A new approach to probabilities in mechanics

Richard Healey

Abstract Some physicists but many philosophers believe that standard Hilbert space quantum mechanics faces a serious measurement problem, whose solution requires a new theory or at least a novel interpretation of standard quantum mechanics. Itamar Pitowsky did not. Instead, he argued in a major essay (Pitowsky 2006) that quantum mechanics offers a new theory of probability. In these and other respects, his views paralleled those of QBists (quantum Bayesians): but their views on the objectivity of measurement outcomes diverged radically. Indeed, Itamar’s view of quantum probability depended on his subtle treatment of the objectivity of outcomes as events whose collective structure underlay this new theory of probability. His thesis is puzzling—that quantum mechanics requires a new theory of probability, as distinct from new ways of calculating probabilities that play the same role as other probabilities in physics and daily life. This essay articulates sources of puzzlement.

James G. Glimm, Arthur Jaffe

Norbert Wiener, Armand Siegel

A new type of representation of the wave function of quantum mechanics, and a new form of the statistical interpretation of the wave function, are presented. The variable in the new representation is a mapping of points of differential space, closely related to Hilbert space, but having an associated measure for suitably defined sets of points in it. In the new statistical interpretation, an eigenvalue of every observable defined in the system is associated with each point in differential space. This can be done in such a way that, for each observable, the measure (=probability) of the set of all differential-space points belonging to a given set of eigenvalues is equal to the quantum-mechanical probability, calculated in the usual way (Born statistical postulate), that an experiment will yield an eigenvalue in this set. Thus we obtain an interpretation of quantum mechanics in terms of probability densities or probabilities instead of probability amplitudes.

Vladimir I. Man’ko

The tomographic invertable map of the Wigner function onto the positive probability distribution function is studied. Alternatives to the Schrödinger evolution equation and to the energy level equation written for the positive probability distribution are discussed. Instead of the transition probability amplitude (Feynman path integral) a transition probability is introduced. A new formulation of the conventional quantum mechanics (without wave function and density matrix) based on the “probability representation” of quantum states is given. An equation for the propagator in the new formulation of quantum mechanics is derived. Some paradoxes of quantum mechanics are reconsidered.

Gian Fabrizio De Angelis