# A Geometric Model in 3+1D Space-Time for Electrodynamic Phenomena

## Abstract

**:**

## 1. Introduction

## 2. Formulation of the Model

#### 2.1. Geometry

#### 2.1.1. Scalar Field

#### 2.1.2. Vector Field = Connection Field

#### 2.1.3. Curvature Field

#### 2.2. Lagrangian

## 3. Soliton Solutions

#### 3.1. Hedgehog Solution

#### 3.2. Quantum Numbers of Solitons

#### 3.2.1. Charge

#### 3.2.2. Topological Charge

#### 3.2.3. Spin as Angular Momentum

## 4. General Equations of Motion

## 5. Electrodynamic Limit

#### 5.1. Comparison to Maxwell Equations

#### 5.2. Coulomb and Lorentz Forces

#### 5.3. U(1) Gauge Invariance

#### 5.4. Goldstone Bosons

## 6. Open Questions and Conjectures

#### 6.1. Running of the Coupling

#### 6.2. Orthogonality of $\mathbf{E}$ and $\mathbf{B}$

#### 6.3. Quantum Effects

#### 6.4. A Possible Mechanism of Cosmic Inflation

#### 6.5. Cosmological Constant

## 7. Conclusions

#### 7.1. Comparison to Other Models

#### 7.2. Comparison to Maxwell’s Electrodynamics

- The Lagrangian (19) is Lorentz covariant, thus the laws of special relativity are respected.
- Charges have Coulombic fields fulfilling Gaußes law (67).
- Charges interact via $\frac{1}{{r}^{2}}$ electric fields (27), they feel Coulomb and Lorentz forces, see Section 5.2.
- A local U(1) gauge invariance is respected, see Section 5.3.
- There are two dofs of massless excitations for photons, see Section 5.4.

- Electric charges are quantized, see Section 3.2.1, like the magnetic charges of Dirac monopoles. The charge is a topological quantum number.
- The topological construction explains the astonishing mirror properties of particles and antiparticles [5], see Section 3.2.
- The mass of solitons is completely due to field energy, see Equation (29), and finite.
- The self-energy of charges is finite and does not need regularisation and renormalization, see Equation (34).
- These SO(3) dofs can be interpreted as orientations of spatial Dreibeins, see Section 2. Thus, besides the properties of space-time, no additional fields are necessary to describe electromagnetism and gravitation.
- Spin appears with usual quantization properties known from quantum mechanics, see Section 3.2.3, and respects the combination rules of representations of SU(2). Spin is attributed to field configurations with values in the group manifold, whereas in quantum field theories it is usually attributed to algebra-valued fields acting on vectors in the corresponding representation.
- Solitons and antisolitons have opposite internal parity, see Table 1, as is well-known for the description of electrons and positrons with the Dirac equation.
- Solitons are characterized by a chirality quantum number which can be related to the sign of the magnetic quantum number.
- Spin contributes to angular momentum due to internal rotations of soliton centers, see Figure 5.
- The canonical energy-momentum tensor (56) does not need additional symmetrization. It is automatically symmetric.
- Due to the dual representation (15) of the field strength tensor static charges are described by the spatial components of vector fields (25). For time-dependent phenomena, like moving charges, electric currents and magnetic fields, non-vanishing time components of the connection ${\overrightarrow{\mathsf{\Gamma}}}_{\mu}$ are necessary, see Equations (12) and (15).
- The finite size of solitons leads to a modification of the Coulomb’s law at distances of the order of the soliton radius [28]. This effect is known in quantum field theories as running of the coupling, see Section 6.1.
- The local U(1) gauge invariance is explained by the free choice of the bases in the tangential spaces of the ${\mathbb{S}}^{2}$ manifold, see Section 5.3.
- Photons are characterized by a topological quantum number, the Gaußian linking number (84) of fibers of the ${\mathbb{S}}^{2}$-field, the soliton field in the electrodynamic limit.
- The photon number can be modified by interaction of photons with charges, see Section 5.4.

- Spin and magnetic moment are dynamical properties. They are consequences of movements of solitons or time-dependent field variations in their surroundings.
- Electric and magnetic field vectors are perpendicular to each other, see Section 6.2. This is in obvious contradiction to experimental realizations of parallel electric and magnetic fields. An excuse may be, that this property may be realized on a microscopic level and allows for parallelity of these fields on a macroscopic level.
- The existence of unquantized magnetic currents is allowed. These currents do not directly appear in Lorentz forces. They contribute only via their magnetic fields.
- $\alpha $-waves with oscillating values of ${q}_{0}=cos\alpha $ feel non-vanishing values of the potential energy density and contribute therefore to matter density. Such non-topological excitations were not directly detected. If they exist, they contribute to dark matter, see Section 6.3.
- $\alpha $-waves lead to additional forces on particles and are a possible origin of quantum fluctuations.
- The potential term allows for a mechanism of cosmic inflation.
- The potential term contributes to dark energy.

## 8. Aftermath

## Funding

## Acknowledgments

## Conflicts of Interest

## Note

1 | With bold characters we indicate vectors in 3D. |

## References

- Odom, B.; Hanneke, D.; D’Urso, B.; Gabrielse, G. New Measurement of the Electron Magnetic Moment Using a One-Electron Quantum Cyclotron. Phys. Rev. Lett.
**2006**, 97, 030801. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bohm, D. David Bohm’s Critique of Modern Physics: Letters to Jeffrey Bub, 1966–1969; Springer: Cham, Switzerland, 2020. [Google Scholar]
- Smolin, L. The Trouble with Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next; Houghton Mifflin: Boston, MA, USA, 2006. [Google Scholar]
- Woit, P. Not Even Wrong: The Failure of String Theory and the Search for Unity in Physical Law; Basic Books: New York, NY, USA, 2006. [Google Scholar]
- Borchert, M.J.; Devlin, J.A.; AU Erlewein, S.R.; Fleck, M.; Harrington, J.A.; Higuchi, T.; Latacz, B.M.; Voelksen, F.; Wursten, E.J.; Abbass, F.; et al. A 16-parts-per-trillion measurement of the antiproton-to-proton charge–mass ratio. Nature
**2022**, 601, 53–57. [Google Scholar] [CrossRef] [PubMed] - Remoissenet, M. Waves Called Solitons: Concepts and Experiments; Advanced Texts in Physics; Springer: New York, NY, USA, 2003. [Google Scholar]
- Pietschmann, H. Consequences of Minkowskis Unification of Space and Time for a Philosophy of Nature. In Minkowski Spacetime: A Hundred Years Later; Petkov, V., Ed.; Springer: Dordrecht, The Netherlands, 2010; pp. 319–326. [Google Scholar] [CrossRef] [Green Version]
- Cabaret, D.M.; Grandou, T.; Grange, G.M.; Perrier, E. Elementary Particles: What are they? Substances, elements and primary matter. arXiv
**2021**, arXiv:2103.05522. [Google Scholar] - Cabaret, D.M.; Grandou, T.; Perrier, E. Status of the wave function of Quantum Mechanics, or, What is Quantum Mechanics trying to tell us? arXiv
**2021**, arXiv:2103.05504. [Google Scholar] - Couder, Y.; Fort, E.; Gautier, C.H.; Boudaoud, A. From Bouncing to Floating: Noncoalescence of Drops on a Fluid Bath. Phys. Rev. Lett.
**2005**, 94, 177801. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Anderson, P.W. More Is Different: Broken symmetry and the nature of the hierarchical structure of science. Science
**1972**, 177, 393–396. [Google Scholar] [CrossRef] [Green Version] - Hooft, G. Past and Future of Gauge Theory. In One Hundred Years of Gauge Theory; Past, Present and Future Perspectives; Fundamental Theories of Physics; De Bianchi, S., Kiefer, C., Eds.; Springer International Publishing: New Yrok, NY, USA, 2020; Volume 199, pp. 301–313. [Google Scholar]
- Faber, M. A Model for topological fermions. Few Body Syst.
**2001**, 30, 149–186, arXiv:9910221. [Google Scholar] [CrossRef] [Green Version] - Kouneiher, J. Conceptual Foundations of Soliton Versus Particle Dualities Toward a Topological Model for Matter. Int. J. Theor. Phys.
**2016**, 55, 2949–2968. [Google Scholar] [CrossRef] - Bush, J.W. Pilot-Wave Hydrodynamics. Annu. Rev. Fluid Mech.
**2015**, 47, 269–292. [Google Scholar] [CrossRef] [Green Version] - Skyrme, T.H.R. A Nonlinear theory of strong interactions. Proc. R. Soc. Lond. A
**1958**, 247, 260–278. [Google Scholar] [CrossRef] - Skyrme, T.H.R. A Nonlinear field theory. Proc. R. Soc. Lond. A
**1961**, 260, 127–138. [Google Scholar] [CrossRef] - Hobart, R. On the Instability of a Class of Unitary Field Models. Proc. Phys. Soc. Lond.
**1963**, 82, 201–203. [Google Scholar] [CrossRef] - Derrick, G. Comments on Nonlinear Wave Equations as Models for Elementary Particles. J. Math. Phys.
**1964**, 5, 1252–1254. [Google Scholar] [CrossRef] [Green Version] - Dirac, P.A.M. Quantised singularities in the electromagnetic field. Proc. Roy. Soc. Lond.
**1931**, A133, 60–72. [Google Scholar] - Dirac, P.A.M. The Theory of magnetic poles. Phys. Rev.
**1948**, 74, 817–830. [Google Scholar] [CrossRef] - Wu, T.T.; Yang, C.N. Some remarks about unquantized nonabelian gauge fields. Phys. Rev.
**1975**, D12, 3843–3844. [Google Scholar] - Wu, T.T.; Yang, C.N. Dirac’s Monopole Without Strings: Classical Lagrangian Theory. Phys. Rev.
**1976**, D14, 437–445. [Google Scholar] [CrossRef] - Wabnig, J. Interaction in the Model of Topological Fermions. Diploma Thesis, Physics, Technische Universitat Wien, Wien, Austria, 2001. [Google Scholar]
- Resch, J. Numerische Analyse an Dipolkonfigurationen im Modell topologischer Fermionen. Diploma Thesis, Physics, Technische Universitat Wien, Wien, Austria, 2011. [Google Scholar]
- Theuerkauf, D. Charged Particles in the Model of Topological Fermions. Diploma Thesis, Physics, Technische Universitat Wien, Wien, Austria, 2016. [Google Scholar]
- Anmasser, F. Running Coupling Constant in the Model of Topological Fermions. Diploma Thesis, Physics, Technische Universitat Wien, Wien, Austria, 2021. [Google Scholar]
- Anmasser, F.; Theuerkauf, D.; Faber, M. About the solution of the numerical instability for topological solitons with long range interaction. Few-Body Syst.
**2021**, 62, 84, arXiv:2108.07309. [Google Scholar] [CrossRef] - Faber, M.; Kobushkin, A.P. Electrodynamic limit in a model for charged solitons. Phys. Rev.
**2004**, D69, 116002, arXiv:0207167. [Google Scholar] [CrossRef] [Green Version] - Chan, H.; Tsou, S. Some Elementary Gauge Theory Concepts; World Scientific Lecture Notes in Physics; World Scientific: Singapore, 1993. [Google Scholar]
- Borisyuk, D.; Faber, M.; Kobushkin, A. Electro-Magnetic Waves within a Model for Charged Solitons. J. Phys.
**2007**, A40, 525–531, arXiv:0708.3173. [Google Scholar] [CrossRef] [Green Version] - Jech, M. Die Hopfzahl in einer SU(2)-Feldtheorie. Master’s Thesis, Technische Universitat Wien, Wien, Austria, 2014. [Google Scholar]
- Faber, M. Charges and Electromagnetic radiation as topological excitations. Adv. High Energy Phys.
**2017**, 2017, 9340516, arXiv:1707.03196. [Google Scholar] [CrossRef] [Green Version] - Ferreira, L.; Sawado, N.; Toda, K. Static Hopfions in the extended Skyrme-Faddeev model. J. High Energy Phys.
**2009**, 2009, 124. [Google Scholar] [CrossRef] [Green Version] - Shen, Y.; Hou, Y.; Papasimakis, N.; Zheludev, N. Supertoroidal light pulses as electromagnetic skyrmions propagating in free space. Nat. Commun.
**2021**, 12, 5891. [Google Scholar] [CrossRef] [PubMed] - Peskin, M.; Schroeder, D. An Introduction To Quantum Field Theory; Perseus Books Publishing, L.L.C.: New York, NY, USA, 1995; p. 864. [Google Scholar]
- Faber, M.; Kobushkin, A.; Pitschmann, M. Shape vibrations of topological fermions. Adv. Stud. Theor. Phys.
**2008**, 2, 11–22, arXiv:0812.4225. [Google Scholar] - de Broglie, L. La mécanique ondulatoire et la structure atomique de la matière et du rayonnement. Journal de Physique et le Radium
**1927**, 8, 225–241. [Google Scholar] [CrossRef] - De la Peña, L.; Cetto, A.M.; Valdés-Hernández, A. The Emerging Quantum; Springer International Publishing: New York, NY, USA, 2015; p. 366. [Google Scholar] [CrossRef] [Green Version]
- Guth, A.H. Inflationary universe: A possible solution to the horizon and flatness problems. Phys. Rev. D
**1981**, 23, 347–356. [Google Scholar] [CrossRef] [Green Version] - Turner, M.S. ΛCDM: Much more than we expected, but now less than what we want. arXiv
**2021**, arXiv:2109.01760. [Google Scholar] - Aghanim, N.; Akrami, Y.; Arroja, F.; Ashdown, M.; Aumont, J.; Baccigalupi, C.; Ballardini, M.; Banday, A.J.; Barreiro, R.B. Planck2018 results. Astron. Astrophys.
**2020**, 641, A1. [Google Scholar] [CrossRef] [Green Version] - ’t Hooft, G. Magnetic Monopoles in Unified Gauge Theories. Nucl. Phys. B
**1974**, 79, 276–284. [Google Scholar] [CrossRef] [Green Version] - Polyakov, A.M. Particle Spectrum in Quantum Field Theory. JETP Lett.
**1974**, 20, 194–195. [Google Scholar] - Polyakov, A.M. Quark Confinement and Topology of Gauge Groups. Nucl. Phys. B
**1977**, 120, 429–458. [Google Scholar] [CrossRef] - Dietz, K.; Filk, T. Critical Higgs mass for the (2 + 1) dimensional Georgi-Glashow model. Nucl. Phys. B
**1980**, 164, 536–545. [Google Scholar] [CrossRef] - Manton, N.S. The Force Between ’t Hooft-Polyakov Monopoles. Nucl. Phys. B
**1977**, 126, 525–541. [Google Scholar] [CrossRef] - Bogomol’nyi, E.B. Stability of classical solutions. Sov. J. Nucl. Phys.
**1976**, 24, 449. [Google Scholar] - Prasad, M.K.; Sommerfield, C.M. An Exact Classical Solution for the ’t Hooft Monopole and the Julia-Zee Dyon. Phys. Rev. Lett.
**1975**, 35, 760–762. [Google Scholar] [CrossRef] - Vachaspati, T. An Attempt to construct the Standard Model with monopoles. Phys. Rev. Lett.
**1996**, 76, 188–191, arXiv:9509271. [Google Scholar] [CrossRef] [Green Version] - Wheeler, J.A.; Ford, K. Geons, Black Holes, and Quantum Foam: A Life in Physics; W.W.Norton & Company: New York, NY, USA, 1998; p. 235. [Google Scholar]

**Figure 1.**Depicted is the imaginary part $\overrightarrow{q}\left(x\right)=\overrightarrow{n}\left(x\right)sin\alpha \left(x\right)$ of the soliton field $Q\left(x\right)$ of Equation (1) in a symmetry plane.

**Figure 2.**Radial energy densities of the stable soliton solution according to Equation (33). The prefactor in h is omitted.

**Figure 3.**Schematic diagrams for the Q-field (arrows) and the flux lines (lines) of two opposite unit charges, $Z={Q}_{\mathrm{el}}=0$. The configurations are rotational symmetric around the axis through the two charge centers. The red/green arrows symbolise values of $\overrightarrow{q}=\overrightarrow{n}sin\alpha $ for postive/negative values of ${q}_{0}=cos\alpha $. For ${q}_{0}\to 0$ the arrows are getting darker or black. The left configuration belongs to the topological quantum numbers $\mathcal{Q}=s=0$ and the left one to $\mathcal{Q}=s=1$. Observe that field lines connect points of constant $\overrightarrow{n}$-field.

**Figure 4.**The rotating inner sphere is connected by an arbitrary number of wires to the outer sphere. Rigid spheres of the increasing radius with fixed piercings rotate around varying axes. The inner sphere performs a $4\pi $ rotation around the axis indicated by the black dot. The red dot shows the status of the positive rotation in the sequence $0,\pi ,2\pi ,3\pi ,4\pi $. The set of these rotations corresponds to a big circle on ${\mathbb{S}}^{3}$. For spheres from inside to outside, these sets build smaller and smaller circles contracting at the outer sphere to no rotation.

**Figure 5.**Two schematic diagrams for a rotating dipole which differ by rotations of 45${}^{\circ}$ and 90${}^{\circ}$ from Figure 3b. The imaginary part of the SU(2)-field are indicated by arrows. The visibility of the positions of the two charges is enhanced by surrounding circles. Red/green/black arrows indicate ${q}_{0}>0/{q}_{0}<0/{q}_{0}=0$.

**Table 1.**The transformations $\mathcal{T}$ of the hedge-hog configuration in the first column, see Equation (24), modify the fields $\overrightarrow{n}$ and ${q}_{0}$ and the topological quantum numbers Z and $\mathcal{Q}$. The diagrams show the imaginary components $\overrightarrow{q}=\overrightarrow{n}sin\alpha $ of the soliton field, in full red for the hemisphere with ${q}_{0}>0$ and in dashed green for ${q}_{0}<0$.

$\mathcal{T}=1$ | $\mathcal{T}=\mathit{z}$ | $\mathcal{T}={\mathbf{\Pi}}_{\mathit{n}}$ | $\mathcal{T}=\mathit{z}{\mathbf{\Pi}}_{\mathit{n}}$ |
---|---|---|---|

$\overrightarrow{n}=\overrightarrow{r}/r$ | $\overrightarrow{n}=-\overrightarrow{r}/r$ | −$\overrightarrow{n}=\overrightarrow{r}/r$ | $\overrightarrow{n}=\overrightarrow{r}/r$ |

${q}_{0}\ge 0$ | ${q}_{0}\le 0$ | ${q}_{0}\ge 0$ | ${q}_{0}\le 0$ |

$Z=1$ | $Z=-1$ | $Z=-1$ | $Z=1$ |

$\mathcal{Q}=\frac{1}{2}$ | $\mathcal{Q}=\frac{1}{2}$ | $\mathcal{Q}=-\frac{1}{2}$ | $\mathcal{Q}=-\frac{1}{2}$ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Faber, M.
A Geometric Model in 3+1D Space-Time for Electrodynamic Phenomena. *Universe* **2022**, *8*, 73.
https://doi.org/10.3390/universe8020073

**AMA Style**

Faber M.
A Geometric Model in 3+1D Space-Time for Electrodynamic Phenomena. *Universe*. 2022; 8(2):73.
https://doi.org/10.3390/universe8020073

**Chicago/Turabian Style**

Faber, Manfried.
2022. "A Geometric Model in 3+1D Space-Time for Electrodynamic Phenomena" *Universe* 8, no. 2: 73.
https://doi.org/10.3390/universe8020073