Paul Budnik paul@mtnmath.com

Einstein did not believe that God plays dice and thought a more complete theory would predict the actual outcome of experiments. He argued that quantities that are conserved absolutely (such as momentum or energy) must correspond to some objective element of physical reality. Because QM does not model this he felt it must be incomplete[Note 1].

It is possible that events are the result of objective physical processes that we do not yet understand. These processes may determine the actual outcome of experiments and not just their probabilities. Certainly that is the natural assumption to make. Any one who does not understand QM and many who have only a superficial understanding naturally think that observations come about from some objective physical process even if they think we can only predict probabilities.

There have been numerous attempts to develop such alternatives. These are often referred to as `hidden variables' theories. Bell proved that such theories cannot deal with quantum entanglement without introducing explicitly nonlocal mechanisms [Note 2]. Quantum entanglement refers to the way observations of two particles are correlated after the particles interact. It comes about because the conservation laws are exact but most observations are probabilistic. Nonlocal operations in hidden variables theories might not seem such a drawback since QM itself must use explicit nonlocal mechanism to deal with entanglement. However in QM the non-locality is in a wave function which most do not consider to be a physical entity. This makes the non-locality less offensive or at least easier to rationalize away.

It might seem that the tables have been turned on Einstein. The very argument he used in EPR to show QM must be incomplete requires that hidden variables models have explicit nonlocal operations. However it is experiments and not theoretical arguments that now must decide the issue. Although all experiments to date have produced results consistent with the predictions of QM, there is general agreement that the existing experiments are inconclusive [Note 3]. There is no conclusive experimental confirmation of the nonlocal predictions of QM. If these experiments eventually confirm locality and not QM Einstein will be largely vindicated for exactly the reasons he gave in EPR. Final vindication will depend on the development of a more complete theory.

Most physicists (including Bell before his untimely death) believe QM is correct in predicting locality is violated. Why do they have so much more faith in the strange formalism of QM than in basic principles like locality or the notion that observations are produced by objective processes? I think the reason may be that they are viewing these problems in the wrong conceptual framework. The term `hidden variables' suggests a theory of classical-like particles with additional hidden variables. However quantum entanglement and the behavior of multi-particle systems strongly suggests that whatever underlies quantum effects it is nothing like classical particles. If that is so then any attempt to develop a more complete theory in this framework can only lead to frustration and failure. The fault may not be in classical principles like locality or determinism. They failure may only be in the imagination of those who are convinced that no more complete theory is possible.

One alternative to classical particles is to think of observations as focal points in state space of nonlinear transformations of the wave function. Attractors in Chaos theory provide one model of processes like this. Perhaps there is an objective physical wave function and QM only models the average or statistical behavior of this wave function. Perhaps the structure of this physical wave function determines the probability that the wave function will transform nonlinearly at a particular location. If this is so then probability in QM combines two very different kinds of probabilities. The first is the probability associated with our state of ignorance about the detailed behavior of the physical wave function. The second is the probability that the physical wave function will transform with a particular focal point.

A model of this type might be able to explain existing experimental results and still never violate locality. I have advocated a class of models of this type based on using a discretized finite difference equation rather then a continuous differential equation to model the wave function [Note 4]. The nonlinearity that must be introduced to discretize the difference equation is a source of chaotic like behavior. In this model the enforcement of the conservation laws comes about through a process of converging to a stable state. Information that enforces these laws is stored holographic-like over a wide region.

Most would agree that the best solution to the measurement problem would be a more complete theory. Where people part company is in their belief in whether such a thing is possible. All attempts to prove it impossible (starting with von Neumann [Note 5]) have been shown to be flawed [Note 6]. It is in part Bell's analysis of these proofs that led to his proof about locality in QM. Bell has transformed a significant part of this issue to one experimenters can address. If nature violates locality in the way QM predicts then a local deterministic theory of the kind Einstein was searching for is not possible. If QM is incorrect in making these predictions then a more accurate and more complete theory is a necessity. Such a theory is quite likely to account for events by an objective physical process.

References:

[1] A. Einstein, B. Podolsky and N.
Rosen, Can quantum-mechanical descriptions of physical reality be
considered complete?, *Physical Review*, 47, 777 (1935).
Reprinted in *Quantum Theory and Measurement*, p. 139,
(1987).

[2] J. S. Bell, On the Einstein
Podolosky Rosen Paradox, *Physics*, 1, 195-200 (1964).
Reprinted in *Quantum Theory and Measurement*, p. 403,
(1987).

[3] P. G. Kwiat, P. H. Eberhard, A. M.
Steinberg, and R. Y. Chiao, Proposal for a loophole-free Bell
inequality experiment, *Physical Reviews A*, 49, 3209
(1994).

[4] P. Budnik, Developing a local
deterministic theory to account for quantum mechanical effects,
hep-th/9410153, (1995).

[5] J. von Neumann, *The
Mathematical Foundations of Quantum Mechanics*, Princeton
University Press, N. J., (1955).

[6] J. S. Bell, On the the problem of
hidden variables in quantum mechanics, *Reviews of Modern
Physics*, 38, 447-452, (1966). Reprinted in *Quantum Theory
and Measurement*, p. 397, (1987).

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