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 Schrödinger propagation. It is shown that the multiplicative scheme’s global features lead to the conclusion that no non-trivial, time-finite event can occur.
Highlights
One motivation for reformulating quantum mechanics beyond the Copenhagen interpretation is to understand quantum mechanics without observers or measurements
A promising starting point for a realist and relativistic formulation of quantum mechanics is the path integral or sum-over-histories approach to quantum mechanics, initiated by P.A.M
Our result shows that the Multiplicative Scheme for the n-site hopper has a rather severe “infrared” problem stemming from the existence of null events with arbitrarily large defining times
Summary
One motivation for reformulating quantum mechanics beyond the Copenhagen interpretation is to understand quantum mechanics without observers or measurements Another is to bring quantum theory into harmony with relativity and the four dimensional world view of General Relativity. The path integral shifts the focus from the state vectors, Hilbert space, Schrodinger evolution, and state vector collapse of canonical quantum mechanics to spacetime histories, spacetime events and their amplitudes. This allows us to consider statements about the quantum world without appealing to the notion of collapse and measurement, and without having to perform the split of the universe into quantum system and classical measuring system necessary in Copenhagen quantum mechanics.
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