Some clarifications on the relation between Bohmian quantum potential and Mach's principle

TL;DR

This paper explores how Machian principles relate to Bohmian quantum mechanics, deriving the quantum potential and higher order terms for the Klein-Gordon particle using Machian ideas without traditional quantum postulates.

Contribution

It provides a novel derivation of Bohmian quantum potential from Machian considerations, bridging classical Machian ideas with quantum mechanics.

Findings

Derivation of Bohmian quantum potential from Machian principles

Identification of conditions where Machian effects influence quantum behavior

Extension to higher order terms for Klein-Gordon particles

Abstract

Mach's principle asserts that the inertial mass of a body is related to the distribution of other distant bodies. This means that in the absence of other bodies, a single body has no mass. In this case, talking about motion is not possible, because the detection of motion is possible only relative to other bodies. But in physics we are faced with situations that are not fully Machian. As in the case of general theory of relativity where geodesics exist in the absence of any matter, the motion has meaning. Another example which is the main topic of our discussion, refers to Bohmian quantum mechanics, where the inertial mass of a single particle does not vanish, but is modified. We can call such situations in which motion or mass of a single particle has meaning, pseudo-Machian situations. In this paper, we use the Machian or pseudo-Machian considerations to clarify under what circumstances and how a Machian effect leads us to Bohmian quantum mechanics. Then, we shall get the Bohmian quantum potential and its higher order terms for the Klein-Gordon particle through Machian considerations, without using any quantum mechanical postulate or operator formalism.