Projective Limits of State Spaces I. Classical Formalism

TL;DR

This paper clarifies the classical symplectic structures underlying the projective framework for quantum state spaces, aiming to facilitate the implementation of dynamics in this formalism.

Contribution

It establishes the classical foundations of the projective state space formalism, focusing on symplectic limits to address dynamical challenges.

Findings

Clarified classical structures underlying the projective quantum state formalism.

Identified issues hindering dynamics implementation.

Proposed strategies for overcoming dynamical challenges.

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 (field) theory as projective families of density matrices. The present first paper aims at clarifying the classical structures that underlies this formalism, namely projective limits of symplectic manifolds. In particular, this allows us to discuss accurately the issues hindering an easy implementation of the dynamics in this context, and to formulate a strategy for overcoming them.