# A "problem of time" in the multiplicative scheme for the $n$-site hopper

## TL;DR

This paper examines the multiplicative scheme in Quantum Measure Theory applied to the n-site hopper model, revealing that it predicts no non-trivial, time-finite events can occur, highlighting a 'problem of time' in this framework.

## Contribution

It investigates the multiplicative scheme within Quantum Measure Theory using the n-site hopper model, exposing fundamental issues with event occurrence over finite times.

## Key findings

- No non-trivial, time-finite event can occur under the scheme.
- The global features of the multiplicative scheme lead to the 'problem of time'.
- Highlights limitations of the multiplicative scheme in discrete quantum models.

## Abstract

Quantum Measure Theory (QMT) is an approach to quantum mechanics, based on the path integral, in which quantum theory is conceived of as a generalised stochastic process. One of the postulates of QMT is that events with zero quantum measure do not occur, however this is not sufficient to give a full picture of the quantum world. Determining the other postulates is a work in progress and this paper investigates a proposal called the Multiplicative Scheme for QMT in which the physical world corresponds, essentially, to a set of histories from the path integral. This scheme is applied to Sorkin's $n$-site hopper, a discrete, unitary model of a single particle on a ring of $n$ sites, motivated by free Schr\"odinger propagation. It is shown that the multiplicative scheme's global features lead to the conclusion that no non-trivial, time-finite event can occur.