Discrete binning: correspondence limit consistent analysis of classical probabilities using faux angular momenta and multi-polar harmonic moments to give 'quantum-like' probabilities

This paper develops discrete binning (DB) for N(i) slices for classical probability functions for an arbitrary number of classical continuous variables, x(i), where 0 <= x(i) <= 1 or -1 <= x(i) <= 1. Faux angular momenta, j(i), are introduced where 2j(i) + 1 = N(i), and the discrete probabilities for the various vertical bar j(i)m(i)> are calculated with a generalisation of the theory of Anderson and Aquilanti. Discrete probabilities are calculated from Legendre moments of the classical intensities with Clebsch-Gordan moments. The m(i) may represent vibrational quanta, rotational angular momenta, or discrete values of the impact parameter, scattering angles and other variables. DB directly yields probabilities for different mi, but in the correspondence limit (large j(i)) the discrete probabilities correspond to classical probabilities, I({x(mi)}) at known discrete values {x(mi)}. DB probabilities sum to unity, but some may be negative. Since the Clebsch-Gordan coefficients appropriate for this work are actually Gram ( discrete Chebyshev) polynomials, DB is equivalent to compression and/or smoothing of data using Gram polynomials. For large Ni, DB and histogram binning (HB) provide equivalent probabilities and statistical errors. However, smoothing can often reduce the statistical errors for DB probabilities. DB is related to Legendre moment binning (LMB), but DB guides the most consistent implementation of LMB. The rule of three is introduced to provide finer resolution for DB, HB, and LMB analysis. This also leads to fractional slice binning (FSB), which is equivalent to Gaussian binning. The paper presents one-, two-, and three-dimensional examples, and spectroscopic plots are very useful for summarising the results.