Probabilistic observables, conditional correlations, and quantum physics

Abstract

We discuss the classical statistics of isolated subsystems. Only a small part of the information contained in the classical probability distribution for the subsystem and its environment is available for the description of the isolated subsystem. The "coarse graining of the information" to micro-states implies probabilistic observables. For two-level probabilistic observables only a probability for finding the values one or minus one can be given for any micro-state, while such observables can be realized as classical observables with sharp values on a substate level. For a continuous family of micro-states parameterized by a sphere all the quantum mechanical laws for a two-state system follow under the assumption that the purity of the ensemble is conserved by the time evolution. The non-commutative correlation functions of quantum mechanics correspond to the use of conditional correlation functions in classical statistics. We further discuss the classical statistical realization of entanglement within a system corresponding to four-state quantum mechanics. We conclude that quantum mechanics can be derived from a classical statistical setting with infinitely many micro-states.Comment: substantially extended discussion of conditional correlations and probabilistic observables, new references 28 pages, 2 figure

Quantum Particles from Classical Probabilities in Phase Space

“…The property of incomplete statistics [17], [3] where the measurement correlation is not based on joint probabilities, is a key point [4], [3] for an understanding of quantum mechanics. Indeed, complete statistical systems, for which the measurement correlation employs the joint probabilities, have to obey Bell's inequalities [18].…”

“…However, on the level of the one-particle subsystem a description in terms of complete statistics is typically no longer possible [3]. For incomplete statistics the EPR-paradoxon [20] can be resolved satisfactorily [3], [4].…”

“…However, it does not necessarily take a fixed value in every one-particle state. In general, on the level of the one-particle subsystem the observables become probabilistic observables [4], [21]. For any state ψ C (x, p) of the subsystem they have a probabilistic distribution of the values in their spectrum.…”

“…In ref. [3][4][5] the basic conceptual settings how quantum physics can emerge from a classical statistical ensemble have been described in detail. It was shown how the quantum formalism with non-commuting operators arises form a description of a subsystem.…”

“…It was shown how the quantum formalism with non-commuting operators arises form a description of a subsystem. We have discussed explicitly quantum mechanically entangled states [6], [4], [5] and shown that the measurement correlation AB m is equivalent to the usual quantum correlation and violates Bell's inequalities [7].…”