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Journal of Mathematical Physics

Canonical Formulation of Relativistic Mechanics

Robert Nyden Hill

Published Sep 1, 1967• 67 citations
Publisher

Abstract

Instantaneous action-at-a-distance relativistic mechanics of the type considered by Currie and by Hill is cast into a Hamiltonian form wherein the transformations of the inhomogeneous Lorentz group are canonical. The Currie-Jordan-Sudarshan zero-interaction theorem is circumvented by renouncing the demand that physical positions be canonical; the implications for measurement theory of renouncing this demand are discussed. Examples are given.

DOI
10.1063/1.1705419

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PSA Proceedings of the Biennial Meeting of the Philosophy of Science Association•Jan 1, 1992•19 citations
Instantaneous action-at-a-distance relativistic mechanics of the type considered by Currie and by Hill is cast into a Hamiltonian form wherein the transformations of the inhomogeneous Lorentz group are canonical. The Currie-Jordan-Sudarshan zero-interaction theorem is circumvented by renouncing the demand that physical positions be canonical; the implications for measurement theory of renouncing this demand are discussed. Examples are given.
The diffculties of relativistic particle theories formulated my means of canonical quantization, such as Klein-Gordon and Dirac theories, ultimately led theoretical physicists to turn on quantum field theory to model elementary particle physics. The aim of the present work is to pursue a method alternative to canonical quantization that avoids these dfficulties. In order to guarantee this result, the present approach is constrained to be developed deductively from physical principles. The physical principles assumed for a free particle consist of the symmetry properties of the particle with respect to the Poincaré group and of the transformation properties of the position observable, expressed by means of a suitably conceived notion of quantum transformation. In so doing, the effectiveness of group theoretical methods is exploited. Our work has pointed out the necessity of new classes of irreducible representations of the Poincaré group the theory can be based on. For spin 0 particle, four inequivalent theories are completely determined, that differ from Klein-Gordon theory for the physical interpretation but also for the mathematical structures. In particular, two of these theories requires irreducible representations in the new classes. The present lack of the explicit transformation properties of position with respect to boosts forbids the complete determination of non zero spin particle theories. A particular form of these transformation properties was in past adopted by Jordan and Mukunda. We check its consistency within the present approach; it is found that for spin 1/2 particles there is only one consistent theory; it is unitarily related to Dirac theory, but, once again, it requires the new classes of irreducible representations.
New relativistic mechanics of two-particle system is set forth, where the two constituent particles are interacting by an arbitrary (central) action-at-a-distance. The fundamental equations are presented in a form covariant under general transformation of parameters parametrizing the world lines of constituent particles. The theory represents the proper relativistic generalization of the usual Newtonian mechanics in the sense that it tends in the non-relativistic (and weak interaction) limit to the usual mechanics of two particles moving under a corresponding non-relativistic potential. For the analysis of theory it is convenient to choose a certain particular gauge (i.e., parametrization) fixed by two gauge relations. This brings the theory to a canonical formalism accompanied by two weak equations, and in this gauge quantization can be performed. The result verifies that the relativistic quantum mechanics for two particles interacting by an action-at-a-distance is just represented by a bilocal wave equation and a subsidiary condition, with the clarification of its correspondence-theoretical fundation and internal dynamics. As an example the case of Hooke-type force is illustrated, where the internal motions are elliptic oscillations in the center-of-mass frame. Its quantum theory just reproduces the original form of bilocal theory giving bound states lying on a straightly rising trajectory and on its daughter trajectories.
The construction of the inhomogeneous Lorentz group is discussed from the viewpoint of relativistic kinematics, within the limitations of the construction with the characteristic invariants P/sub mu P/sup mu > 0. The construction given depends essentially on an explicit calculation of Wigner considerations. The problem of reducing this construction of the homogeneous Lorentz group is solved, explicitly for the construction of the inhomogeneous group, by a transformation from the canonical basis to another basis into which the construction operators of the homogeneous Lorentz transformation is decomposed. The analogs to the Clebsch-Gordon series for the product construction of the rotational group were determined for the product of the two irreducible constructions of the inhomogeneous Lorentz group. (J.S.R.)
The formal mathematical structure of the truly Hamiltonian formulations of nonrelativistic and relativistic quantum mechanics is critically examined for both spin 0 and spin 12 particles with nonzero mass. It is shown that the relativistic quantum mechanics can, in principle, at least be fitted into a truly covariant Hamiltonian procedure where the Hamiltonian is a Lorentz-invariant world scalar (and is the negative of 12 the rest energy of the particle, regardless of its spin). In particular, a truly covariant Hamiltonian theory of the electron is presented. It will be seen that the Lorentz-invariant Dirac theory of the electron does not fall within the framework of a truly covariant Hamiltonian procedure. Further, the dynamical variables associated with the translational degree of freedom of the particle (such as the position, the canonical momentum, and the velocity four-vectors of the relativistic quantum mechanics) correspond precisely to the same variables of the truly covariant formulation of relativistic classical mechanics. The mathematical structure of relativistic quantum mechanics is seen to be rather similar to that of nonrelativistic quantum mechanics for both spin 0 and spin 12 particles with nonzero mass.
What is relativistic quantum theory? How does it differ from non-relativistic quantum mechanics? Is there something more involved than the transition from the Galilean group to the Lorentz group? These questions are evidently of great importance to the understanding of relativistic quantum theory (RQT), in particular to the distinction between the basic principles of quantum theory, and the characterization of a particular class of dynamical evolutions. Within non-relativistic quantum mechanics (NRQM) a reasonably precise response is possible (differences concern the number of degrees of freedom, the existence of internal symmetries, and the form of the Hamiltonian as a function of the canonical variables); what appears characteristic of RQT is the appearance of a dynamics of a qualitatively different character (involving the creation and annihilation of particles), involving new kinds of constraints (renormalization theory: anomalies). Nevertheless, it seems we must know how to pass from the inhomogeneous Galilean group (IHGG) to the Lorentz group (IHLG), if only because we have successfully constructed a RQT, surely, then, we know what it is that we hold invariant in this transition.