads to a reparametrization-invariant reduced theory specified by a well-defined nonzero local Hamiltonian. This method is based on introducing a global (dependent only on time) conformal variable. The physical and geometric meaning of the variables in the reduced action functional is investigated. It is shown that, within the theory, the method of small perturbations is self-consistent. It is demonstrated that, in the theory of gravity, there are no wavelike excitations that make a negative contribution to the Hamiltonian. From an analysis of the reduced classical theory in the linear approximation, it follows that, at the first instants from the birth of the Universe, the extremely rigid equation of state appeared to be the effective equation of the state of gravity matter.     

Canonical structure of tetrad bimetric gravity

We perform the complete canonical analysis of the tetrad formulation of bimetric gravity and confirm that it is ghost-free describing the seven degrees of freedom of a massless and a massive gravitons. In particular, we find explicit expressions for secondary constraints, one of which is responsible for removing the ghost, whereas the other ensures the equivalence with the metric formulation. Both of them have a remarkably simple form and, being combined with conditions on Lagrange multipliers, can be written in a covariant way.     

Matter induced bimetric actions for gravity

The gravitational effective average action is studied in a bimetric truncation with a nontrivial background field dependence, and its renormalization group flow due to a scalar multiplet coupled to gravity is derived. Neglecting the metric contributions to the corresponding beta functions, the analysis of its fixed points reveals that, even on the new enlarged theory space which includes bimetric action functionals, the theory is asymptotically safe in the large N expansion.     

Hamiltonian formalism of fractional systems

In this paper we consider a generalized classical mechanics with fractional derivatives. The generalization is based on the time-clock randomization of momenta and coordinates taken from the conventional phase space. The fractional equations of motion are derived using the Hamiltonian formalism. The approach is illustrated with a simple-fractional oscillator in a free motion and under an external force. Besides the behavior of the coupled fractional oscillators is analyzed. The natural extension of this approach to continuous systems is stated. The interpretation of the mechanics is discussed.     

Algebraic programming of Hamiltonian formalism in general relativity: Application to inhomogeneous space-times

We describe the structure and the use of a program written in the algebraic programming languagereduce 2, giving the super-Hamiltonian and supermomenta constraints, as well as Hamilton's canonical equations in terms of the canonical variables, for vacuum relativistic space-times. The program uses as input the components of the spatial metric tensor and of the corresponding canonically conjugate momenta in a coordinate or in a spatial Cartan basis. The results of the application of the program to a series of inhomogeneous (cosmological as well as noncosmological) space-times are given: in particular, the constraints, the Dirac Hamiltonian and the canonical equations are explicitly written for axisymmetric space-times, constituting the starting point for the study of the dynamics and of the canonical quantization of these configurations.     

Hamiltonian formalism for non-invariant dynamics

In a completely Hamiltonian dynamical system, there will be a generating functionH _Y for each infinitesimal space-time transformationY. In the non-autonomous case, theH _Y depend on the observer. This dependence is here described by a system of commutation relations. It is also shown that these relations can be made to mirror exactly the commutation relations of theY's in the Lorentz-invariant case.The preparation of this paper was supported in part by NSF grant GP-33696X. It is a pleasure to acknowledge discussions with Professor H. Bacri, D. Kastler, J. M. Souriau and others at CNRS, CPT, Marseilles.     

Quantum Hamiltonian formalism with constraints

A quantum system with constraints that does not necessarily correspond to a classical system with constraints is described in the Hamiltonian formalism.     

Becchi-Rouet-Stora-Tyutin quantization and Hamiltonian formalism

An introductory review of BRST hamiltonian formalism is presented. The method of quantization of gauge and string theories is discussed. A few simple examples are presented to illustrate the BRST techniques.     

Elementary particles in bimetric general relativity

A classical model of an elementary particle is considered in the framework of the bimetric general relativity theory. The particle is regarded as a spherically symmetric object filling its Schwarzschild sphere and made of matter having mass density, pressure, and charge density. The mass is taken to be the Planck mass, and possible values of the charge are taken as zero, ±1/3e, ±2/3e, and ±e, with e the electron charge.     

Nonextensive formalism and continuous Hamiltonian systems

A recurring question in nonequilibrium statistical mechanics is what deviation from standard statistical mechanics gives rise to non-Boltzmann behavior and to nonlinear response, which amounts to identifying the emergence of “statistics from dynamics” in systems out of equilibrium. Among several possible analytical developments which have been proposed, the idea of nonextensive statistics introduced by Tsallis about 20 years ago was to develop a statistical mechanical theory for systems out of equilibrium where the Boltzmann distribution no longer holds, and to generalize the Boltzmann entropy by a more general function Sq while maintaining the formalism of thermodynamics. From a phenomenological viewpoint, nonextensive statistics appeared to be of interest because maximization of the generalized entropy Sq yields the q-exponential distribution which has been successfully used to describe distributions observed in a large class of phenomena, in particular power law distributions for q>1. Here we re-examine the validity of the nonextensive formalism for continuous Hamiltonian systems. In particular we consider the q-ideal gas, a model system of quasi-particles where the effect of the interactions are included in the particle properties. On the basis of exact results for the q-ideal gas, we find that the theory is restricted to the range q<1, which raises the question of its formal validity range for continuous Hamiltonian systems.     

Covariant canonical formalism for gravity

We continue the previous discussion (A. D'Adda, J. E. Nelson, and T. Regge, Ann. Phys. (N.Y.)165) of the covariant canonical formalism for the group manifold and relate it to the standard canonical vierbein formalism as pioneered by Dirac. The form bracket is related to the Poisson bracket of classical mechanics. We utilise systematically the calculus of differential forms and a compound notation which labels Poincaré multiplets. In this way we obtain a particularly clear and compact expression for the Hamiltonian and the constraints algebra of the vierbein formalism.     

Conformally equivalent metrics in bimetric general relativity

Space-times conformal to physical space-time are considered, assuming the nongravitational energy is conservative after the conformal transformation. Necessary and sufficient conditions satisfied by the conformal factor axe found for a given type of transformation of the energy tensor. The weak gravitational field is defined and the coordinate conditions for the existence of conformal factors in such a field are obtained.     

A charged particle in bimetric general relativity

The gravitational field of a charged particle is investigated on the basis of the bimetric general relativity theory. It is found that the field differs from the Reissner-Nordström field only very close to the sphereR=m+(m ² –Q ²)^1/2. This sphere is impenetrable, and its interior is unphysical.     

Hamiltonian path integral formalism with higher derivatives

We study the Hamiltonian path integral formalism for systems containing higher derivatives. First we show the consistency of the formalism in applications involving only scalar fields. Later we use the Maxwell electromagnetic theory with a higher order regularization term to show that the Batalin-Fradkin-Vilkovisky (BFV) theory can also be consistently described.     

The cosmic field tensor in bimetric general relativity

We construct all cosmic field tensors which are symmetric rank-two tensor concomitants of a metric and a background metric and which have zero divergence when the background metric satisfies the generalized De Donder condition. The resulting background cosmic field represents an Einstein space-time.