Separable interactions in classical relativistic Hamiltonian mechanics

We show the existence of separable interactions in classical relativistic Hamiltonian mechanics of particles interacting at a distance. The framework which is used is that of manifestly covariant systems with constraints.Laboratoire associé au CNRS.     

Relativistic Hamiltonian dynamics

A manifestly covariant relativistic hamiltonian dynamics is presented for a closed system of N particles in mutual interaction. The “no-interaction theorem” is overcome by use of relativistic center-of-mass variables instead of individual particle variables. The theory permits canonical quantization.     

Relativistic action-at-a-distance interactions: Lagrangian and Hamiltonian to terms of second order

Relativistic systems of particles interacting pairwise at a distance (interactions not mediated by fields) in flat spacetime are studied. It is assumed that the interactions propagate at the speed of light in vacuum and that all masses are scalars under Poincaré transformations. The action functional of the theory depends on multiple times (the proper times of the particles). In the static limit, the theory has three components: a linearly rising potential, a Coulomb-like interaction and a dynamical component to the Poincaré invariant mass. In this Letter we obtain explicitly, to terms of second order, the Lagrangian and the Hamiltonian with all the dynamical variables depending on a single time. Approximate solutions of the relativistic two-body problem are presented.     

Relativistic non-Hamiltonian mechanics

Relativistic particle subjected to a general four-force is considered as a nonholonomic system. The nonholonomic constraint in four-dimensional space-time represents the relativistic invariance by the equation for four-velocity u_μu^μ + c² = 0, where c is the speed of light in vacuum. In the general case, four-forces are non-potential, and the relativistic particle is a non-Hamiltonian system in four-dimensional pseudo-Euclidean space-time. We consider non-Hamiltonian and dissipative systems in relativistic mechanics. Covariant forms of the principle of stationary action and the Hamilton’s principle for relativistic mechanics of non-Hamiltonian systems are discussed. The equivalence of these principles is considered for relativistic particles subjected to potential and non-potential forces. We note that the equations of motion which follow from the Hamilton’s principle are not equivalent to the equations which follow from the variational principle of stationary action. The Hamilton’s principle and the principle of stationary action are not compatible in the case of systems with nonholonomic constraint and the potential forces. The principle of stationary action for relativistic particle subjected to non-potential forces can be used if the Helmholtz conditions are satisfied. The Hamilton’s principle and the principle of stationary action are equivalent only for a special class of relativistic non-Hamiltonian systems.     

Relativistic Hamiltonian dynamics. II. Momentum-dependent interactions, confinement and quantization

The previously developed covariant classical relativistic N-particle dynamics is extended to momentum-dependent interactions and generalized to a gauge-independent constraint reduction. This reduction is made via center-of-momentum variables as well as via the more conventional individual particle variables. A canonical quantization is then carried out. The two-body problem is discussed in detail for the case of momentum-dependent interactions. It is demonstrated that such interactions can give rise to dynamical confinement both classically and quantum mechanically. The prototype interaction −β²(ξ · π)² has a harmonic oscillator type spectrum and shows a linear dependence of the binding energy on the angular momentum for small particle rest masses (m₁, m₂ β).     

Synthetic Hamiltonian mechanics

This paper deals with some infinitesimal aspects of Hamiltonian mechanics from the standpoint of synthetic differential geometry. Fundamental results concerning Hamiltonian vector fields, Poisson brackets, and momentum mappings are discussed. The significance of the Lie derivative in the synthetic context is also consistently stressed. In particular, the notion of an infinitesimally Euclidean space is introduced, and the Jacobi identity of vector fields with respect to Lie brackets is established naturally for microlinear, infinitesimally Euclidean spaces by using Lie derivatives instead of a highly combinatorial device such as P. Hall's 42-letter identity.     

Geometric Quantization of Relativistic Hamiltonian Mechanics

A relativistic Hamiltonian mechanical system is seen as a conservative Dirac constraint system on the cotangent bundle of a pseudo-Riemannian manifold. We provide geometric quantization of this cotangent bundle where the quantum constraint serves as a relativistic quantum equation.     

Relativistic quantum mechanics of fermions

We present a modification of the Dirac equation that allows us to formulate a relativistic quantum mechanics for spin-1/2 fermions in an external electromagnetic field, with a probabilistic interpretation similar to that in nonrelativistic quantum mechanics and based on an indefinite charge density. We find that stationary states cannot be interpreted in this manner, and we replace them by quasistationary states. We also include a general discussion of the difficulties and possible generalizations of this approach.     

Relativistic Hamiltonian dynamics and few-nucleon systems

We present a preliminary calculation of the electromagnetic form factors of ³He and ³H, performed within the light-front Hamiltonian dynamics. Relativistic effects show their relevance even at the static limit, increasing at higher values of momentum transfer, as expected.     

A generalization of Hamiltonian mechanics

For a symplectic manifold the Poisson bracket on the space of functions is (uniquely) extended to a graded Lie bracket on the space of differential forms modulo exact forms. A large portion of the Hamiltonian formalism is still working.     

Relativistic quantum mechanics of supersymmetric particles

The quantum mechanics of an electron in an external field is developed by Hamiltonian path integral methods. The electron is described classically by an action which is invariant under gauge supersymmetry transformations as well as worldline reparametrizations. The simpler case of a spinless particle is first reviewed and it is pointed out that a strictly canonical approach does not exist. This follows formally from the gauge invariance properties of the action and physically it corresponds to the fact that particles can travel backwards as well as forward in coordinate time. However, appropriate application of path integral techniques yields directly the proper time representation of the Feynman propagator. Next we extend the formalism to systems described by anticommuting variables. This problem presents some difficulty when the dimension of the phase space is odd, because the holomorphic representation does not exist. It is shown, however, that the usual connection between the evolution operator and the path integral still holds provided one indludes in the action the boundary term that makes the classical variational principle well defined. The path integral for the relativistic spinning particle is then evaluated and it is shown to lead directly to a representation for the Feynman propagator in terms of two proper times, one commuting, the other anticommuting, which appear in a symmetric manner. This representation is used to derive scattering amplitudes in an external field. In this step the anticommuting proper time is integrated away and the analysis is carried in terms of one (commuting) proper time only, just as in the spinless case. Finally, some properties of the quantum mechanics of the ghost particles that appear in the path integral for constrained systems are developed in an appendix.     

Relativistic models of nonlinear quantum mechanics

I present and discuss a class of nonlinear quantum-theory models, based on simple relativistic field theories, in which the parameters depend on the state of the system via expectation values of local functions of the fields.     

Relativistic celestial mechanics of binary stars

We present the results of a systematic study of the dynamics of realistic binary systems in the post-Newtonian approximation (PNA) of general relativity. We propose definitions valid in the PNA for the self-angular-momenta of the binary's members, as well as for the angular momentum of their relative orbital motion, and we examine under which conditions they can be considered as constant in the PNA. This enables us to define to the same approximation the plane relative orbital motion. Then we find the form of the differential equations of motion from an integration of which we prove that in the PNA the relative motion is a processing ellipse composed of a basic orbit and a correction, both of which are of post-Newtonian character. Moreover, using the polar equation of the above ellipse we define the elements of the post-Newtonian, relative, basic orbit, we generalize to the PNA the three well-known laws of classical celestial mechanics of Kepler, and we derive the precessional motion of the relative orbit's pericenter. Finally, we compare our method with other methods existing in the literature, and we expose its theoretical and conceptual differences with them.     

Relativistic quantum statistical mechanics in two-dimensional space-time

We construct for a boson field in two-dimensional space-time with polynomial or exponential interactions and without cut-offs, the positive temperature state or the Gibbs state at temperature 1/β. We prove that at positive temperatures i.e. β<∞, there is no phase transitions and the thermodynamic limit exists and is unique for all interactions. It turns out that the Schwinger functions for the Gibbs state at temperature 1/β is after interchange of space and time equal to the Schwinger functions for the vacuum or temperature zero state for the field in a periodic box of length β, and the lowest eigenvalue for the energy of the field in a periodic box is simply related to the pressure in the Gibbs state at temperature 1/β.     

Relativistic fluid dynamics as a Hamiltonian system

The equations of ideal relativistic fluid dynamics in the laboratory frame form a noncanonical hamiltonian system with the same Poisson bracket as for nonrelativistic fluids, but with dynamical variables and hamiltonian obtained via a regular deformation of their nonrelativistic counterparts.     

Relativistic classical mechanics and canonical formalism

The analysis of interacting relativistic many-particle systems provides a theoretical basis for further work in many diverse fields of physics. After a discussion of the nonrelativisticN-particle systems we describe two approaches for obtaining the canonical equations of the corresponding relativistic forms. A further aspect of our approach is the consideration of the constants of the motion.