Abstract
We present a unified field theory of wave and particle in quantum mechanics. This emerges from an investigation of three weaknesses in the de Broglie–Bohm theory: its reliance on the quantum probability formula to justify the particle-guidance equation; its insouciance regarding the absence of reciprocal action of the particle on the guiding wavefunction; and its lack of a unified model to represent its inseparable components. Following the author’s previous work, these problems are examined within an analytical framework by requiring that the wave–particle composite exhibits no observable differences with a quantum system. This scheme is implemented by appealing to symmetries (global gauge and spacetime translations) and imposing equality of the corresponding conserved Noether densities (matter, energy, and momentum) with their Schrödinger counterparts. In conjunction with the condition of time-reversal covariance, this implies the de Broglie–Bohm law for the particle where the quantum potential mediates the wave–particle interaction (we also show how the time-reversal assumption may be replaced by a statistical condition). The method clarifies the nature of the composite’s mass, and its energy and momentum conservation laws. Our principal result is the unification of the Schrödinger equation and the de Broglie–Bohm law in a single inhomogeneous equation whose solution amalgamates the wavefunction and a singular soliton model of the particle in a unified spacetime field. The wavefunction suffers no reaction from the particle since it is the homogeneous part of the unified field to whose source the particle contributes via the quantum potential. The theory is extended to many-body systems. We review de Broglie’s objections to the pilot-wave theory and suggest that our field-theoretic description provides a realization of his hitherto unfulfilled ‘double solution’ programme. A revised set of postulates for the de Broglie–Bohm theory is proposed in which the unified field is taken as the basic descriptive element of a physical system.
Highlights
The supposition that the wavefunction ψ provides the most complete description of the state of a physical system that is in principle possible has permeated interpretational discourse since quantum theory’s inception
Treating ψ and qi as u’s constituents, its equation unifies their interaction: the homogeneous part is the Schrödinger equation obeyed by the unmodified ψ and, as we show using the solution (3.41), the inhomogeneous equation is equivalent to the particle guidance equation: Guidance Theorem 3
Our initial aim was to find a nonstatistical vindication of the de Broglie–Bohm law that incorporates the nonreactive character of the particle on the ψ wave
Summary
The supposition that the wavefunction ψ provides the most complete description of the state of a physical system that is in principle possible has permeated interpretational discourse since quantum theory’s inception. It is an arbitrary and unproven conjecture adopted out of theoretical choice rather than empirical imperative. The most successful example of such a completion is the de Broglie–Bohm causal interpretation, or pilot-wave theory, where in addition to ψ, conceived as a physically real-guiding field, the state comprises a material corpuscle traversing a well-defined spacetime trajectory. Whilst the virtues of the de Broglie–Bohm theory cannot nowadays be gainsaid, its standard exposition is open to question. The remainder of the paper works out this programme in detail and culminates in a proposed reformulation of the de Broglie–Bohm theory based on a unified field whose law of motion combines those of the field and the particle
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