The distribution postulate in Bohm's theory

Erik Sjöqvist, Henrik Carlsen

C. Colijn, Edward R. Vrscay

The evolution of probability distributions for quantum systems defined by the de Broglie–Bohm causal interpretation of quantum mechanics is examined in the context of whether, in general, a non-|ψ|2 distribution will evolve to approximate a |ψ|2 distribution, as has been suggested by Bohm and others. This study focusses on hydrogen eigenstates and two-state hydrogen transitions stimulated by semi-classical radiation. It is shown that this process of ‘relaxation’ will not in general occur in such hydrogen systems, and thus that the de Broglie–Bohm theory must retain the |ψ|2 postulate if it is to be consistent with the standard quantum theory of individual systems.

Jitendra Subramanyam

If there's one lesson that philosophers seem to have learned from quantum mechanics it is that the quantum realm is governed by indeterministic laws. However, in 1952 David Bohm constructed a deterministic theory and proved that it is empirically equivalent to quantum mechanics. One of the three conditions that are together sufficient for the empirical adequacy of Bohm's theory is the statistical postulate. This postulate demands a very particular initial condition to hold for ensembles of quantum systems. I argue that the statistical postulate requires a justification. A justification would consist of showing how the dynamics of Bohm's theory is able to generate the initial conditions specified by the statistical postulate. I examine two deterministic justifications of the statistical postulate, one by Valentini and the other by Durr, Goldstein and Zanghi. I explain precisely the ways in which they are lacking. The basic problem is that crucial dynamical details are missing, ones needed to complete the justification. Whether these missing bits can be filled in is an open question. Until a justification is at hand, and its application to real-world ensembles is clarified, the empirical adequacy of Bohm's theory remains an open question. Bohm himself proposed a stochastic or indeterministic justification for the statistical postulate. It has been shown that the dynamics of this stochastic theory does not have the resources to justify the statistical postulate. In addition I show that the mathematical argument Bohm uses in his justification is flawed. Other stochastic theories do justify the statistical postulate but have other problems. The issue of indeterminism in the quantum realm is far from settled. Bohm's deterministic dynamics enables an investigation into the limits (if any) of a deterministic dynamics for the quantum realm. The properties of Bohm's deterministic equations of motion remain largely unknown. What we do know is that if the quantum realm is indeterministic it will not be for any of the reasons we have so far been given.

Jeffrey A. Barrett, Eddy Keming Chen, Josiah Lopez-Wild

In order to make the right empirical predictions Bohmian mechanics requires a special statistical boundary condition -- the distribution postulate -- but it is unclear how best to understand this condition. We show how one might use the theory of algorithmic randomness to formulate the distribution postulate as an objective constraining law. The framework requires us to say something about admissible quantum-mechanical states and measurements. In return, algorithmic Bohmian mechanics (aBM) guarantees the standard Born statistics for a collection of canonical quantum experiments in the limit, not just with high probability. The algorithmic distribution postulate provides a sharp typicality condition, clarifies the status of quantum probabilities in the deterministic theory, and provides a concrete example of how notions provided by the theory of algorithmic randomness can aid in specifying the content of a physical law.

Mariangiola Dezani-ciancaglini A, Stefano Guerrini, Adolfo Piperno

Jean Bricmont