A transformational approach to collective behavior

TL;DR

This paper introduces a new theoretical framework for analyzing collective systems using a generalized canonical transformation approach, leading to the Heisenberg Scattering Transformation, which aids in understanding system behavior and coupling.

Contribution

It develops a novel theory extending canonical transformations to collective fields and introduces the Heisenberg Scattering Transformation for analyzing collective behavior.

Findings

Derivation of the Heisenberg Scattering Transformation (HST) as a localized Fourier Transform.

Formalization of how different collective systems couple into systems-of-systems.

Connection between measurement limitations and quantization of stochastic probabilities.

Abstract

This paper presents a revolutionary approach to the characterization, forecast, and control of collective systems. Collective systems are an ensemble of conservatively interacting entities. The evolution of the entities are determined by symmetries of the entities. Collective systems take many different forms. A plasma is a collective of charged particles, a fluid is a collective of molecules, a elementary field is a collective of elementary particles, and a cosmos is a collective of celestial bodies. Our new theory builds on the canonical transformation approach to dynamics. This approach recognizes that the symmetry leads to the conservation of a real function, that is the infinitesimal generator of a Lie group. The finite generator of the canonical transformation is derived from the infinitesimal generator by the solution of the Hamilton-Jacobi equation. This generating function is also known as the action, the entropy, and the logarithmic likelihood. The new theory generalizes this generating function to the generating functional of the collective field. Finally, this paper derives the formula for the Mayer Cluster Expansion, or the S-matrix expansion of the generating functional. We call it the Heisenberg Scattering Transformation (HST). Practically, this is a localized Fourier Transformation, whose principal components give the singularity spectrums, that is the solution to the Renormalization Group Equations. Limitations on the measurement of the system (that is the Born Rule and the Heisenberg Uncertainty Principle) lead to quantization of the stochastic probabilities of the collective field. How different collective systems couple together to form systems-of-systems is formalized. The details of a practical implementation of the HST will be presented.