Quantum Mechanics from Relational Properties, Part III: Path Integral Implementation

TL;DR

This paper advances relational quantum mechanics by implementing a path integral approach to clarify measurement probabilities, entanglement, and the physical meaning of relational amplitudes, connecting it with quantum reference frame theory.

Contribution

It provides a concrete path integral implementation of relational probability amplitudes, enhancing understanding of measurement, entanglement, and the physical basis of quantum probabilities.

Findings

Clarifies the calculation of measurement probability using path integrals.

Provides a physical interpretation of relational probability amplitudes.

Explains entanglement entropy and measurement processes within the relational framework.

Abstract

Relational formulation of quantum mechanics is based on the idea that relational properties among quantum systems, instead of the independent properties of a quantum system, are the most fundamental elements to construct quantum mechanics. In a recent paper (J. M. Yang, Sci. Rep. 8:13305, 2018), basic relational quantum mechanics framework is formulated to derive quantum probability, Born's Rule, Schr\"{o}dinger Equations, and measurement theory. This paper further extends the reformulation effort in three aspects. First, it gives a clearer explanation of the key concepts behind the framework to calculate measurement probability. Second, we provide a concrete implementation of the relational probability amplitude by extending the path integral formulation. The implementation not only clarifies the physical meaning of the relational probability amplitude, but also allows us to elegantly explain the double slit experiment, to describe the interaction history between the measured system and a series of measuring systems, and to calculate entanglement entropy based on path integral and influence functional. In return, the implementation brings back new insight to path integral itself by completing the explanation on why measurement probability can be calculated as modulus square of probability amplitude. Lastly, we clarify the connection between our reformulation and the quantum reference frame theory. A complete relational formulation of quantum mechanics needs to combine the present works with the quantum reference frame theory.