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arXiv:quant-ph/0301119

The continuum limit of the Bell model

Colin, Samuel

Published Jan 22, 2003• 7 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0301119

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•Jan 1, 2007•80 citations
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.
Examines a suggestion by Bell (1984) that a deterministic model might be furnished by the continuum limit of an objective stochastic model which he has proposed for quantum field theory on a lattice. The virtues of models of Bell's type for general quantum systems are discussed, and it is shown that for a system of a fixed number of particles such a model does yield the deterministic de Broglie-Bohm model in an appropriate limit. It is argued, however, that a deterministic limit is unlikely to emerge from a quantum field theory capable of describing particle production and decay. An interpretation of quantum field theory is proposed which has the advantages of Bell's interpretation but is closer to conventional quantum mechanics.
An interpretation of quantum mechanics in terms of classical concepts, ``beables,'' due to de Broglie, Bohm, and Bell (BBB) is generalized and further developed. By assuming that all physical quantities take discrete values on sufficiently small scales, we can use this interpretation to give trajectories for all possible quantities, including the position of a particle, its spin, etc. When applied to position, it is shown that, in the continuum limit, this interpretation reduces to the causal one of Bohm. As an illustration, the BBB trajectories are computed explicitly in two simple models.
In his paper (1986 Beables for quantum field theory Phys. Rep. 137 49–54) John S Bell proposed how to associate particle trajectories with a lattice quantum field theory, yielding what can be regarded as a |Ψ| 2 -distributed Markov process on the appropriate configuration space. A similar process can be defined in the continuum, for more or less any regularized quantum field theory; we call such processes Bell-type quantum field theories. We describe methods for explicitly constructing these processes. These concern, in addition to the definition of the Markov processes, the efficient calculation of jump rates, how to obtain the process from the processes corresponding to the free and interaction Hamiltonian alone, and how to obtain the free process from the free Hamiltonian or, alternatively, from the one-particle process by a construction analogous to ‘second quantization’. As an example, we consider the process for a second quantized Dirac field in an external electromagnetic field.
We present a pilot-wave model for quantum field theory in which the Dirac sea is taken seriously. The model ascribes particle trajectories to all the fermions, including the fermions filling the Dirac sea. The model is deterministic and applies to the regime in which fermion number is superselected. This work is a further elaboration of work by Colin, in which a Dirac sea pilot-wave model is presented for quantum electrodynamics. We extend his work to non-electromagnetic interactions, we discuss a cut-off regularization of the pilot-wave model and study how it reproduces the standard quantum predictions. The Dirac sea pilot-wave model can be seen as a possible continuum generalization of a lattice model by Bell. It can also be seen as a development and generalization of the ideas by Bohm, Hiley and Kaloyerou, who also suggested the use of the Dirac sea for the development of a pilot-wave model for quantum electrodynamics.
Bell's theorem rules out developing a locally causal theory to describe quantum phenomena. Many take this to imply that any model of quantum entanglement must employ variables (called beables by Bell) which follow nonlocal rules, even though signaling is local. The alternative is to adopt an all-at-once (block universe) approach, with beables which may depend on both past and future inputs, even though signaling is causal. Within this lenient-causality approach (a.k.a. retrocausal), simple cases of entanglement have been successfully described by locally mediated stochastic toy models, i.e.,~toy models which are local in a sense which generalizes Bell's local causality. Developing a widely applicable reformulation of quantum mechanics along these lines is a grand challenge. This work presents a general framework for such models and theories, and identifies the corresponding ontic and epistemic states. The epistemic state is closely analogous to the quantum state, yielding an explanation for the collapse of the wavefunction. In the case of the models of the framework, it is clear what the information is about. The expression for the empirically verifiable predictions of the models in terms of the ontic and epistemic states displays remarkable parallels to the Born rule. A toy-model example is discussed.
We present a pilot-wave model for quantum field theory in which the Dirac sea is taken seriously. The model ascribes particle trajectories to all the fermions, including the fermions filling the Dirac sea. The model is deterministic and applies to the regime in which fermion-number is super-selected. This work is a further elaboration of work by Colin, in which a Dirac sea pilot-wave model is presented for quantum electrodynamics. We extend his work to non-electromagnetic interactions, we discuss the regularization of the pilot-wave model and study how it reproduces the standard quantum predictions. The Dirac sea pilot-model can be seen as a possible continuum generalization of a lattice model by Bell. It can also be seen as a development and generalization of the ideas by Bohm, Hiley and Kaloyerou, who also suggested the use of the Dirac sea for the development of a pilot-wave model for quantum electrodynamics. 1