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European Journal of Physics

Addendum on ‘Natural number dynamics: reconstructing physics from generalized atomicity’ (2021 Eur. J. Phys. 42 025402)

T. R. Robinson

Published Aug 28, 2024• 1 citation
Publisher

Abstract

Abstract The derivation of the Schrödinger equation is revisited with particular emphasis on the role of the Riccati equation and the relationship between the Hamiltonian and natural number operators.
DOI
10.1088/1361-6404/ad74ba

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Abstract A novel approach is presented in which some key basic equations of physics are reconstructed from a generalized atomicity viewpoint i.e., one in which a system is characterised simply by the number of items it contains and hence by the natural numbers alone. The natural numbers may then be associated with the eigenvalue spectrum of an operator. The non-negative nature of the natural numbers leads to the properties of creation and annihilation operators. By examining the parastatistical properties of the number operator, it is also shown that there is only one linear representation that allows a primitive system to be created. An interpretation of the nonlinear factorization of the number operator is also given in terms of the Hamiltonian for bound states of the Schrödinger equation for the case of general one-dimensional potentials. Introducing multiple categories of items yields the standard equations of many-body systems with interaction and scattering, and the existence of bosons and fermions.
In this paper we revisit the problem of decoherence by applying the block operator matrices analysis. The Riccati algebraic equation associated with the Hamiltonian describing the process of decoherence is studied. We prove that if the environment responsible for decoherence is invariant with respect to the antilinear transformation then the antilinear operator solves the Riccati equation in question. We also argue that this solution leads to neither a linear nor an antilinear operator similarity matrix. Therefore, we cannot use the standard procedure for solving a linear differential equation (e.g., Schrödinger equation). Furthermore, the explicit solution of the Riccati equation is found for the case where the environmental operators commute with each other. We discuss the connection between our results and the standard description of decoherence (one that uses the Kraus representation). We show that the reduced dynamics we obtain does not have the Kraus representation if the initial correlations between the system and its environment are present. However, for any initial state of the system (even when the correlations occur) reduced dynamics can be written in a manageable way.
Abstract The idea presented here of a general quantization rule for bound states is mainly based on the Riccati equation which is a result of the transformed, time-independent, one-dimensional Schrödinger equation. The condition imposed on the logarithmic derivative of the ground state function W 0 allows to present the Riccati equation as the unit circle equation with winding number equal to one which, by appropriately chosen transformations, can be converted into the unit circle equation with multiple winding number. As a consequence, a completely new quantization condition, which gives exact results for any quantum number, is obtained.