The Time Operator of Reals

Abstract

The purpose of the paper is to establish the continuum in terms of the complex systems physics. It is based upon the intuitionistic mathematics of Brouwer, implying a processual definition of real numbers that concerns the measurement problem. The Brouwerian continuum is proved to be a categorical skeleton of complex systems whose determining feature is the existence of time operator. Acting on continuous signals, the time operator represents multiresolution hierarchy of the measurement process. The wavelet domain hidden Markov model, that recapitulates statistical properties of the hierarchy, is applicable to a wide range of signals having experimental verification. It indicates a novel method that has already been proved tremendously useful in applied mathematics.