The origin of quantum mechanics and time from incomplete classical statistics

The quantum mechanical concepts of states and operators arise naturally from the question how expectation values of observables can be computed in classical statistical systems where only incomplete local information about the probability distribution is available. The notions of evolution and time are related to the translation of the available information between neighboring local regions. The complex structure of quantum mechanics, the superposition of states and the interference effects characteristic for quantum statistics find a simple origin. This suggests the emergence of quantum mechanics from classical statistics with infinitely many degrees of freedom. 1 Incomplete statistics The reason why the physical laws can be described in terms of quantum mechanics has remained a mystery from its early beginnings up to now. Physicists have become used to quantum mechanics because of overwhelming experimental evidence. Yet we do not understand the “why” of its basic principles like the formulation in terms of states and non-commuting operators, the superposition of “probability amplitudes” and associated interference effects. An important ingredient in the conceptual foundations of quantum mechanics is the emphasis on measurable quantities while discarding classical notions not accessible to measurement. On the other hand, the understanding of classical statistical systems with infinitely many degrees of freedom has made tremendous progress in the past decades. Important links to quantum mechanics have been established by the use of path integrals [1] for the description of the quantum mechanical evolution. This process has diminished considerably the distance between classical statistics and quantum mechanics or quantum field theory [2]. One wonders if quantum mechanics cannot be understood as a particular structure of classical statistical systems with infinitely many degrees of freedom or general statistics. Within general statistics [3] the notions of distance, geometry and topology can be formulated in terms of properties of correlation functions [4]. The question arises if the notions of time and quantum mechanical evolution can find their origin within the same framework. The formulation of the basic partition function for classical statistical systems with infinitely many degrees of freedom uses implicitly an assumption of “completeness of the statistical information”. This means that we assign a probability to everyone of the infinitely many configurations. The specification of the probability distribution contains therefore an “infinite amount of information”. This contrasts with the simple observation that only a finite amount of information is available in practice for the computation of the outcome of any physical measurement. A concentration on measurable quantities suggests that the assumption of completeness of the statistical information may have to be abandoned. In this note we explore consequences of “incomplete statistics” which deals with situations where only partial information about the probability distribution is available. In particular, we consider extended systems for which only local information about the probability distribution is given. We will see that the quantum mechanical concepts of states, operators, evolution and interference emerge naturally in this setting. As an example we consider a classical statistical system where the infinitely many degrees of freedom φn (n ∈ ZZ) are ordered in an infinite chain. We concentrate on a “local region” |ñ| < n̄ and assume that the probability distribution p[φ] has a “locality property” in the sense that the probability for any configuration of the “local variables” φñ is independent of the values that take the variables φm with |m| > n̄. Furthermore, we assume that the probability distribution for the φñ is known for given values of the variables φn̄, φ−n̄ at the border of the local interval. This statistical system cannot be reduced to a system with a finite number of degrees