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Astronomische Nachrichten

On the possibility of a finite model describing the universe II. The case of a relativistic free particle

G. Järnefelt

Published Jan 1, 1977• 3 citations
Publisher

Abstract

Abstract The ultimate purpose of this investigation is the development of a world model or at least a skeleton for a world model, consisting of a finite number of material particles and of a finite number of geometrical points and time instants. The fundamental argument in the present article is the fact, that in the quantum theory we have to do with separable HILBERT spaces. The separable HILBERT spaces can be represented by means of linear operators with discrete eigenvalues and their eigenvectors. It is possible to replace the discrete eigenvalues with, for instance, non‐negative integers, and these in their turn with numbers of a GALOIS field. In such a way a finitized physical system seems possible, or, if not a complete system, in any case a “skeleton” for a system. How far such a system might be developed remains an open question.

DOI
10.1002/asna.19772980203

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Abstract The ultimate purpose of this investigation is the development of a world model or at least a skeleton for a world model, consisting of a finite number of material particles and of a finite number of geometrical points and time instants. The fundamental argument in the present article is the fact, that in the quantum theory we have to do with separable HILBERT spaces. The separable HILBERT spaces can be represented by means of linear operators with discrete eigenvalues and their eigenvectors. It is possible to replace the discrete eigenvalues with, for instance, non‐negative integers, and these in their turn with numbers of a GALOIS field. In such a way a finitized physical system seems possible, or, if not a complete system, in any case a “skeleton” for a system. How far such a system might be developed remains an open question.
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.
Continuous degrees of freedom The quantum systems we have discussed so far have been described by finite-dimensional Hilbert spaces. Basic measurements on such systems have a finite number of possible outcomes, and a quantum state predicts a discrete probability distribution over these. Now we wish to extend our theory to handle systems with one or more continuous degrees of freedom, such as the position of a particle that can move in one dimension. This will require an extension of our theory to Hilbert spaces of infinite dimension, and to systems with continuous observables. There is a philosophical issue here. How do we know that there really are infinitely many distinct locations for a particle? The short answer is, we don't. It might be that space itself is both discrete (at the tiniest scales) and bounded (at the largest), so that the number of possible locations of a particle is some very large but finite number. If this is the case, then the continuum model for space is nothing more than a convenient approximation. Infinity is just a simplified way of describing a quantity that is immense, but still finite. In this section, we will adopt this view of infinity. We will imagine that any continuous variable is really an approximation of a “true” discrete variable. This idea will motivate the continuous quantities and operations that we need.
A physical picture of macroscopic and microscopic worlds is presented. In the space-time description, both the macroscopic and microscopic worlds are described simultaneously in space-time being a single boundless four dimensional continuum in which the motion of the observer is represented as a world line. In our picture, the space-time description is valid only for the macroscopic world and the microscopic world is local in the sense that it is described in a finite space associated with each point of the macroscopic space-time. In this case, space-time does not exist in the microscopic world in which the observer can not move. For the description of our picture, the language of the non-space-time description is necessary in which space-time is not presupposed. We can utilize the space-time description for the macroscopic world and, at the same time, we have a theory of finite degree of freedom describing creations and annihilations of elementary particles in the microscopic world free from the divergence difficulty, instead of the divergent quantum field theory in the space-time description which is considered in our picture a limit of an approximation like the ideal gas in gas theories. The finite degree of freedom is given by a one-particle state-vector space of a finite dimension from which the second quantization leads to a state-vector space of a countable dimension (a Hilbert space) for the theory of creations and annihilations of elementary particles describing a local microscopic world. The relation between microscopic worlds associated with different macroscopic space-time points is given by a dynamical principle up to a phase factor due to possible fluctuations of phases of local state-vector spaces. The experiment on interference of light performed recently by Franson and Potocki is interpreted in terms of fluctuation of the local phases and is considered to support the theory of finite degree of freedom which leads to the idea of local microscopic worlds.
The tachyon model of neutrino is constructed, basing on the statement that quantum description is a statistical description of stochastically moving particles. Besides, the tachyon model contains two conceptual points: (1) universal formalism of particle dynamics, describing uniformly all particles: deterministic, stochastic and quantum, (2) discrete space-time geometry and skeleton conception of particle dynamics. The universal formalism is a result of a logical reloading, when the statistical ensemble becomes to be the basic object of particle dynamics instead of a single particle. Such a reloading admits one to describe uniformly the quantum, stochastic and deterministic particles in terms of a statistical ensemble without a reference to principles of quantum mechanics. Besides, one uses a relativistic state of a particle, when the state is described by the particle skeleton (several space-time points) instead of the point in the phase space, what is nonrelativistic concept of the particle state. Representing the Dirac equation in terms of the statistical ensemble, one concludes that in the deterministic approximation the world line of the Dirac particle may be a spacelike helix with timelike axis. The rotational component of the relativistic Dirac particle is described nonrelativistically. It shows that the world line may be spacelike, and the Dirac particle may be a tachyon. Neutrino is a Dirac particle, and it is a tachyon.
Based on a number of experimentally verified physical observations, it is argued that the standard principles of quantum mechanics should be applied to the Universe as a whole. Thus, a paradigm is proposed in which the entire Universe is represented by a pure state wavefunction contained in a factorisable Hilbert space of enormous dimension, and where this statevector is developed by successive applications of operators that correspond to unitary rotations and Hermitian tests. Moreover, because by definition the Universe contains everything, it is argued that these operators must be chosen self-referentially; the overall dynamics of the system is envisaged to be analogous to a gigantic, self-governing, quantum computation. The issue of how the Universe could choose these operators without requiring or referring to a fictitious external observer is addressed. The processes by which conventional physics might be recovered from this fundamental, mathematical and global description of reality are particularly investigated. Specifically, it is demonstrated that by considering the changing properties, separabilities and factorisations of both the state and the operators as the Universe proceeds though a sequence of discrete computations, familiar notions such as classical distinguishability, particle physics, space, time, special relativity and endo-physical experiments can all begin to emerge from the proposed picture. A pregeometric vision of cosmology is therefore discussed, with all of physics ultimately arising from the relationships occurring between the elements of the underlying mathematical structure.
A review is given of a new relativistic classical mechanics for N-particle systems compatible with relativistic bound states. It is formulated in global non-inertial frames in Minkowski space-time by means of parametrized Minkowski theories and then restricted to the intrinsic inertial rest frame of the isolated system. There is a complete control on the relativistic collective variables (Newton-Wigner center of mass, Fokker-Pryce center of inertia, M/oller center of energy) and on the realization of the Poincare' algebra (with the explicit form of the interaction-dependent Lorentz boosts). The particle world-lines are found to correspond to the ones of predictive mechanics and localization problems are clarified. The model can be consistently quantized avoiding the instantaneous spreading of the center-of-mass wave packets (Hegerfeldt theorem), because the non-local non-covariant center of mass is a non-measurable quantity. This implies that the Hilbert space must be the tensor product of the center-of-mass Hilbert space with the one of relative motions. This spatial non-separability (due to the Lorentz signature of space-time) makes relativistic entanglement much more involved than the non-relativistic one.