The wave function and the distribution of the observables in relativistic quantum theory

TL;DR

This paper defines a relativistic invariant probability density for particles in quantum mechanics, clarifies energy-momentum distribution, and derives new operators for free fields, grounding uncertainty relations in relativistic context.

Contribution

It introduces a unique, relativistically invariant probability density for particles, including photons and electrons, and develops new operators for quantum fields in this framework.

Findings

Established a relativistic invariant probability density for particles.

Derived operators for free quantum fields based on the new distribution.

Grounded Heisenberg's uncertainty relations in relativistic quantum theory.

Abstract

Here the probability density of relativistic particles coordinates, satisfying the formal conditions of the quantum mechanics and the special relativity, is determined (under textbooks view, such density does not exist). It is specified for the boson, neutral and charged, scalar and vector, including the photon, the electron. Uniqueness of this design has been proved. This is a relativistic invariant, describing the probability of the particle observation in points of space-time. The time is deprived of his role of a dynamic parameter of distribution and formally equalized with other coordinates. Also the meaning of the energy-momentum distribution is clarified. Conditions of such distributions observation are described. As applied to the quantum field these structures are transformed into new characteristics of particles distribution in space-time in addition to those for momentum states. Their operators for free fields are obtained. Under such model the Heisenberg's uncertainty relations are grounded for the relativistic particle; both particles and antiparticles turn out to be different states of a single particle; vacuum state of boson quantum field has zero energy.