A Possible Explanation of the Superposition Principle

I tentatively suggest that the superposition principle of quantum mechanics is explicable in a mathematically natural way if it is possible to understand probability amplitudes as complex-valued logarithms. This notion is inspired by the fact that the quantum state may be interpreted as a measure of information. The object of this note is to sketch what J. A. Wheeler might call an “idea for an idea” [8]. The notion I present here points the way toward a possible new interpretation of quantum mechanics, although the development of this approach that I am able to offer in this paper is in itself certainly not yet complete enough to deserve such a lofty description. I hope to produce a more thorough treatment soon; but in the meantime I dare to think that the bare notion I describe here may be of sufficient interest to merit an airing in preprint form, and I commend it to the attention of those who may be more mathematically skilled than I am. I. The Feynman Problem Quantum mechanics is beset by a number of interpretational challenges arising from spectacularly non-classical phenomena such as nonlocality and superfluidity, as well as the host of difficulties surrounding the measurement problem. But the deepest mysteries surround our lack of understanding of origin of the basic rules of the theory. We can frame the problem very clearly by going back to the statement of the basic principles of quantum mechanics given by Richard Feynman and his co-authors in The Feynman Lectures on Physics [6]: (1) The probability of an event in an ideal experiment is given by the square of the absolute value of a complex number φ which is called the probability amplitude: P = probability (1) φ = probability amplitude (2) P = |φ|. (3) (2) When an event can occur in several alternative ways, the probability amplitude for the event is the sum of the probability amplitudes for each way considered separately. There is Email: [email protected]