# Polar solutions with tensorial connection of the spinor equation

## Abstract

Dirac field equations are studied for spinor fields without any external interaction and when they are considered on a background having a tensorial connection with a specific non-vanishing structure some solution can be found in polar form displaying a square-integrable localized behaviour.

## 1 Introduction

Physics as seen from a very general perspective consists in writing a system of field equations and then finding the corresponding solutions. Once found, these solutions are applied to particular cases so to make specific predictions that are later compared to experiments and observations.

Because finding solutions is a very difficult endeavour, it has become customary to make predictions by exploiting properties that do not require having the explicit form of a special solution. Yet, the quest for solutions, though not central in contemporary fashion, is still fundamental in general because explicit solutions do have a complete information content. Just the same, due to the difficulty of the task, explicit solutions are rare, and they generally possess properties that may be undesirable. For example, in quantum field theory explicit solutions are in form of plane waves, which are not square-integrable, hence their conserved quantities are infinite and this is to be regarded as giving rise to meaningless interpretations of results.

Dirac fields [1, 2, 3, 4, 5, 6, 7, 8] are one of the most important fields in nature, but so far as we are aware, there is no explicit solution that has been found and which has the desirable property of square-integrability. To be precise, there are explicit solutions that are square-integrable, such as the electronic orbitals in a Coulomb potential. Nevertheless, these solutions are based on external fields that are kept fixed, so they do not describe a complete solution, like a spinor and its surrounding electrodynamic field as given in the form of simultaneous solutions of the fully-coupled system of the Dirac and the Maxwell field equations.

And in any case, we known no solution for spinors having no external interactions, or even self-interaction, and that is in the free situation. However, by taking considerable advantage of the methods that we have drawn in references [9, 10, 11], some solutions, explicit, square-integrable and completely self-sustained, could be obtained.

This is what we will be showing in this work.

## 2 Geometry of spinors

We shall consider a geometry having both torsion and curvature as background for the spinor fields [1], and as a start we recall that \(\varvec{\gamma }^{a}\) are the Clifford matrices, from which \(\left[ \varvec{\gamma }_{a},\varvec{\gamma }_{b}\right] =4\varvec{\sigma }_{ab}\) and \(2i\varvec{\sigma }_{ab} =\varepsilon _{abcd}\varvec{\pi }\varvec{\sigma }^{cd}\) are the definitions of the \(\varvec{\sigma }_{ab}\) and the \(\varvec{\pi }\) matrix (this matrix is what is usually indicated as gamma with an index five, but here we prefer to employ a notation with no index because the appearance of any meaningless index might be source of a number of misunderstanding for some readers).

A general discussion on these field equations together with an attempt to find solitonic solutions can be found in reference [7] while for some general treatment of various solutions we refer the reader to the recent reference [8].

Extensive comments on the structure of regular as well as singular spinors may be found in references [9, 10].

Explicit examples are provided in reference [11].

## 3 Restrictions and the case of the hydrogen atom

In general, the above is a system of fully-coupled field equations that have to be simultaneously solved to obtain the complete solution; in practice, this is almost impossible: so we will begin by making some assumption.

A first assumption that comes to mind is to disregard the effects of torsion-gravity: torsion is so far unseen and gravity relatively weak, so it would make sense to search solutions without considering these effects; setting torsion to zero is given by \(W_{a}=0\) but vanishing gravity is more difficult since the condition of flat space-time that is implemented by a null Riemann tensor \(R^{i}_{\phantom {i}j\mu \nu }=0\) provides no further information. In [9] and [10] we have discussed how a reasonable assumption might be \(R_{ijk}=0\) but in a further work [11] we also showed that such an assumption is untenable since the tensors \(R_{ijk}\) can not equal zero in very general cases indeed: so there is no other way apart from finding a specific realization of the condition of vanishing Riemann tensor that does not involve any a priori assumption on the structure of the \(R_{ijk}\) tensor. We have already mentioned that \(R_{ijk}\) are a tensor containing the information about the reference system, which was called tensorial connection, and as such it encodes the information about a force that cannot be vanished despite being inertial, interpretable as a covariant type of inertial force.

*S*orbital: in [11] we have seen that such an orbital is described on the background

*q*is the electric charge and where \(E=m(1-q^{4})^{\frac{1}{2}}\) is the energy of the orbital. This background can be gotten for \(\alpha =\alpha (\theta )\) and \(\gamma =\gamma (\theta )\) with \(\varepsilon =0\) as a special instance of the more general background we have built so far.

However, even more general backgrounds are possible if the constant \(\varepsilon \) is allowed to be different from zero, and a general result is that it becomes possible to have a shift in energy according to \(E=m(1-q^{4})^{\frac{1}{2}}-\varepsilon \) without modifying the structure of the 1*S*-orbital spinor field solution.

The fact that the very same spinor may have different energy levels looks like a degeneracy. The difference with respect to the known solution is that in this modification the change is in the form of the tensorial connection.

The tensorial connection may thus encode some of the information about the energy of the spinorial field.

## 4 Solutions in free case

As we had already remarked in [11] and as we have recalled here above, some of the components of the tensorial connection may contain information about the energy of spinor fields for electronic orbitals: specifically, they give rise to a \(\Delta E=-\varepsilon \) energy shift. For \(\varepsilon >0\) we obtain some negative energy contribution corresponding to having an attraction in the structure of the covariant inertial force.

In the rest of the article, we are going to neglect electrodynamics; for spinors in the form of a spin-eigenstate along the third axis, rotations around the third axis have the same effect as gauge transformations: hence, in this case we can always choose to have \(P_{k}=0\) identically and without spoiling the polar structure of the spinor field.

Therefore, it is essential for this solution to be defined on a background with a structure richer than usual.

Consequently, the existence of some square-integrable polar solution has been established in the instance where a non-zero tensorial connection is present in a free Dirac differential field equation. The square-integrability of the polar solution is a direct consequence of the existence of the negative energy contribution, that is of the existence of the covariant attractive inertial force given as tensorial connection, for the free Dirac differential field equation.

A more general account for such non-trivial underlying background within the polar form of the Dirac differential field equation has been provided in reference [11].

## 5 Some deepening

Notice that the square invariants can be taken for large radial coordinate and in this case they would all be different from zero if the condition \(m\ne \varepsilon \ne 0\) is still respected for the constants; because \(R_{ijk}\) contains all information normally contained within the connection while being a tensor we called \(R_{ijk}\) tensorial connection. It is generally non-zero but it has no curvature tensor and hence it can not have any source, so that it makes sense to think at it as what encodes some sort of covariant inertial force.

We notice that in the velocity, large relevance must be attributed to the azimuthal component, measured by the function \(\alpha \) both in magnitude and sign: since \(\alpha \) is strictly negative, the vorticity has direction opposite to the spin.

Also notice that the constant \(\varepsilon \) has a fundamental part because by lowering the mass parameter from *m* down to an effective mass \(\sqrt{m^{2}-\varepsilon ^{2}}\) it essentially plays the role of a negative energy, which is what we need for localization.

In fact the constraint \(\varepsilon >0\) gives an effective negative energy, as we have seen when in the previous section we have presented the example of the Coulomb potential.

The constraints given by \(m>\varepsilon \) with \(\alpha <0\) respectively imply that solutions behave as hyperbolic functions and with an exponential convergence to zero at infinity.

Negative energies are proper of attractive forces, which in this case are codified through the tensorial connection by the action of covariant inertial forces [11].

## 6 Conclusion

Finding square-integrable solutions is a nice result, but it is even more impressive to think that such a result has been obtained after a relatively large number of restrictions, and we can only guess how complex other solutions may be, if we were to allow an even richer structure of the background. The restrictions we made were three: some were restrictions on the structure of the \(R_{ijk}\) tensor that consisted in having some components set to zero; others were on the structure of the velocity and spin; and a final was about assuming no Yvon-Takabayashi angle at all.

Under these restrictions, we found solutions for spinors in polar form described in terms of the module having an exponential damping with radial distance: this property relied on the fact that an overall negative contribution for the energy was present. These negative energies are the consequence of a specific type of covariant attractive inertial force introduced through the tensorial connection.

From a purely mathematical perspective, the technicalities of the presentation could be summarized by stating the following: the Dirac equation necessitates of a basis of tetrad fields beside the usual spinor structure, and so we might as well take advantage of the fact that tetrads could be non-trivial to get more general spinor solutions.

From a genuinely physical point of view we might say that since it is generally impossible to vanish the tensorial connection \(R_{ijk}\) in the spinorial dynamics then we might as well use them to get more general spinorial dynamics.

The special structure of the tensorial connection entails the presence of some covariantly attractive inertial force, which is a force that despite being sourceless can not be vanished, in the same way in which the spin is a covariant angular momentum, because despite having the algebraic structure of an angular momentum we can not vanish it in an arbitrary way. Both the covariant inertial force and the spin are intrinsically defined only for spinor fields.

Nevertheless, there is a difference between these two in that we have become accustomed to spin while covariant inertial forces are a recently introduced concept.

In this paper we have discussed in what way they can be used in specific cases to find well-behaved solutions of the Dirac differential field equation.

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