Finite Automata, Random Fields, Quantum Fluctuations, and Bell Inequalities

Peter Morgan

Yale University


The obvious question is to ask how my approach may be relevant to an NKS type of approach. I would argue that the NKS book, and everything I have seen on the NKS forum, does not make sufficient contact with the Bell inequalities argument. Much of the resistance of physicists to the NKS approach is because finite automata appear to satisfy the assumptions required to derive Bell inequalities. (I will only marginally address what I see as the second major point of resistance, that model building is generally decried for fundamental physics in favor of a constraint or principle-driven construction of theories. This is nonetheless very significant for NKS.)

I would argue that NKS models, considered statistically, which in the presence of significant thermal or quantum fluctuations they must be, can be formulated as classical random fields. Once an NKS model is formulated as a random field, it presumably is subject to the usual issues of lattice regularized statistical physics, under which fine details of the models are washed out, so that it is possible to discuss NKS models using effective field theory methods. Since Bell-type inequalities are relations between probabilities or expected values, any argument that seeks to justify NKS models to physicists has to construct a probabilistic model, and show why it reasonably can be thought not to satisfy the assumptions required to derive Bell inequalities.

The argument of my paper "Bell inequalities for random fields" is significantly different from the usual Bell inequalities argument, which focuses on particle properties. My approach is to consider that particles are derived concepts, that the field is primary. In my understanding, much of the huge literature on Bell inequalities does not argue carefully enough to be applicable to classical random fields, because there is an almost universal assumption that there are particles and that they have well-defined properties. In quantum field theory, it is generally accepted by mathematical physicists that the idea of a particle has become very subtle, although particle physicists do not generally have to consider these issues. One of the points of contact of NKS models with my approach is that NKS models naturally take particles to be not point-like.

In the field context, it is well known that there are extensive correlations between observables of a classical field at thermal equilibrium, which immediately make it impossible to derive Bell inequalities. Correlations between various components of the field are well known to prevent the derivation of Bell inequalities, but are almost universally dismissed as “conspiracy.” Gerard 't Hooft has maintained that the conspiracy loophole is big enough for his models not to satisfy Bell inequalities, but to my knowledge he has failed to convince physicists that conspiracy is a reasonable way out. Such correlations would certainly be present in a probabilistic presentation of a finite automata model. The arguments that establish that these correlations are very natural for a classical random field, that they are not a conspiracy for random fields, but are a conspiracy for particle property models, are quite simple, but they have to be made with sufficient clarity and simplicity to be ineluctable, which I cannot say that I have achieved. However, the argument I have made may be enough to inspire someone in the NKS community to make the argument in a slightly simpler way, which I would welcome.

A secondary issue, but very important to NKS, is that quantum fluctuations and thermal fluctuations can be distinguished very clearly in the free field context, as Poincaré invariant and non-Poincaré invariant fluctuations. This is a fundamental enough distinction to suggest that it might apply to interacting fields. With the inclusion of Poincaré invariant fluctuations, NKS models would have a chance of reproducing the phenomenology of relativistic quantum field theory—without such fluctuations. I do not believe there is any hope to do so. Obviously it is only in the mathematical detail that any argument can be made convincing to physicists.