Functional methods underlying classical mechanics, relativity and quantum theory

TL;DR

This paper explores a unified mathematical framework that connects classical mechanics, relativity, and quantum theory by embedding classical states into a Hilbert space, revealing deep relations and deriving the Born rule from classical probability distributions.

Contribution

It introduces a novel embedding of classical states into a Hilbert space, unifying the formalisms of classical, relativistic, and quantum physics, and derives the Born rule from classical probability distributions.

Findings

Classical states are represented as delta functions within a Hilbert space.

The embedding explains deep relations between classical, quantum, and relativistic physics.

The Born rule is derived from the probability distribution of classical positions.

Abstract

The paper investigates the physical content of a recently proposed mathematical framework that unifies the standard formalisms of classical mechanics, relativity and quantum theory. In the framework states of a classical particle are identified with Dirac delta functions. The classical space is "made" of these functions and becomes a submanifold in a Hilbert space of states of the particle. The resulting embedding of the classical space into the space of states is highly non-trivial and accounts for numerous deep relations between classical and quantum physics and relativity. One of the most striking results is the proof that the normal probability distribution of position of a macroscopic particle (equivalently, position of the corresponding delta state within the classical space submanifold) yields the Born rule for transitions between arbitrary quantum states.