A New Approach to the Unification of Fundamental Interactions: The Balance Formula for Quantum, Gravitational, and Electromagnetic Fields
Creators: name: Silvio, Corrêa Junior
Publisher: name: Zenodo
Year: 2024

description: A New Approach to the Unification of Fundamental Interactions: The Balance Formula for Quantum, Gravitational, and Electromagnetic Fields

AbstractIn this work, we present an original formula that proposes the unification of quantum, gravitational, and electromagnetic fields in a four-dimensional spacetime, considering the complex interactions among these fundamental forces. The formula integrates concepts from general relativity, electromagnetism, quantum mechanics, and scalar fields to provide a comprehensive description of the fundamental laws of physics. The equation is derived from a relativistic action incorporating various contributions, such as spacetime curvature, quantum fields, and matter-energy interactions. This model may open new perspectives in the quest for a unified theory of fundamental forces, being applicable to the observation and simulation of both cosmic and microscopic phenomena.

1. IntroductionThe unification of fundamental forces is one of the greatest challenges in theoretical physics. Einstein's theory of general relativity describes gravity as the curvature of spacetime, while quantum field theory (QFT) addresses the interactions of fundamental particles, such as electromagnetism. However, these theories are incompatible at extreme scales, such as near black holes or at the universe's inception. Proposals like String Theory and Quantum Gravity attempt to resolve this incompatibility but still face significant challenges.This work proposes a new approach to integrate quantum, gravitational, electromagnetic, and scalar fields, seeking a unified formulation consistent with both relativity principles and quantum laws. The formula is based on a relativistic action that aims to integrate gravitational, quantum, electromagnetic, and scalar fields into a single equation. The model assumes that the interactions among these forces can be described in a four-dimensional spacetime, considering both the relativistic and quantum aspects of fundamental fields.

2. Equation FormulationThe proposed equation can be expressed as the action integral given by:

C = E ∫ [(c⁴/G)(R - 2Λ) + K₁λ(c⁴/G) + ℏc⁸(∇ᵤΨ∇ᵘΨ) + (1/16π)c⁶FᵤᵥFᵘᵥ + (α/4)ℏc⁹(∇ᵤφ∇ᵘφ - V(φ)) + Lₘₐₜₜₑᵣ]√(-g) d⁴x

Where:

E is a scaling or energy factor modulating the equation.c is the speed of light.G is the gravitational constant, representing gravitational interaction.R is the curvature scalar, related to spacetime geometry.Λ is the cosmological constant, associated with dark energy.λ is an interaction constant among the fields.K₁ is a scaling parameter relating the different interactions.ℏ is the reduced Planck constant, representing quantum scales.Ψ is the quantum field, describing fundamental particles.Fᵤᵥ is the electromagnetic field tensor, governing electromagnetic interactions.φ is the scalar field, responsible for weak force interactions or dark energy modulation.V(φ) is the scalar field potential.Lₘₐₜₜₑᵣ is the Lagrangian associated with matter.g is the determinant of the spacetime metric, fundamental for relativistic formulation.d⁴x is the volume element in four-dimensional spacetime.

Equation Terms

Gravitational Term:c4G(R−2Λ)\frac{c^4}{G}(R - 2Λ)

This term represents spacetime curvature and dark energy (Λ), essential for gravitational dynamics and its interaction with other fundamental forces. The cosmological constant (Λ) is associated with the acceleration of the universe's expansion, a characteristic of dark energy.

Quantum Interaction Term:K1λc4GK₁λ\frac{c^4}{G}

The parameter K₁ modulates the interaction between gravitational fields and other fundamental fields, such as quantum and scalar fields, establishing a link between various fundamental forces at quantum scales.

Quantum Field Term:

ℏc8(∇uΨ∇uΨ)ℏc^8(∇ᵤΨ∇ᵘΨ)This term describes the dynamics of the quantum field Ψ, representing fundamental particles and their interactions according to quantum mechanics and special relativity principles.

Electromagnetic Term:116πc6FuvFvu\frac{1}{16π}c^6FᵤᵥFᵘᵥ

This term relates the electromagnetic field tensor Fᵤᵥ to general relativity, describing electromagnetic interactions and the behavior of electromagnetic fields in curved spacetime.

Scalar Field Term:α4ℏc9(∇uφ∇uφ−V(φ))\frac{α}{4}ℏc^9(∇ᵤφ∇ᵘφ - V(φ))

The scalar field φ may represent phenomena such as cosmic inflation or dark energy modulation, with the potential V(φ) determining its dynamics. This term is crucial for describing the universe's accelerated expansion and its interaction with dark energy.

Matter Term:LmatterLₘₐₜₜₑᵣ

The matter-associated Lagrangian describes interactions of matter in the model, including known particles and exotic forms like dark matter.

Volume Factor:√(−g)d4x√(-g)d⁴x

This factor ensures the equation has the correct form in curved spacetime, preserving invariance under relativistic transformations.

3. Motivation and JustificationThe need for unifying physical interactions arises from the quest for a cohesive theory that explains both quantum and gravitational phenomena. The proposed formula aims to integrate the following fundamental aspects:Gravitation: General relativity describing spacetime curvature.Electromagnetism: Maxwell's model and interactions among charged particles.Quantum Fields: Quantum field theory and interactions among fundamental particles.Scalar Fields: Fields like the inflaton and dark energy, pivotal in modern cosmology.

4. Dimensional Analysis and SymmetryDimensional analysis ensures the formula is physically consistent, balancing the units correctly. The equation maintains key symmetries, including:Lorentz Symmetry: Covariance under Lorentz transformations ensures relativistic invariance.Gauge Symmetry: Quantum and electromagnetic fields preserve gauge symmetries, essential for consistent particle interaction descriptions.

5. Applications and Numerical SimulationsSimplified numerical simulations were conducted, focusing on interactions between quantum and electromagnetic fields. The results showed patterns consistent with known models, such as the Higgs theory and general relativity, revealing field interactions under different initial conditions. These simulations, though preliminary, confirm the model's potential to describe observed phenomena like mass generation through the Higgs mechanism and spacetime curvature in the presence of strong electromagnetic fields.

6. ConclusionsThe proposed formula offers a new approach to unifying fundamental interactions, integrating aspects of general relativity, electromagnetism, and quantum mechanics. While still in its early stages, the model provides a promising starting point for future theoretical and experimental investigations. It is expected that technological advances and new experiments will test the model's predictions and refine our understanding of fundamental forces.

ReferencesEinstein, A. (1915). "Die Feldgleichungen der Gravitation." Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften.Feynman, R., et al. (1964). The Feynman Lectures on Physics. Addison-Wesley.Peskin, M. E., & Schroeder, D. V. (1995). An Introduction to Quantum Field Theory. Addison-Wesley.Weinberg, S. (1995). The Quantum Theory of Fields, Vol. 1: Foundations. Cambridge University Press.Corrêa, S. (2024). A Jornada da Consciência: Explorando a Criação e a Relatividade do Universo.