# On a Mass-Charge Structure of Gauge Invariance

## TL;DR

This paper explores a generalized gauge invariance framework involving mass-charge structures, proposing new equations and transformations that link photon symmetry, particle lifetimes, and inertial mass generation.

## Contribution

It introduces a generalized Klein-Gordon equation and a new Dirac equation to account for different photon components and their masses, revealing a novel gauge transformation responsible for inertial mass.

## Key findings

- Different photon components have distinct masses and lifetimes.
- Quantum mass, energy, and momentum operators can act independently on wave functions.
- A new gauge transformation explains the origin of inertial mass in matter fields.

## Abstract

The mathematical logic of a true nature of mirror symmetry expresses, in the case of the Dirac Lagrangian, the ideas of the left- and right-handed photons referring to long- and short-lived particles, respectively. Such a difference in lifetimes says about the photons of the different components having the unidentical masses, energies, and momenta. This requires the generalization of the classical Klein-Gordon equation to the case of all types of bosons with a nonzero spin. The latter together with a new Dirac equation admits the existence of the second type of the local gauge transformation responsible for origination in the Lagrangian of an interaction Newton component, which gives an inertial mass to all the interacting matter fields. The quantum mass operator and the mirror presentation of the classical Schr\"odinger equation suggest one more highly important equation. Findings show clearly that each of the quantum mass, energy, and momentum operators can individually act on the wave function. They constitute herewith the Euler-Lagrange equation at the level of the mass-charge structure of gauge invariance.