#### Abstract

We discuss a model with stable topological solitons in Minkowski space with only three degrees of freedom, the rotational angles of a spatial Dreibein. This model has four types of solitons differing in two topological quantum numbers which we identify with electric charge and spin. The vacuum has a two-dimensional degeneracy leading to two types of massless excitations, characterised by a topological quantum number which could have a physical equivalent in the photon number.

#### 1. Introduction

In our mathematical description of nature we use two different concepts. Following Einstein we formulate gravity in a geometrical language, whereas particle physics uses the algebraic formulae of quantum field theory. To unify the two very successful theories most physicists try quantized gravity. I go here into the opposite direction and follow first steps in a geometrical formulation of particle physics. We should find out from experiments the direction which we should take in our investigations. Nature may give us some hints about the mechanisms. I get my first intuition from the sine-Gordon model and its experimental realisation with a pendulum model. The mathematics of the sine-Gordon model is nicely described in [1]. Here I want only to repeat the most interesting physical pictures of this model. It is a fully relativistic model in 1 + 1D, where the velocity of light corresponds to the propagation velocity of small amplitude waves. In the experimental realisation (see [1]), this velocity is of the order of 1 m/s. Besides waves, we find two types of particle-like excitations, kinks, and antikinks. They behave in many ways like particles. Their energy density is concentrated in a certain region in space with a total energy defining the mass. The three contributions to the energy, stress energy, potential energy, and kinetic energy have different dependencies on the velocity of a moving kink. The potential energy is decreasing; stress energy and kinetic energy are increasing with . As expected for a relativistic model, the three contributions nicely sum up to a mass increasing with . A moving kink is Lorentz contracted. In the mechanical model, we can easily imagine what nature does to decrease the size of the kink. To accelerate the pendula at the front of the kink the angle between them has to increase, leading to a smaller size of the kink. The pendulum model gives me some idea on how nature could work to realise the phenomena of special relativity. Even more impressive is the existence of two types of kinks, kinks and antikinks and their interaction. They behave like charged particles: kinks and kinks repel; kinks and antikinks attract each other. In soft collisions kinks behave similarly to billiard balls. In hard collisions the diameters of kinks shrink proportional to and get point-like. Further, we can observe how kinks and antikinks annihilate. In the mechanical model the annihilation due to friction effects gives rise to the emission of waves. In the abstract theoretical model solitons and antisolitons get through each other with a small time delay. Mathematically we can separate the various kink configurations in homotopy classes differing in their winding number. This is condensed in the relation of homotopy theory.

I get a second hint from a simple model teaching us about the nature of -rotations. I saw it for the first time in Figure on page 1149 of “Gravitation” by Misner et al. [2]. A ball is attached with several wires to the surrounding, for example, with eight wires to the corners of a cube. Rotating the ball one or two times around some axis leads to a complete mess of the strings. But after a -rotation one can disentangle the wires without moving the ball. We can learn from this model that a body which is connected to the surrounding returns only after a -rotation to his original state. For a disconnected body this happens already after a -rotation. This is mathematically formulated in the relation . There is a continuous transition between -rotations and no rotation. This ball model gives me a hint on how possibly nature realises particles with spin 1/2 just by connecting them with the surrounding.

Thirdly, I want to mention that, observing phenomena at and below the atomic scale, we always observe particles or clicks and never waves. Remember the double-slit experiment. In this interference experiment of electrons or photons, the wave-picture appears only after several hundreds or thousands of particles have been registered on the screen.

In the main part of this article, I describe a Lorentz covariant model which has stable topological excitations with properties of particles. In a certain sense, it is a generalisation of the sine-Gordon model to 3 + 1D. Several features of this model were already described in a few articles [3–8]. In this article, I will mainly concentrate on topological questions.

#### 2. Definition of the Model

We are using a scalar SO()-field in 3 + 1D Minkowski space. The only degrees of freedom of this model are therefore three rotational angles, for example, the three Euler angles, describing the rotations of a spatial Dreibein. Work with SU() is simpler than using SO()-matrices, that is, with -matrices. Since SU() is the double covering group of SO() there is an essential difference between SU()- and SO()-fields. Every field configuration of an SO()-field is twice realised by SU()-fields. The two realisations differ by a transformation with the nontrivial center element, by a -rotation. We have to remember this property using SU()-matrices:at every site in . The symbol we are using reminds us of quaternions, invented by Rodrigues [9] in the year 1840, to describe active rotations with the three imaginary quaternionic units . In (1), they are represented by Pauli matrices . is a three-component unit vector and is the component of the Pauli matrices in direction of . Rotations are unit quaternions with . Their manifold is isomorphic to .

The idea for the definition of the dynamical part of the Lagrangian is its identification with the square of the area density on in appropriate units of an action density. We start defining tangential vectors:(tangential one-forms) to . We would like to emphasise that is a trivial connection but is not. With the cross-product we can get the square of the area density and define the Lagrangian in appropriate SI-units with :Up to a proportionality factor, the kinetic term of this model agrees with the Skyrme term in the Skyrme model [10]. To get stable solitons, Skyrme suggested to use a mass term for the vector field leading to Skyrmions with short-range forces which are accepted as approximations for nucleons. We want to describe particles with long-range Coulombic forces and have therefore to avoid the Skyrme term. The Hobart-Derrick theorem [11, 12] allows as additional terms only terms without derivative, a potential term, which we chose asTherefore, we have a two-dimensional manifold of degenerate vacua; the equatorial sphere is defined by . The choice of the potential term (4) has two immediate physical consequences. There are two Goldstone bosons which we can identify with the two photon degrees of freedom. Further, we get nontrivial field configurations of finite energy which can be classified by , by the map of to the sphere at spatial infinity.

#### 3. Stable Solitons of Finite Energy

Inserting the time-independent hedgehog ansatz into Lagrangian (3), we get the Euler-Lagrange equation: a nonlinear differential equation, which we can most easily solve for . It has the simple solution leading to the radial energy density:

The radial dependencies of its three contributions are shown in Figure 1. At large distances the radial energy density agrees with the Coulombic energy density of a point charge. As one can clearly see, the singularity at the origin that point-charges usually are suffering from is removed. The tangential and the potential energy density have equal shapes and decay faster. They lead to a deviation from the Coulomb interaction at distances in the order of and smaller. The total energy sums up to . Comparing this result with the rest energy of an electron we get a value for the scale of 2.21 fm.

The hedgehog configuration, defined in (5), is schematically depicted in the upper diagram of Figure 2. It maps to half of . This half-sphere is indicated in the lower diagram of Figure 2. By the electric field, the hedgehog is wired to the surrounding. If the center of the hedgehog is rotated by , the tangled lines of constant -field can be disentangled without further rotation of the center, and the original configuration can get restored. This behaviour reminds us of the rotational property of spin-1/2 particles.

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#### 4. Topological Quantum Numbers

We can find a further relation to spin in the number of coverings of , the topological charge which we define in spherical coordinates by

The configuration of (5) and Figure 2 results in . Continuous modifications of the soliton field do not change and the homotopy class of the configuration. The homotopy class can be changed by parity transformations in the internal space and by transformations with the nontrivial center element . The four types of solitons which we get by these transformations are shown in Table 1. -values with are indicated by full blue arrows and with by dashed green arrows. Besides the topological charge, the four configurations differ in the direction of the rotational axis of the -field at infinity which we will relate in (16)–(18) with the electric charge . This field at large distances from the center determines the interaction with other solitons, attraction, or repulsion. The pairs of configurations with the same electric charge differ in the value of the topological charge, in the chirality. We can combine pairs of solitons with either to or to . Since SU() is the spin group and due to the above described properties of the soliton configurations under rotations we dare to identify the absolute value of the topological charge with the spin quantum number:We would like to remember that, within SO(), the two configurations which differ in the sign of are identical.

Traversing the center of a soliton we follow a rotation of the local Dreibeins by . This may answer a question, posed by Tsung-Dao Lee in a talk given in Vienna in the 1980s: “why does the mass break chiral symmetry?”

Besides its group theoretical properties, spin is a contribution to the total angular momentum. We will investigate these angular momentum properties in a dipole configuration. Since a dipole is uncharged, the field at infinity is independent of the direction, it approaches, for example, . The symmetry of the vacuum is broken. We can combine the first configuration in Table 1 with the second to total spin or with the third to . The energy for is slightly lower than that for , the configuration shown in Figure 3. During a rotation of the dipole, the vacuum has to remain unchanged; a rigid rotation is not possible. If the dipole axis is rotated, for example, by , as shown in Figure 4, the centers of the solitons have to rotate by the same angle. These rotations contribute to the total angular momentum.

Dipole configurations are not stable due to the attractive interaction between the charges. Their time evolution follows the general equations of motion [3]:derived from the Lagrangian (3). The energy-momentum tensorturns out to be symmetric. There is no special symmetrisation necessary as in Maxwell’s elecrodynamics [13]. Since there are no external forces, the force density is vanishing:

To determine a static potential one has to fix the positions of the centers at a chosen distance . For distances the Coulombic behaviour is nicely reproduced, as can be seen in the diploma theses [14–16]. For distances the interaction strength increases.

#### 5. Electrodynamic Limit

One gets a pure Coulombic behaviour in the limit where one arrives at the Wu and Yang description [17] of dual Dirac monopoles by two degrees of freedom which we can choose as a normalised three-dimensional vector field . In this limit, we get This is a description where the singularity of the Dirac string is removed, but the singularity of the Coulomb field is still present. The Lagrangian (3) reduces to (The same degrees of freedom but a different Lagrangian is used in the Fadeev-Niemi = Skyrme-Fadeev = Baby-Skyrme model [18–20]. The same Lagrangian was used in [21].) with the dual field strength tensor In this limit hedgehogs are characterised by point-like singularities in space and closed world-lines of line-like singularities in space-time:Charges and fields are related by the inhomogeneous Maxwell equations:Already here we see an essential difference to Maxwell’s theory. Charges are quantized; there are no other charges possible than integer multiples of the elementary charge . We get two further differences from the equations of motion:They allow for nonvanishing magnetic currents:The solutions of the homogeneous Maxwell equations fulfil the equations of motion (19). But there are further solutions possible which fulfil the relations: equivalent to the equations of motion (19). The presence of unquantized closed magnetic currents may be a discrepancy to experiments. But we could explain it with the observation, mentioned in the introduction, that in the experiment we only detect particles and never waves. We can speculate that such currents contribute to the recently intensively discussed dark matter. Further, we read from these equations that there are no solutions possible where and are parallel. This seems obviously also in contradiction to experiments, where it is rather simple to produce static parallel electric and magnetic fields. In this case it is more difficult to find an excuse. It could be that and are locally perpendicular and they appear to be parallel only in the average over space or time. This is the price to pay for restricting charges to integer multiples to the elementary charge and the fields to the two degrees of freedom of the -field.

#### 6. Coulomb and Lorentz Forces

By the artificial splitting (16) of a single field in particles and their fields we reduce to the symmetric energy-momentum tensor,and split the force density in two contributionsshowing clearly the presence of Coulomb and Lorentz forces:Here we would like to underline that the magnetic currents do not contribute to electromagnetic forces.

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

A U() gauge invariance appears as a rotational invariance by around the -axis. By a rotation in colour space withWe can rotate the -field in 3-direction:Under this transformation the vector field transforms to withThe curvature gets rotor form and turns out to be invariant against the rotations with : The far field of a hedgehog soliton is the electric field strength of a classical electron and the vector field of a dual Dirac monopole: with a Dirac string along the 3-direction.

#### 8. Hopf Number

In Figure 2(a), we realised that the field lines of a point charge are lines of constant -field. We find also this relation for the dipole fields in Figures 3 and 4. Since in our model the vacuum has broken symmetry the field at infinity, “” is independent of the direction. Thus we get the isomorphism . Due to the topological relation there is an additional quantum number for the -field, the Hopf number, or Gauß linking number of fibres defined by = const and thus by certain values and . This linking number is especially interesting in regions where we can neglect the influence of charges, in regions of pure -field, where the Lagrangian reduces to of (16). According to the Hobart-Derrick theorem [11, 12] such configurations are unstable, if they move with a velocity slower than the speed of light . Moving with they have an action constant in time and can be used to describe electromagnetic waves. For simplicity we choose the -axis in the direction of motion; then it is sufficient to describe these configurations in given by the coordinates , , and . The linking number is a topological invariant. Its natural physical equivalent is the number of photons in this configuration.

As an example of such a configuration in with we defined in the diploma thesis [22] a field by an area preserving map of to a circle rotating with around the line , defined in cylindrical coordinates by For a given -field one can get the linking number in by the famous formula of Carl Friedrich Gauß:by a double integral. We get from the observation of the neighbourhood of a single fibre another determination of , by a single integral. We parametrize the position along the fibre with some parameter . We determine the tangential vector to the fibre and a unit vector to some neighbouring fibre, for example, defined by the perpendicular component of the gradient . We get a local Dreibein with . Then we determine the rotational velocity along the fibre byIt turns out that the integrated rotational angle depends on the radius of the torus surrounded by the fibreand is not an integer. Its dependence on is depicted in Figure 6. This is understandable from the Gauß map and to the unit sphere; see Figure 5. , plotted from the origin of by , draws a curve on (With bold symbols we indicate the vectors and with the set of its coordinates.). and are defined as vectors in the tangential space at by the coordinate equalities and . Their parallel transport along is nontrivial, except for that is a great circle. Deviations from a great circle lead to an additional contribution of the curved geometry on to the rotational angle. Indicating the position on in spherical coordinates , we can derive, for the corresponding spherical coordinate bases and , the affine U() connection:which is trivial for meridians and the equator of . The additional contribution readsThe sum of both contributions gives then the expected value:Both contributions of our example are shown in Figure 6; they give .

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With the conjecture that the photon number of a configuration is given by the Gauß linking number of fibres, configurations with higher linking numbers correspond to states with several photons. In such configurations, see Figure 7; the linked fibres are spiraling several times. The spiral in the right diagram of Figure 7 reminds us of circular polarised waves.

It may be helpful to have a look at experiments. The Na-D-line in a strong magnetic field splits into three lines, the Lorentz triple (see Figure 8), according to . It is interesting to observe the three spectral lines parallel and perpendicular to the magnetic field, as depicted in Figure 10. The lines with , the “-components,” are linearly polarised with an azimuth dependency of the intensity . The -lines, the “-components,” are circularly polarised with . If the circular polarised lines are observed in the direction perpendicular to the -field they appear linearly polarised. This reminds us of different projections of circular motion, of a spiral in the comoving frame; see Figure 9.

Here a problem may appear. Rotations of a spiral do not change their chirality. After a rotation by a right-handed spiral remains right-handed as one can easily check with a right-handed thread of a screw. It is well-known that right polarised light can be easily transformed to linear polarised and to left polarised light by quarter-wave plates. These plates have perpendicular fast and slow axes.

This looks as a counterargument for the interpretation of the Hopf number as photon number. But it may also be an unknown feature of polarised light? Looking carefully at polarisation filters, one realises that filters exist for linear polarised light only! Right and left circular polarised light is detected by transforming it with quarter-wave plates, by interaction with matter, to linear polarisation.

#### 9. Conclusion

The model described in this article is based on a field of SO()-matrices in 3 + 1D Minkowski space-time. Therefore, the model has 3 degrees of freedom only, corresponding to the 3 Euler angles. These rotational matrices can be interpreted as describing the field of orientations of spatial Dreibeins at the points of space-time. In this sense, this simple model needs only the degrees of freedom of space and time to describe the various phenomena discussed in this article. To formulate the algebra of this model we are using the simpler representation of rotations by SU()-matrices. As a consequence of this treatment we have to take into account the fact that for every SO()-configuration there exist two SU()-configurations representing the same field. Vector fields (potentials) and tensor fields (field strength) are derived from the basic soliton field .

The Lagrangian (3) of the model contains two terms. The kinetic term is proportional to the square of the curvature or to the dual field strength. Such a term is well-known from electrodynamics and QCD. The potential term reminds us of the Higgs potential and the cosmological constant. According to the Hobart-Derrick theorem a quarter of the mass of solitons would contribute with its average to the cosmological constant. The transition from and to releases an energy density of and could have contributed to inflation. Further, the potential term allows for a two-dimensional degeneracy of the vacuum states.

In this model, there appear particle-like excitations as topological solitons. Their mass is field energy only. The field configurations are characterised by three topological quantum numbers which could find a physical realisation as electric charge, spin quantum number, and photon number. The topological structure does not allow two solitons to occupy the same space. This could be the topological origin of the Pauli principle. Charges appear as integer multiples of an elementary charge. No fractional charges are possible for stable excitations. Charges are characterised by regions where the spatial Dreibein rotates by . Such regions interact via Coulomb and Lorentz forces. The electric force follows the behaviour at large distances and increases for high momentum transfers. The distinction between charges and their fields is unnecessary. Such a distinction can be introduced as an approximation and simplifies the comparison with our physical experience. Eigenangular momentum (spin) appears as a consequence of orbital motion. The degenerate vacuum allows for two types of massless excitations, which could be related to the two polarisations of photons. U() gauge invariance emerges in the electrodynamic limit as rotational invariance of the above-mentioned Dreibein around the -axis.

Despite the small number of degrees of freedom there are further nonquantized disturbances possible. Magnetic currents as violations of the homogeneous Maxwell equations propagate with the speed of light and contribute to force fields only via their field strengths. Further we find waves in the rotational angle with energy contributions from the potential term in the Hamiltonian. The cosmological discoveries of recent years allow attributing these two types of disturbances to dark energy and dark matter contributions.

The investigations seem to support the conjecture that the particles we find in experiments are topological solitons characterised by topological quantum numbers. They produce only spots on films and clicks in detectors. Waves seem to escape and are not directly detected. Waves could disturb the paths of particles and could contribute to a subquantum medium leading to quantum mechanics. In analogy to Couder’s silicon oil drop experiments [23–26], the interference of waves created by the particle themselves with the vacuum waves could serve as a guiding wave field for particles.

If this model reproduced some features of nature, the two types of long-range forces which we find, gravitational and Coulomb forces, would be described in a geometrical manner with the degrees of freedom of space-time only. This would give a hint that particle physics could be closely related to gravitation.

#### Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.