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RELATIVISTIC QUANTUM THEORY AS A GROUP PROBLEM. II. WORLD GEOMETRY AND THE ELEMENTARY PARTICLES

Yrjö Ahmavaara

Published Jan 1, 1962• 1 citation
OpenAlex

Abstract

The world geometry underlying the theory of elementary particles is investigated. The elementary, non-relativistic geometry of the physical space is given a general formulation, sufficient to yield the quanta of momentum and energy when the group quantization process is applied to the physical space. The elementary geometry is represented by a commutative (finite or continuous) group of translations to which Pontrjagin's theorem concerning the duality between a commutative group and its character group applies. The principle of uncertainty is given a group theoretical formulation in terms of the Pontrjaginian duality. This formulation allows a consequent and meaningful extension of quantum theory to a finite geometry. The role of the relativistic (finite or continuous) geometry of space-time in the theory of elementary particles is investigated. A formulation of the problem of elementary particles is given, and an eventual connection between the observed elementary particles and a geometry of the Lorentz group space is suggested. (auth)

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The world geometry underlying the theory of elementary particles is investigated. The elementary, non-relativistic geometry of the physical space is given a general formulation, sufficient to yield the quanta of momentum and energy when the group quantization process is applied to the physical space. The elementary geometry is represented by a commutative (finite or continuous) group of translations to which Pontrjagin's theorem concerning the duality between a commutative group and its character group applies. The principle of uncertainty is given a group theoretical formulation in terms of the Pontrjaginian duality. This formulation allows a consequent and meaningful extension of quantum theory to a finite geometry. The role of the relativistic (finite or continuous) geometry of space-time in the theory of elementary particles is investigated. A formulation of the problem of elementary particles is given, and an eventual connection between the observed elementary particles and a geometry of the Lorentz group space is suggested. (auth)
Elementary Cycles Theory is a self-consistent, unified formulation of quantum and relativistic physics. Here we introduce its basic quantum aspects. On one hand, Newton's law of inertia states that every isolated particle has persistent motion, i.e. constant energy and momentum. On the other hand, the wave-particle duality associates a space-time recurrence to the elementary particle energy-momentum. Paraphrasing these two fundamental principles, Elementary Cycles Theory postulates that every isolated elementary constituent of nature (every elementary particle) must be characterized by persistent intrinsic space-time periodicity. Elementary particles are the elementary reference clocks of Nature. The space-time periodicity is determined by the kinematical state (energy and momentum), so that interactions imply modulations, and every system is decomposable in terms of modulated elementary cycles. Undulatory mechanics is imposed as constraint "overdetermining" relativistic mechanics, similarly to Einstein's proposal of unification. Surprisingly this mathematically proves that the unification of quantum and relativistic physics is fully achieved by imposing an intrinsically cyclic (or compact) nature for relativistic space-time coordinates. In particular the Minkowskian time must be cyclic. The resulting classical mechanics are in fact fully consistent with relativity and reproduces all the fundamental aspects of quantum-relativistic mechanics without explicit quantization. This "overdetermination" just enforces both the local nature of relativistic space-time and the wave-particle duality. It also implies a fully geometrodynamical formulation of gauge interactions which, similarly to gravity and general relativity, is inferred as modulations of the elementary space-time clocks. This brings novel elements to address most of the fundamental open problems of modern physics.
An interpretation and re-formulation of modern physics which removes the presumption of the space-time continuum, and bases physical theory on a small number of rational and empirical principles. After briefly describing the philosophical principles underlying the theory, we rigorously construct a discrete model of quantum mechanics. Kets are names, or labels which categorise states of matter but do not directly describe them. The principle of superposition is seen as a definitional truism in the categorisation of states. This resolves the measurement problem of quantum mechanics by attributing the collapse of the wave function to information. The probability interpretation has a natural meaning in which the configuration of interacting particles plays the role of a hidden variable. The model supports a form of relativistic quantum field theory which does not depend on quantisation or second quantisation from classical mechanics. Continuous laws of wave mechanics are found in a discrete metaphysic which does not involve waves. Classical law is the expected behaviour of many elementary particles. The constraints on the theory are sufficient to establish that particles are point-like entities with specific properties found in nature. Newton's first law and conservation of momentum are established from the principle of homogeneity. Maxwell's equations are derived from the simple interaction in which a Dirac particle emits or absorbs a photon. Feynman rules are calculated for the discrete theory and differ from the standard rules by the removal of the ultraviolet divergence and the use of proper loop integrals. They give finite and unambiguous results.
In a previous paper, we showed that the Hamilton equations of motion of a quantum particle are obtained as group velocities of the wave packet describing this particle only if the time dependent phase of a particle wave is proportional to the Lagrangian, not to the Hamiltonian as in a solution of the conventional Schrodinger equation. When a Lagrangian of relativistic form is considered, the wave packet of a quantum particle takes a physical form, with a finite spectrum of a cut off velocity (c). Based on a relativistic quantum principle, asserting the invariance of the time dependent phase for an arbitrary change of coordinates, we obtained the relativistic kinematics and dynamics, the electromagnetic field equations, the spin and the electromagnetic and gravitational interactions. When the Lagrangian is considered as a function of the Hamiltonian, we obtain a Schrodinger type equation with an additional term depending on the velocity and the momentum operator. Based on this equation, we investigate the dynamics of a relativistic quantum particle. In this framework, such a particle is described as a continuous distribution of conservative matter, according to the general theory of relativity. In an electromagnetic field, any time dependent phase variation is modified with a term proportional to a vector potential conjugated to the spatial coordinates and a scalar potential conjugated to time. However, this classical description tells us nothing about the structure of the physical world. Only Quantum Mechanics tells us something about the structure of this world. Namely, that this world is composed of species of identical quantum particles. On one hand, experimentally, it has been found that these particles are of a wavy nature. On the other hand, one could find that the simplest way to define a quantum particle is by a wave packet, with momenta conjugated to coordinates and energy conjugated to time Consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrodinger equation. We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves. We consider a quantum particle as a distribution of conservative matter propagating according to the General Theory of Relativity. We obtain the dynamics of this matter in a gravitational field, the propagation in plane waves perpendicular to geodesic tracks, and equations of conservation. We describe the matter dynamics as a positively defined density and show that, according to the general theory of relativity, such a distribution can be conceive only as of a fragment of matter with a finite mass equal to a mass , as a characteristic of the matter dynamics, – the matter quantization. The group velocities of the Fourier conjugate representations in the coordinate and momentum spaces describe the dynamics of a quantum particle in agreement with the Hamiltonian equations. Under the action of an external (non-gravitational) field, the acceleration of the quantum matter has two components: 1) A component perpendicular to the velocity, given by the relativistic mechanical component of the time dependent phase, and 2) A component given by the additional field terms of this phase. A free quantum particle is described by a non-dispersing wave function , contrary to the solution of the Schrodinger equation. A coherent electromagnetic field, in resonance with a system of active quantum particles in a Fabry-Perot cavity, has a wave vector approximately proportional to the metric elements, as the resonance frequency is approximately constant – a gravitational wave can be detected by the transmission characteristics of an active Fabry-Perot cavity. In a constant gravitational field, a quantum particle undertakes a velocity and an acceleration, which, at the boundary of a black hole are null – absorption and evaporation processes at the boundary of a black hole arise only by gravitational perturbations. Generally, a quantum particle is described by a time-space volume, called graviton, with a spin 2, and a distribution of a specific matter in this volume, with a half-integer spin for Fermions and an integer spin for Bosons. A graviton Lagrangian is obtained as a curvature integral on a graviton volume, and a Hamiltonian tensor is obtained for the gravitational coordinates and velocities. In a gravitational field, the time space coordinates are deformed. In such a field, any plane wave remains perpendicular geodesic, while an additional acceleration is possible in the wave plane.
Nonrelativistic Galilean quantum mechanics and the standard transition to relativistic Poincar\'e quantum mechanics is analyzed in terms of group theory. Special emphasis is given to the discussion of the relation between dynamics and geometry. Certain unsatisfactory features are pointed out and a new relativistic group G5{\mathcal{G}}_{5}G5​ is suggested as the symmetry group of dynamics. G5{\mathcal{G}}_{5}G5​ contains both the nonrelativistic Galilei group and the Poincar\'e group as subgroups, and it is a group extension of the restricted Lorentz group. For use in relativistic quantum mechanics, the central extension of G5{\mathcal{G}}_{5}G5​ by a phase group must be employed. The Lie algebra of this relativistic quantum-mechanical Galilei group ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{G}}}_{5}$ contains an acceptable covariant space-time position operator and a nontrivial relativistic mass operator. The latter also serves to describe dynamical development. The irreducible unitary projective representations of ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{G}}}_{5}$ correspond to infinite towers of states with increasing spin.
The skeleton conception of elementary particles is considered in the paper. Conventional particle dynamics is formulated in an unaccomplished form, which is adequate only in the continuous space-time geometry. The conventional differential equations of the particle motion cannot be written in the discrete space-time geometry. In the discrete space-time geometry the particle world line is replaced by the world chain. The world chain links has a finite length (not infinitesimal). The world chain appears to be stochastic. Statistical description of stochastic world chains leads to the Schrödinger equation, if the elementary length of the geometry is chosen in a proper way. The quantum principles are founded by existence of the discrete (and multivariant) space-time geometry and lose the role of prime physical principles. In the skeleton conception the particle is described by its skeleton (several rigidly connected space-time points). Skeleton conception of the elementary particles realizes a proper description of the particle state, which appears to be adequate in a discrete space-time geometry. The particle dynamics takes the form of a monistic conception, which is described completely in terms of the world function of the space-time geometry. The skeleton conception accomplishes the transition from nonrelativistic physics to the relativistic one and realizes the total geometrization of particle dynamics.
One uses the investigation strategy: ”Find a mistake in foundation of physics and correct it”. Two fundamental defects are corrected: (1) nonrelativistic concept of the particle state, (2) inadequacy of the differential geometry formalism at the metric approach to geometry. Usage of relativistic concept of the particle state admits one to create united formalism for description of deterministic and stochastic particles motion. The quantum mechanics is founded as a statistical description of the stochastic particles motion. Stochasticity of the quantum particles motion is explained by multivariance of the space-time geometry, which leads to the world lines wobbling of elementary particles. Relativistic concept of the particle state and the metric approach to the space-time geometry admit one to construct the skeleton conception of elementary particles. The skeleton conception admits one to investigate the arrangement of elementary particles (but not only to systamatize them, ascribing quantum numbers to them).