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Fortschritte der Physik

Outline of a Nonlinear, Relativistic Quantum Mechanics of Extended Particles

Eckehard W. Mielke

Published Jan 1, 1981• 7 citations
Publisher

Abstract

Abstract A quantum theory of intrinsically extended particles similar to de Broglie's Theory of the Double Solution is proposed. A rational notion of the particle's extension is enthroned by realizing its internal structure via soliton‐type solutions of nonlinear, relativistic wave equations. These droplet‐type waves have a quasi‐objective character except for certain boundary conditions which may be subject to stochastic fluctuations. More precisely, this assumption amounts to a probabilistic description of the center of a soliton such that it would follow the conventional quantum‐mechanical formalism in the limit of zero particle radius. At short interaction distances, however, a promising nonlinear and nonlocal theory emerges. This model is not only capable of achieving a conceptually satisfying synthesis of the particle‐wave dualism, but may also lead to a rational resolution of epistemological problems in the quantum‐theoretical measurement process. Within experimental errors the results for, e.g., the hydrogen atom can be reproduced by appropriately specifying the nature of the nonlinear self‐interaction. It is speculated that field theoretical issues raised by such notions as identical particles, field quantization and renormalization are already incorporated or resolved by this nonlocal theory, at least in principle.

DOI
10.1002/prop.19810291103

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Abstract A quantum theory of intrinsically extended particles similar to de Broglie's Theory of the Double Solution is proposed. A rational notion of the particle's extension is enthroned by realizing its internal structure via soliton‐type solutions of nonlinear, relativistic wave equations. These droplet‐type waves have a quasi‐objective character except for certain boundary conditions which may be subject to stochastic fluctuations. More precisely, this assumption amounts to a probabilistic description of the center of a soliton such that it would follow the conventional quantum‐mechanical formalism in the limit of zero particle radius. At short interaction distances, however, a promising nonlinear and nonlocal theory emerges. This model is not only capable of achieving a conceptually satisfying synthesis of the particle‐wave dualism, but may also lead to a rational resolution of epistemological problems in the quantum‐theoretical measurement process. Within experimental errors the results for, e.g., the hydrogen atom can be reproduced by appropriately specifying the nature of the nonlinear self‐interaction. It is speculated that field theoretical issues raised by such notions as identical particles, field quantization and renormalization are already incorporated or resolved by this nonlocal theory, at least in principle.
The theoretical foundations of quantum mechanics and de Broglie–Bohm mechanics are analyzed and it is shown that both theories employ a formal approach to microphysics. By using a realistic approach it can be established that the internal structures of extended particles comply with a wave equation. Including external potentials yields the Schrödinger equation in this context, is arbitrary due to internal energy components. The statistical interpretation of wave functions in quantum theory as well as Heisenberg’s uncertainty relations are shown to be an expression of this, fundamental arbitrariness. Electrons and photons can be described by an identical formalism, providing formulations equivalent to the Maxwell equations. Electrostatic interactions justify the initial assumption of electron–wave stability: the stability of electron waves can be referred to vanishing intrinsic fields of interaction. The theory finally points out some fundamental difficulties for a fully covariant formulation of quantum electrodynamics, which seem to be related to the existing infinity problems in this field.
We discuss the statistical origin of quantum theoretical description and show that within the scope of the Mie-Einstein field approach, with particles being considered as lumps of some material field satisfying nonlinear equations, such a description appears to be natural and inevitable since it is impossible to specify initial conditions for the extended particle-soliton. Beside this, it is shown that on solitons’ basis, the special stochastic representation of quantum mechanics can be constructed, the latter one being equivalent to the theory of nonlocal hidden variables.
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.
In this work, we review and extend a version of the old attempt made by Louis de broglie for interpreting quantum mechanics in realistic terms, namely the double solution. In this theory quantum particles are localized waves, i.e, solitons, that are solutions of relativistic nonlinear field equations. The theory that we present here is the natural extension of this old work and relies on a strong time-symmetry requiring the presence of advanced and retarded waves converging on particles. Using this method, we are able to justify wave-particle duality and to explain the violations of Bell's inequalities. Moreover, the theory recovers the predictions of the pilot-wave theory of de Borglie and Bohm, often known as Bohmian mechanics. As a direct consequence, we reinterpret the nonlocal action at a distance presents in the pilot-wave theory. In the double solution developed here there is fundamentally no action at a distance but the theory requires a form of superdeterminism driven by time-symmetry.
In this work, we review and extend a version of the old attempt made by Louis de Broglie for interpreting quantum mechanics in realistic terms, namely, the double solution. In this theory, quantum particles are localized waves, i.e., solitons, that are solutions of relativistic nonlinear field equations. The theory that we present here is the natural extension of this old work and relies on a strong time-symmetry requiring the presence of advanced and retarded waves converging on particles. Using this method, we are able to justify wave–particle duality and to explain the violations of Bell’s inequalities. Moreover, the theory recovers the predictions of the pilot-wave theory of de Broglie and Bohm, often known as Bohmian mechanics. As a direct consequence, we reinterpret the nonlocal action-at-a-distance in the pilot-wave theory. In the double solution developed here, there is fundamentally no action-at-a-distance but the theory requires a form of superdeterminism driven by time-symmetry.
Problems with the extension to relativity. The Klein–Gordon equation The attempt to extend the causal interpretation to cover relativistic quantum mechanics raises several problems of technique and principle. These relate not only to difficulties peculiar to the quantum theory of motion but have to do with still unresolved issues in the underlying quantum mechanical formalism (such as the appearance of infinities in the computation of physical quantities and the lack of a relativistic theory of measurement). Some problems are common to both, such as the difficulty of defining positive densities. The subject has not been extensively investigated and in this chapter we shall discuss only some of the relevant issues. There are two particularly important, overlapping problems: Can the trajectory concept be retained in the relativistic quantum domain? And: how do we interpret relativistic quantum fields? The unresolved issue underlying both questions is the extent to which we should require a quantum theory to be ‘relativistic’. It is known that Lorentz covariant wave equations can have consequences in conflict with relativity. This, as we shall see, is indeed the case with the current relativistic formalism as interpreted, or completed, by the quantum theory of motion. Especially in field theory, the individual processes that build up and explain the covariant statistical predictions of the formalism exhibit features that are in conflict with our entrenched ideas regarding covariance and locality as necessary concomitants of relativity theory. That is, relativity is statistically valid but the individual events do not have an intrinsically relativistic character.