A quantum field theory as emergent description of constrained supersymmetric classical dynamics

TL;DR

This paper demonstrates how a supersymmetric classical dynamical system can be reformulated as an emergent quantum field theory through constraints and symmetry breaking, linking classical and quantum descriptions.

Contribution

It introduces a novel approach to derive quantum field theory from classical supersymmetric dynamics using the Hilbert space framework and constraints.

Findings

Classical Liouville equation can be transformed into a functional Schrödinger equation.

A constraint removes unstable negative spectrum, leading to a positive-definite Hamiltonian.

Emergent quantum behavior arises from symmetry breaking and phase space choice.

Abstract

Deterministic dynamical models are discussed which can be described in quantum mechanical terms. In particular, a local quantum field theory is presented which is a supersymmetric classical model. -- The Hilbert space approach of Koopman and von Neumann is used to study the evolution of an ensemble of such classical systems. With the help of the supersymmetry algebra, the corresponding Liouville operator can be decomposed into two contributions, with positive and negative spectrum, respectively. The unstable negative part is eliminated by a constraint on physical states, which is invariant under the Hamiltonian flow. In this way, choosing suitable phase space coordinates, the classical Liouville equation becomes a functional Schroedinger equation of a genuine quantum field theory. Quantization here is intimately related to the constraint, which selects the part of Hilbert space where the Hamilton operator is positive. This is interpreted as dynamical symmetry breaking in an extended model, introducing a mass scale which discriminates classical dynamics beneath from emergent quantum mechanical behaviour.