Projective Limits of State Spaces III. Toy-Models

TL;DR

This paper explores projective state space frameworks in quantum theory through simple toy-models, demonstrating how classical and quantum dynamics can be represented and quantized within this approach.

Contribution

It introduces a method to implement and test dynamics in the projective state space formalism using toy-models, including a classical reformulation of the Schrödinger equation and its second quantization.

Findings

Successfully reformulated Schrödinger equation as a classical field theory within the projective framework

Reproduced the physical content of Fock quantization using the toy-model approach

Demonstrated the potential of the projective formalism for quantum dynamics

Abstract

In this series of papers, we investigate the projective framework initiated by Jerzy Kijowski and Andrzej Oko{\l}\'ow, which describes the states of a quantum theory as projective families of density matrices. A strategy to implement the dynamics in this formalism was presented in our first paper, which we now test in two simple toy-models. The first one is a very basic linear model, meant as an illustration of the general procedure, and we will only discuss it at the classical level. In the second one, we reformulate the Schr\"odinger equation, treated as a classical field theory, within this projective framework, and proceed to its (non-relativistic) second quantization. We are then able to reproduce the physical content of the usual Fock quantization.