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arXiv:quant-ph/0310056 · Annales de la fondation Louis de Broglie/Annales de la Fondation Louis de Broglie

Beables for Quantum Electrodynamics

Colin, Samuel

Published Oct 8, 2003• 9 citations
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Abstract

We show that it is possible to obtain a realistic and deterministic model, based on a previous work of John Bell, which reproduces the experimental predictions of the orthodox interpretation of quantum electrodynamics.
DOI
10.48550/arxiv.quant-ph/0310056

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In a paper entitled Beables for Quantum Field Theory, John Bell has shown that it was possible to build a realistic interpretation of any hamiltonian lattice quantum field theory involving Fermi fields. His model is stochastic but Bell thought that it would become deterministic in the continuum limit. We show that it is indeed the case, under an assumption about the state of the universe, namely that the universe is in a state obtained from the positronic sea (all positron states occupied) by creating a finite number of negative charges. Moreover, the continuum model can be established directly. The physical interpretation is the following: the negative charges are in motion in the positronic sea and their positions are the beables of the Bell model.
John Bell once argued that one ought to select, out of the 'observables' of quantum theory, some subset of 'beables' that can be consistently ascribed determinate values. Moreover, this subset should be selected so as to guarantee (among other things) that we can dispense with the orthodox interpretation's loose talk about 'measurement values': "...the probability of a beable being a particular value would be calculated just as was formerly calculated the probability of observing that value". Working in the framework of C*-algebras (in particular, Segal algebras), I propose an algebraic characterization of those subsets of bounded observables of a quantum system that can have beable status with respect to any (fixed) state of the system. It turns out that observables with beable status in a state need not all commute (a possibility Bell himself does not consider), but they must at least form a certain kind of 'quasicommutative' subalgebra determined by the state. A virtue of the analysis is that it applies to beables with continuous spectra, usually neglected in discussions of the no-hidden-variables theorems. In the (very) special case where the algebra of observables for a system is representable on a finite-dimensional Hilbert space, I give a complete characterization of the maximal beable subalgebras determined by any state of the system; the infinite-dimensional case remains open. These results are discussed in relation to previous results of a similar nature, to 'no-collapse' interpretations of quantum mechanics, and to algebraic relativistic quantum field theory.
We present a way to construct a pilot-wave model for quantum electrodynamics. The idea is to introduce beables corresponding only to the bosonic and not to the fermionic degrees of freedom of the quantum state. We show that this is sufficient to reproduce the quantum predictions. The beables will be field beables corresponding to the electromagnetic field and will be introduced in a way similar to that of Bohm's model for the free electromagnetic field. Our approach is analogous to the situation in non-relativistic quantum theory, where Bell treated spin not as a beable but only as a property of the wave function. After presenting this model, we also discuss a simple way for introducing additional beables that represent the fermionic degrees of freedom.