Probabilities in Quantum Mechanics

This note seeks to survey crucial issues and to pinpoint difficulties in the current controversy over the meaning of measurement [1] and its proper mathematical description in quantum physics. It deals pri­ marily with those philosophic and mathematical considerations which arise from the statistical nature of the results of physical measurements. The first part reviews and aims to clarify matters of more or less general interest; the second and third are devoted to some specific recent re­ sults concerning joint probabilities and phase-space distribution func­ tions. 1. One of the important features of quantum mechanical description is the scattering, the dispersion of measured values in repeated observa­ tions of a given quantity even when the physical state (1J'-function) of the system, which is the carrier of the quantity, is as precise and determinate as human ability permits. This fact has occasioned a variety of philoso­ phical explanations: BOHR'S and HEISENBERG'S early belief that the statistical dispersion inherent in a quantum mechanical state is the reflection of the uncertainties introduced by the necessary interaction with the measuring apparatus which occurs at the time of measurement; DE BROGLIE'S and BOHM'S suggestion that obscure factors not appearing in the analysis (hidden variables) account for these fluctuations; HEISEN­ BERG'S later appeal to the Aristotelean theory of potentia [2], according to which the measurement of an observable in a state other than its eigenstate converts a potential quantity into an actually existing one; the distinction between "possessed" and "latent" observables [3], which formulates a new version of the philosophic contrast between primary and second ary qualities.