$\hbar$ as parameter of Minkowski metric in effective theory

TL;DR

This paper proposes a framework where fundamental constants like ar and c are parameters of a family of solutions in effective theory, emphasizing dimensionless quantities and redefining the role of spacetime geometry.

Contribution

It introduces a novel perspective where ar and c are parameters of Minkowski vacuum solutions, focusing on dimensionless and energy-scale-dependent quantities in effective theory.

Findings

Constants ar and c parametrize Minkowski vacuum solutions.

Physical quantities are expressed as dimensionless or energy-dependent parameters.

The effective interval s characterizes particle dynamics, not spacetime geometry.

Abstract

With the proper choice of the dimensionality of the metric components, the action for all fields becomes dimensionless. Such quantities as the vacuum speed of light c, the Planck constant \hbar, the electric charge e, the particle mass m, the Newton constant G never enter equations written in the covariant form, i.e., via the metric g^{\mu\nu}. The speed of light c and the Planck constant are parameters of a particular two-parametric family of solutions of general relativity equations describing the flat isotropic Minkowski vacuum in effective theory emerging at low energy: g^{\mu\nu}=diag(-\hbar^2, (\hbar c)^2, (\hbar c)^2, (\hbar c)^2). They parametrize the equilibrium quantum vacuum state. The physical quantities which enter the covariant equations are dimensionless quantities and dimensionful quantities of dimension of rest energy M or its power. Dimensionless quantities include the running coupling `constants' \alpha_i; topological and geometric quantum numbers (angular momentum quantum number j, weak charge, electric charge q, hypercharge, baryonic and leptonic charges, number of atoms N, etc). Dimensionful parameters include the rest energies of particles M_n (or/and mass matrices); the gravitational coupling K with dimension of M^2; cosmological constant with dimension M^4; etc. In effective theory, the interval s has the dimension of 1/M; it characterizes the dynamics of particles in the quantum vacuum rather than geometry of space-time. We discuss the effective action, and the measured physical quantities resulting from the action, including parameters which enter the Josepson effect, quantum Hall effect, etc.