The interpretation of quantum theory does neither influence its theoretical predictions nor the experimentally observed data. Nevertheless it is extremely important because it determines the direction of future research. One of the many controversial interpretations of quantum mechanics is the “statistical interpretation” or “ensemble interpretation” [2]. It presents a point of view, which is in opposition to most variants of the Copenhagen interpretation [3], but has been advocated by a large number of eminent physicists, including Einstein. It claims that quantum mechanics is incomplete with regard to the description of single events and that all its dynamic predictions are of a purely statistical nature. This means that, in general, a large number of measurements on identically prepared systems have to be performed in order to verify a (dynamical) prediction of quantum theory.
The origin of the time-dependent Schrödinger equation is of course an essential aspect for the interpretation of quantum mechanics. Recently a number of derivations of Schrödinger’s equation have been reported which use as a starting point not a particle Hamiltonian but a statistical ensemble. The basic assumptions underlying these works include special postulates about the structure of momentum fluctuations [15], the principle of minimum Fisher information [36, 46], a linear time-evolution law for a complex state variable [26], or the assumption of a classical stochastic force of unspecified form [21]. The work reported in this paper belongs to this class of theories, which do not “quantize” a single particle but a statistical ensemble. It is shown that Schrödinger’s equation may be derived from a small number of very general and simple assumptions - which are all essentially of a statistical nature. In a first step an infinite class of statistical theories is derived, containing a classical statistical theory as well as quantum mechanics. In a second step quantum mechanics is singled out as “most reasonable statistical theory” by imposing an additional requirement. This additional requirement is the principle of maximal disorder as realized by the principle of minimal Fisher information.
We begin in section 2 with a general discussion of the role of probability in physical theories. In section 3 the central ’statistical condition’ (first assumption) of this work is formulated. The set of corresponding statistical theories is derived in section 5. In sections 4 and 7 structural differences between quantum theory and classical statistical theories are investigated. The quantum mechanical rule for calculating expectation values is derived from the requirement of conservation of energy in the mean in section 6. In sections 7-9 the principle of maximal disorder is implemented and Fisher’s information measure is derived in section 10. Section 11 contains a detailed discussion of all assumptions and results and may be consulted in a first reading to obtain an overview of this work; questions of interpretation of the quantum theoretical formalism are also discussed in this section. In the last section 12 open questions for future research are listed.