On classical systems and measurements in quantum mechanics

Abstract

The recent rigorous derivation of the Born rule from the dynamical law of quantum mechanics Allahverdyan et al. (Phys Rep 525:1–166. https://doi.org/10.1016/j.physrep.2012.11.001 , 2013) is taken as incentive to reexamine whether quantum mechanics has to be an inherently probabilistic theory. It is shown, as an existence proof, that an alternative perspective on quantum mechanics is possible where the fundamental ontological element, the ket, is not probabilistic in nature and in which the Born rule can also be derived from the dynamics. The probabilistic phenomenology of quantum mechanics follows from a new definition of statistical state in the form of a probability measure on the Hilbert space of kets that is a replacement for the von Neumann statistical operator to address the lack of uniqueness in recovering the pure states included in mixed states, as was pointed out by Schrodinger. From the statistical state of a quantum system, classical variables are defined as collective variables with negligible dispersion. In this framework, classical variables can be chosen to define a derived classical system that obeys, by Ehrenfest’s theorem, the laws of classical mechanics and that describes the macroscopic behavior of the quantum system. The Born rule is derived from the dynamics of the statistical state of the quantum system composed of the observed system interacting with the measurement system and the role of the derived classical system in the process is exhibited. The approach suggests to formulate physical systems in second quantization in terms of local quantum fields to ensure conceptually equivalent treatment of space and time. A real double-slit experiment, as opposed to a thought experiment, is studied in detail to illustrate the measurement process.