Histories quantisation of parameterised systems: I. Development of a general algorithm

TL;DR

This paper presents a new algorithm within the History Projection Operator framework for quantising parameterised systems, effectively addressing the problem of time in classical and quantum contexts.

Contribution

It introduces a general algorithm for quantising constrained systems that preserves the temporal structure, avoiding the traditional problem of time.

Findings

Algorithm successfully applied to simple models

Preserves the two laws of time transformation in HPO theory

Addresses the problem of time in parameterised systems

Abstract

We develop a new algorithm for the quantisation of systems with first-class constraints. Our approach lies within the (History Projection Operator) continuous-time histories quantisation programme. In particular, the Hamiltonian treatment (either classical or quantum) of parameterised systems is characterised by the loss of the notion of time in the space of true degrees of freedom (i.e. the `problem of time'). The novel temporal structure of the HPO theory (two laws of time transformation that distinguish between the temporal logical structure and the dynamics) persists after the imposition of the constraints, hence the problem of time does not arise. We expound the algorithm for both the classical and quantum cases and apply it to simple models.