Relating the wave-function collapse with Euler's formula, with applications to Classical Statistical Field Theory

TL;DR

This paper links the wave-function collapse in quantum mechanics to Euler's formula, proposing a unified approach that addresses mathematical inconsistencies in Classical Statistical Field Theory and offers new insights into quantum interpretations.

Contribution

It introduces a novel wave-function parametrization as a map from an hypersphere to probability distributions, connecting quantum collapse to Euler's formula, and applying this to classical and quantum field theories.

Findings

Wave-function is a probability distribution parametrization.

Wave-function collapse is a recursion of 2D collapses.

Proposes a new mathematical foundation for Classical Statistical Field Theory.

Abstract

One attractive interpretation of quantum mechanics is the ensemble interpretation, where Quantum Mechanics merely describes a statistical ensemble of objects and not individual objects. But this interpretation does not address why the wave-function plays a central role in the calculations of probabilities, unlike most other interpretations of quantum mechanics. On the other hand, Classical Statistical Field Theory suffers from severe mathematical inconsistencies (specially for Hamiltonians which are non-polynomial in the fields, e.g. General relativistic statistical field theory). We claim that both problems are related to each other and we propose a solution to both. We prove: 1) the wave-function is a parametrization of any probability distribution of a statistical ensemble: there is a surjective map from an hypersphere to the set of all probability distributions; 2) for a quantum system defined in a 2-dimensional real Hilbert space, the role of the (2-dimensional real) wave-function is identical to the role of the Euler's formula in engineering, while the collapse of the wave-function is identical to selecting the real part of a complex number; 3) the collapse of the wave-function of any quantum system is a recursion of collapses of 2-dimensional real wave-functions; 4) the wave-function parametrization is key in the mathematical definition of Classical Statistical Field Theory we propose here. The same formalism is applied to Quantum Yang-Mills theory and Quantum Gravity in another article.