*4.1. Massive Cases*

We start from the analysis of the consequences of massive torsion: assuming also that the torsion mass is quite large, we will study the effective approximation. Finally, some comments about low-speed conditions will be given from the perspective of the non-relativistic limit.

### 4.1.1. Effective Approximation

To begin our investigation, we remark that torsion had a first property that was unlike what any other space–time or gauge fields had, and that it comes as a general feature of the geometry and not from a symmetry principle, with the consequence that there is no symmetry protecting it from being massive. Thus, the torsional field Equations (141) are such that, in the presence of a massive field, they can be taken in the approximation in which the dynamical term is negligible compared to the mass term. Thus, we may write

$$M^2 \mathcal{W}^\mu \approx \mathbf{X} \overline{\psi} \gamma^\mu \pi \psi \tag{165}$$

yielding an algebraic equation that can be used to have torsion substituted in all other field equations in terms of the spin of the spinor, so that all torsional contributions can effectively be converted into spin–spin interactions.

This is the so-called effective approximation.

Let us now move back to the physical field equations, which consist of expressions (141)–(144). These equations, by employing the variational formalism, can be derived from a dynamical action whose Lagrangian is

$$\begin{aligned} \mathcal{L}^{\mathcal{L}} &= -\frac{1}{4} (\partial \mathcal{W})^2 + \frac{1}{2} M^2 \mathcal{W}^2 - \frac{1}{k} \mathcal{R} - \frac{2}{k} \Lambda - \frac{1}{4} F^2 + \\ &+ i \overline{\psi} \gamma^\mu \nabla\_\mu \psi - X \overline{\psi} \gamma^\mu \pi \psi \mathcal{W} \mu - m \overline{\psi} \psi \end{aligned} \tag{166}$$

where torsion is already decomposed. Equivalently,

$$\begin{split} \mathcal{L}\mathcal{C} &= -\frac{1}{4} (\overline{\partial} \mathcal{W})^2 + \frac{1}{2} M^2 \mathcal{W}^2 - \frac{1}{k} \mathcal{R} - \frac{2}{k} \Lambda - \frac{1}{4} F^2 + \\ &+ i \overline{\psi}\_L \gamma^\mu \nabla\_\mu \psi\_L + i \overline{\psi}\_R \gamma^\mu \nabla\_\mu \psi\_R + \\ &+ X \overline{\psi}\_L \gamma^\mu \psi\_L \mathcal{W}\_\mu - X \overline{\psi}\_R \gamma^\mu \psi\_R \mathcal{W}\_\mu - \\ &- m \overline{\psi}\_R \psi\_L - m \overline{\psi}\_L \psi\_R \end{split} \tag{167}$$

in which the chiral split is already done.

In effective approximation, the Lagrangian becomes

$$\begin{aligned} \mathcal{L}^{\rho}\_{\text{effective}} &= -\frac{1}{k} \mathcal{R} - \frac{2}{k} \Lambda - \frac{1}{4} F^2 + \\ + i \overline{\psi} \gamma^{\mu} \nabla\_{\mu} \psi + \frac{1}{2} \frac{X^2}{M^2} \overline{\psi} \gamma^{\mu} \psi \overline{\psi} \gamma\_{\mu} \psi - m \overline{\psi} \psi \end{aligned} \tag{168}$$

where (102) was used. Equivalently,

$$\begin{split} \mathcal{L}\_{\text{effective}} &= -\frac{1}{k} \mathcal{R} - \frac{2}{k} \Lambda - \frac{1}{4} F^2 + \\ &+ i \overline{\psi}\_L \gamma^\mu \nabla\_\mu \psi\_L + i \overline{\psi}\_R \gamma^\mu \nabla\_\mu \psi\_R + \\ &+ \frac{\chi^2}{M^2} \overline{\psi}\_L \gamma^\mu \psi\_L \overline{\psi}\_R \gamma\_\mu \psi\_R - \\ &- m \overline{\psi}\_R \psi\_L - m \overline{\psi}\_L \psi\_R \end{split} \tag{169}$$

which is exactly the Lagrangian of the Nambu–Jona–Lasinio model [32,33].

As (102) shows, it is precisely the axial-vector nature of the field that produces the inversion of the sign of the potential, making the contact interaction attractive.

In addition, as it is clear, such an interaction takes place between two chiral projections. In fact, general knowledge of the NJL model shows that the torsionally-induced spin–spin contact interaction is an attraction between the two chiral parts of the spinor. We recall that the role of the Higgs boson is analogous.

This is not surprising since the torsion–spin coupling is the axial-vector analog of the scalar Yukawa coupling. In fact, if the effective Lagrangian (168) is further re-arranged in terms of (102), it can be put in the form

$$\mathcal{L}\_{\text{effective}}^{\text{spinor}} = i\overline{\psi}\gamma^{\mu}\nabla\_{\mu}\psi + \\\\ + \frac{1}{2}\frac{X^2}{M^2}(|\overline{\psi}\psi|^2 + |i\overline{\psi}\,\pi\psi|^2) - m\overline{\psi}\psi\tag{170}$$

as the Lagrangian of the spinor field complemented with the torsionally-induced spincontact interactions. On the other hand, in the standard model of particle physics [34], we might take into account the Lagrangian for the electron in the presence of the Higgs interaction alone. If the Higgs is taken in effective approximation, we have

$$M^2H \approx -\frac{Y}{2}\overline{e}e$$

which is analogous to (165) in scalar form. Plugging it into the standard model Lagrangian gives

$$\mathcal{L}\mathcal{L}\_{\text{effective}}^{\text{electron}} = i\overline{\varepsilon}\gamma^{\mu}\nabla\_{\mu}e + \frac{\chi^{2}}{4M^{2}}|\overline{\varepsilon}e|^{2} - m\overline{\varepsilon}e \tag{172}$$

for the electronic field with the Higgs-induced interaction. The comparison between (170) and (172) shows that

$$\mathcal{V}\_{\text{effective}}^{\text{spinor}} = -\frac{1}{2} \frac{\chi^2}{M^2} (|\overline{\psi}\psi|^2 + |i\overline{\psi}\pi\psi|^2) \tag{173}$$

$$\mathcal{V}\_{\text{effective}}^{\text{electron}} = -\frac{Y^2}{4M^2} |\overline{e}e|^2 \tag{174}$$

meaning that torsion gives a self-interaction with a scalar part and a pseudo-scalar part, so spin dependent, while the Higgs gives rise to a scalar self-interaction only. Apart from this, they are both attractive and occur between the chiral parts.

From the Lagrangian (170), we extract the potential

$$\mathcal{V} = -\frac{\mathbf{x}^2}{2M^2} \left( |\overline{\Psi}\psi|^2 + |i\overline{\Psi}\pi\pi\psi|^2 \right) \tag{175}$$

which is negative, as expected for attractive interactions, and so the energy is the kinetic energy plus the potential energy, given by the general expression according to

$$\mathcal{E} = \mathcal{X}' - \frac{\mathbf{x}^2}{2M^2} \left( |\overline{\psi}\psi|^2 + |i\overline{\psi}\pi\psi|^2 \right) \tag{176}$$

and we recall that all quantities are densities. In fact, a straightforward dimensional analysis shows that we have

$$E = K - \frac{X^2}{2M^2} \frac{1}{V} \tag{177}$$

having interpreted |*ψψ*| <sup>2</sup>+|*iψπψ*<sup>|</sup> <sup>2</sup> =*V* <sup>−</sup><sup>2</sup> as inverse volume, which is reasonable at least on dimensional grounds. On the other hand, it is possible to compute what turns out to be the expression for the internal energy of a van der Waals gas with negative pressure, given by

$$
\mathcal{U} = T - \mathbb{C}^2 \frac{1}{T} \tag{178}
$$

in terms of a generic constant *C*, as it is known from general thermodynamic arguments.

Because thermodynamically the kinetic energy can be interpreted as the temperature, and of course the energy is the internal energy, then the formal similarities of these two apparently unrelated expressions are striking.

In this thermodynamic analogy, we have that the single spinor field can be seen as a matter distribution behaving in the same way in which a van der Waals attractive gas with attractive intermolecular forces would [35].

Consider now the pair of second-order derivative Equations (157) and (158) with *K<sup>µ</sup>* ≈2*XW<sup>µ</sup>* and *G<sup>µ</sup>* ≈0 and implement the torsion effective approximation: (157) becomes

$$\begin{aligned} \nabla^2 \phi - 4X^4 M^{-4} \phi^5 + 2X^2 M^{-2} K \cdot s \phi^3 - \\ -|\nabla \beta / 2|^2 \phi + m^2 \phi &= 0 \end{aligned} \tag{179}$$

with a quintic potential. We see that such a nonlinear potential is attractive.

Summarizing, in the effective approximation, torsional interactions give rise to a contact force much in the same way in which the Higgs field would, with these two forces being similarly attractive and chiral. In addition, we have seen that the torsional potential would also be analogous to the internal energy of an attractive van der Waals gas.

Consequently, insofar as this effective approximation holds, there is a clear indication that torsion is a sort of internal binding force, a tension, localizing the spinor.

### 4.1.2. Non-Relativistic Limit

In the initial section in which we introduced kinematic quantities, it was clear that tensors and gauge fields were characterized by general definition while spinors were defined in a way that was strongly dependent on the background being a (1 + 3)-dimensional space–time. Therefore, in such a space–time, the spinorial transformation law has a total of six parameters while spinor fields defined in terms of this transformation have a total of eight real components, and we have seen how to remove six components from the spinor field leaving it with two physical degrees of freedom.

However, now one might wonder what would happen when we consider the nonrelativistic limit. In such a limit of small velocities, boosts can no longer be viable transformation laws and so time gets frozen, reducing the background to effectively be a three-dimensional space. In this case, spinorial transformation laws would possess a total of three parameters while spinor fields defined by this transformation would have four real components, so that we could remove three of the components from the spinor, hence leaving it with only one physical degree of freedom and nothing more.

To be mathematically precise, in the (1 + 3)-dimensional space–time, the spinor can always be written as (105) like

$$
\psi = \phi e^{-i\alpha} \begin{pmatrix} e^{\frac{i}{2}\beta} \\ 0 \\ e^{-\frac{i}{2}\beta} \\ 0 \end{pmatrix} \tag{180}
$$

in the representation we used throughout this presentation, called chiral representation, with Yvon–Takabayashi angle expressed in terms of imaginary exponentials. It is, however, possible to introduce another representation in which the Yvon–Takabayashi is expressed

in terms of real circular functions, called standard representation, obtained via the unitary transformation

$$\mathbf{U} = \frac{1}{\sqrt{2}} \begin{pmatrix} \mathbb{I} & \mathbb{I} \\ -\mathbb{I} & \mathbb{I} \end{pmatrix} \tag{181}$$

which operates on gamma matrices to give

$$
\gamma^0 = \begin{pmatrix} \mathbb{I} & \mathbf{0} \\ \mathbf{0} & -\mathbb{I} \\ \vdots & \vdots \end{pmatrix} \tag{182}
$$

$$
\gamma^K = \begin{pmatrix} 0 & \sigma^K \\ -\sigma^K & 0 \end{pmatrix} \tag{183}
$$

so that

$$
\sigma^{0A} = \frac{1}{2} \begin{pmatrix} 0 & \sigma^A \\ \sigma^A & 0 \end{pmatrix} \tag{184}
$$

$$
\sigma\_{AB} = -\frac{i}{2} \varepsilon\_{ABC} \begin{pmatrix} \sigma^{\mathbb{C}} & 0 \\ 0 & \sigma^{\mathbb{C}} \end{pmatrix} \tag{185}
$$

and on spinors to give

$$
\psi = \sqrt{2} \phi e^{-i\alpha} \begin{pmatrix} \cos\frac{\beta}{2} \\ 0 \\ -i \sin\frac{\beta}{2} \\ 0 \end{pmatrix} \tag{186}
$$

in general. In (1 + 3)-dimensional space–times, spinors can always be written as above. In three-dimensional space, the spinor can always be written according to

$$
\psi = \phi e^{-i\alpha} \begin{pmatrix} 1 \\ 0 \end{pmatrix} \tag{187}
$$

and the representation is unique.

Upon comparison, it becomes easy to see that the non-relativistic limit requires a small spatial part of the velocity *u <sup>a</sup>* but also a small Yvon–Takabayashi angle *β* and, when this is accomplished, we have that, in standard representation, the spinor reduces to the form

$$
\psi = \sqrt{2} \phi e^{-i\alpha} \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix} \tag{188}
$$

where the lower component has vanished, and the upper component has reduced to

$$
\psi = \phi e^{-i\alpha} \begin{pmatrix} 1 \\ 0 \end{pmatrix} \tag{189}
$$

up to an overall constant, which is irrelevant.

It is also worth noticing that, so far, we have been able to obtain a procedure of nonrelativistic limit that involves no definition of momentum. However, if the momentum in (163) is considered, we would see that only in the case in which all spin contributions are negligible can the explicit form of the momentum (164) reduce to

$$P^\nu \approx m \cos \beta u^\nu \tag{190}$$

so that the non-relativistic limit is given as a small spatial part of *P <sup>a</sup>* as commonly used.

Therefore, we have that the non-relativistic limit is implemented by the requirement that, when written in standard representation, the spinor loses its lower component

$$
\psi \to \sqrt{2} \phi e^{-i\alpha} \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix} \tag{191}
$$

and this is why this component is called small component. Equivalently, we have that the conditions

$$u^a \to \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix} \tag{192}$$

$$
\beta \to 0 \tag{193}
$$

$$\text{are what implements the non-relativistic limit.}$$

In addition, additionally, if the spin is negligible, then

$$P^a \to \begin{pmatrix} \underline{m} \\ \underline{0} \\ 0 \\ 0 \end{pmatrix} \tag{194}$$

is the final form of non-relativistic limit, and the one that is normally employed.

We notice that, because *u a* is the velocity and, as we said, *β* is already linked to the internal dynamics, then, in a non-relativistic limit, the spinor loses both the overall and the internal motions, which is intuitive. In addition, it is remarkable that the spinorial lower component is connected to *Zitterbewegung* effects which are yet another signature of internal dynamics [36]. There seems to be a very tight relation linking the Yvon–Takabayashi angle with effects of *Zitterbewegung* as manifestations of internal dynamics for the Dirac spinorial matter fields in general [37].

### *4.2. Massless Case*

In this part, we will study the complementary situation given when both torsion and spinors are massless.

### Ultra-Relativistic Limit

Let us now consider what happens when torsion as well as the Dirac spinor are both massless. The torsional field Equations (141) become

$$\nabla\_{\rho}(\partial \mathcal{W})^{\rho \mu} = X \overline{\psi} \gamma^{\mu} \pi \psi \tag{195}$$

which are analogous to the electro-dynamic field equations apart from the fact that these above are parity-odd. This aside, both are vector field equations in a massless case, and, as such, we should expect some symmetry to be present. The full Lagrangian in the case of masslessness also for the spinor field is given by the following:

$$\begin{split} \mathcal{L}\mathcal{L}\_{\text{massless}} &= -\frac{1}{4} (\partial \mathcal{W})^2 - \frac{1}{k} R - \frac{2}{k} \Lambda - \frac{1}{4} F^2 + \\ &+ i \overline{\Psi}\_L \gamma^\mu \nabla\_\mu \psi\_L + i \overline{\Psi}\_R \gamma^\mu \nabla\_\mu \psi\_R + \\ &+ X \overline{\Psi}\_L \gamma^\mu \psi\_L \mathcal{W}\_\mu - X \overline{\Psi}\_R \gamma^\mu \psi\_R \mathcal{W}\_\mu \end{split} \tag{196}$$

as it is straightforward to see.

This is invariant for the transformation

$$\mathcal{W}'\_{\nu} = \mathcal{W}\_{\nu} - \partial\_{\nu}\omega \tag{197}$$

with

$$
\psi\_L' = e^{-i\mathbf{X}\omega} \psi\_L \qquad \psi\_R' = e^{i\mathbf{X}\omega} \psi\_R \tag{198}
$$

or in compact form

$$
\psi' = e^{i\mathbf{X}\omega\pi}\psi\tag{199}
$$

known as chiral gauge transformation.

Additionally, expression (105) can also be written as

$$
\psi = \phi e^{-i\alpha} e^{-\frac{i}{2}\beta\pi} \begin{pmatrix} 1 \\ 0 \\ 1 \\ 0 \end{pmatrix} \tag{200}
$$

and, from this expression, it is clear that it is always possible to perform a chiral gauge transformation taking the local parameter to be *β*=2*Xω* and leaving

$$
\psi' = \phi e^{-i\alpha} \begin{pmatrix} 1 \\ 0 \\ 1 \\ 0 \end{pmatrix} \tag{201}
$$

in terms of the module alone: this has to be expected, as symmetries come with redundant information that can be removed by reducing the fields, and because in this case the chiral symmetry is an additional symmetry with one parameter, we have to expect that one degree of freedom be removed. It is clear that the only degree of freedom to remain is the one that cannot be removed in any way whatsoever that is the module.

Because in the massless approximation the two chiral Lagrangians become separable, the two chiral projections are independent, and therefore the Yvon–Takabayashi angle can be vanished, since it carries no information.

This is yet another fact that supports the evidence for which the Yvon–Takabayashi angle can be related to internal dynamics and *Zitterbewegung* for spinor fields.

In addition, this is possible because of the attractiveness that characterizes the axialvector massive torsion mediation of the chiral mutual interaction within the spinor field.

If torsion were not an axial-vector, the chiral interaction would not be attractive, and, if such an attraction were not massive enough, it would not be sufficiently strong to grant stability for the bound-state spinorial field itself.

### *4.3. Two: Basic Applications*

This second chapter will be about applying the above theory to solve or discuss fundamental problems in modern physics: in the first section, we will tackle the problem of gravitational singularity formation. In the second section, we will discuss the problem of positivity of energy.

### **5. Consequences of Spin**

In this first section, our main goal is to take into account the problem of the formation of gravitational singularities and face it in terms of the modifications brought by the presence of torsion interacting with spinors. Some comments on the Pauli exclusion principle will be made.

### *5.1. Singularity Avoidance*

If we consider Einstein gravity on its own, it is remarkably difficult to overestimate its success. From planetary precession, through gravitational waves, to black holes, there is not a single prediction that has not been corroborated yet. In fact, if Dark Matter is just another form of matter, there is not a single effect, whether predicted or not, that has never been confirmed so far. Nevertheless, there is a black spot, theoretically.

The Hawking–Penrose theorem is a very general result showing how, under very general conditions on energy, gravitationally-induced singularities form. If true, such a result would constitute an indication that Einstein gravitation has to be generalized, or at least included in an extended framework. There are, in fact, several attempts at extended models, whether they are simple extensions of Einstein gravity, or major revisions of all Einstein concepts of a geometric theory in itself. All these models and theories are certainly worth our attention. However, at times, the solution to a given puzzle might well be much closer than expected. If we wish to try a solution that is based on the physics we already have, the most straightforward possibility is to use the torsion tensor.

Employing torsion to solve this problem has already been done [38]. However, contrary to the expectation that torsion could solve or at least alleviate this issue, Kerlick found that the issue was actually worsened. This way was then abandoned.

Nevertheless, to a more attentive examination, we may find a possible way out. A closer look at the reasons why torsion would enhance the formation of singularities will reveal that the gravitational field is increased because, in the energy density, there are positive contributions coming from the fact that torsional effects for the spin contact interaction of spinors are taken to be repulsive.

This happens to be the case because Kerlick considers the simplest generalization of Einstein gravity, the original Einstein–Sciama–Kibble theory, where torsion is tied to the spin in terms of the Newton gravitational constant.

However, as discussed above, a more general theory of torsion would, first of all, involve a torsion–spin coupling that is not the Newton gravitational constant, but which can be any possible constant and in particular a constant with the opposite sign. In addition, secondly, in the most general case in which torsion propagates, in the effective approximation, the torsion–spin coupling constant has an opposite sign necessarily.

In fact, in this case, in effective approximation, we found that we do have an attractive torsion effect, resulting in a negative potential in the energy density, decreasing gravitation and making the singularity formation avoidable.

Indeed, the torsional contribution could provide such a negative potential that the whole energy may turn negative, the gravitational field may turn repulsive, and singularity formation would be avoided necessarily.

To put words into expressions, take (142) contracted as

$$-R - 4\Lambda = \frac{k}{2}(-M^2\mathcal{W}^2 + m\Phi) \tag{202}$$

and plug this back into the original equations to get

$$\begin{split} R^{\rho\sigma} + \Lambda \multimap^{\rho\sigma} &= \frac{k}{2} \Big[ \frac{1}{4} F^2 g^{\rho\sigma} - F^\rho \Gamma^\sigma\_a + \frac{1}{4} (\partial \mathcal{W})^2 g^{\rho\sigma} - (\partial \mathcal{W})^{\sigma a} (\partial \mathcal{W})^\rho{}\_a + M^2 \mathcal{W}^\rho \mathcal{W}^\sigma + \\ &\quad + \frac{i}{4} (\overline{\Psi} \gamma^\rho \nabla^\sigma \psi - \nabla^\sigma \overline{\Psi} \gamma^\rho \psi + \overline{\Psi} \gamma^\sigma \nabla^\rho \psi - \nabla^\rho \overline{\Psi} \gamma^\sigma \psi) - \\ &\quad - \frac{1}{2} X (\mathcal{W}^\sigma S^\rho + \mathcal{W}^\rho S^\sigma) - \frac{1}{2} m \Phi g^{\rho\sigma} \Big] \end{split} \tag{203}$$

equivalent to those in the original form. For the singularity theorem in Einstein gravity, we have that the condition

$$R^{\rho\sigma}u\_{\rho}u\_{\sigma} \gtrless 0\tag{204}$$

must be verified, and, when this is the case, then singularity formation will become inevitable. With no cosmological constant and neglecting electro-dynamics, we obtain that the condition to have singularity formation reads

$$\begin{split} \frac{1}{4} (\boldsymbol{\partial}\mathcal{W})^2 \boldsymbol{g}^{\rho\sigma} - (\boldsymbol{\partial}\mathcal{W})^{\sigma\mathfrak{a}} (\boldsymbol{\partial}\mathcal{W})^{\rho}{}\_{a} + \frac{i}{2} (\overline{\boldsymbol{\psi}}\gamma^{\rho}\nabla^{\sigma}\boldsymbol{\psi} - \nabla^{\sigma}\overline{\boldsymbol{\psi}}\gamma^{\rho}\boldsymbol{\psi}) + \\ + M^2 \boldsymbol{W}^{\rho} \boldsymbol{W}^{\sigma} - X \boldsymbol{W}^{\sigma} \boldsymbol{S}^{\rho} - \frac{1}{2} m \boldsymbol{\Phi} \boldsymbol{g}^{\rho\sigma} \big] u\_{\rho} u\_{\sigma} \geqslant 0 \end{split} \tag{205}$$

and this is what we have to study.

[

In effective approximation, it becomes

$$
\frac{1}{2}(\overline{\psi}\gamma^0\nabla\_0\psi - \nabla\_0\overline{\psi}\gamma^0\psi) - \frac{1}{2}m\Phi \ge 0\tag{206}
$$

and because (144) in the effective approximation is

$$i\gamma^0 \nabla\_0 \psi + i\vec{\gamma} \cdot \vec{\nabla} \psi - \frac{X^2}{M^2} S\_\sigma \gamma^\sigma \pi \psi - m\psi = 0\tag{207}$$

we may use this in the above to get

$$\frac{1}{2}(\vec{\nabla}\overline{\psi}\cdot\vec{\gamma}\psi - \overline{\psi}\gamma\vec{\gamma}\cdot\vec{\nabla}\psi) + \frac{X^2}{M^2}S\_\sigma S^\sigma + \frac{1}{2}m\Phi \gtrsim 0\tag{208}$$

whose structure is similar to the condition of Kerlick but with the sign of the nonlinear interaction inverted. We may now follow Kerlick argument by neglecting the derivative term, and, by employing (102), we get that

$$-4\frac{X^2}{M^2}\phi^4 + m\phi^2\cos\beta \ge 0\tag{209}$$

which for for large densities are be violated, and quite easily too.

Therefore, because of the torsion–spin coupling, the energy condition is not verified and gravitational singularity formation is no longer a necessity [39].

We already said that torsion in effective approximation generates interactions which, without the spin-dependent part, are similar to what we would get by using the Higgs potential. Therefore, it is not surprising that singularity avoidance could be achieved also by the Higgs [40]. The difference is in the mass scale: the Higgs potential can only be used to avoid singularities in black holes, as it does not work before symmetry breaking, while torsion can be used to avoid singularities for black holes and the big bang, since torsion is always a massive field even prior to any mass generation mechanism.

Notice that this mechanism is proper to the Einsteinian gravitation. In fact, in order for this mechanism to work, one must have a theory in which gravitation can become repulsive if the energy density switches sign and in which the energy density is allowed to switch its overall sign. None of this would ever be possible in a theory of gravitation in which the source is not the energy but the mass, since the mass can never be negative.

### *5.2. Pauli Exclusion*

The above-commented mechanism with which one may avoid the formation of singularity at a gravitational level is reminiscent of the degeneracy pressure encountered in the usual treatment of neutron stars. Consequently, the correlated Pauli exclusion principle comes to the mind. Such a principle stems from the fact that, in the construction of electronic levels, obtained by solving the non-relativistic matter field equations in a Coulomb potential, the solutions are given in terms of a quantum number *n* giving the energy level of the external shell, accounting for a total of *n* 2 electrons. However, the number of observed electrons 2*n* <sup>2</sup> and hence there must be a two-fold degeneracy. This means that solutions of the matter field equation come in pairs of two, so that each electronic shell can be filled twice by the same state. The exclusion principle presented in this way is the original form by Pauli. Pauli's initial idea to assign a two-fold degeneracy was most straightforwardly that of introducing the concept of spin: the connection is very simple, based on the fact

that irreducible representations of particles of spin *s* have exactly *d*=2*s*+1 independent components. For particles of spin *s*=1/2, this means *d*=2/2+1=2 components, so that it is possible to think that these two components be precisely the two states that account for the double state of multiplicity. Mathematically, this can be seen from the fact that the spinor field has, for each chiral part, two components. Indeed, recalling (105), we have that spinor fields can be written as

$$\psi = \phi e^{-i\alpha} e^{-\frac{i}{2}\beta\pi} \begin{pmatrix} 1 \\ 0 \\ 1 \\ 0 \end{pmatrix} \qquad \text{and} \qquad \psi = \phi e^{-i\alpha} e^{-\frac{i}{2}\beta\pi} \begin{pmatrix} 0 \\ 1 \\ 0 \\ 1 \end{pmatrix} \tag{210}$$

where the first is a spin-up (spin +1/2) eigenstate while the second is a spin-down (spin −1/2) eigenstate. These are the two opposite-spin eigenvalues of the same eigenspinor. As a consequence of this structure, superposition of two opposite-spin spinors is allowed and thus, if the two initial spinors are solutions, then also their superposition is another solution. This mechanism is indeed what happens in the hydrogen atom.

Nevertheless, the Pauli exclusion principle is not only this. Such a principle must also include a mechanism for which no more than two states can superpose. Quantum mechanics does not solve this problem. In quantum field theory, however, a solution is proposed, and the commonly accepted paradigm is described by the spin-statistic theorem: this theorem says that in a theory that is Lorentz covariance and causal, with positive norms and energies, half-integer spin particles cannot occupy more than one state at a time (while integer spin particles can). However, for this result to take place, the theorem must engage, and this is subject to the conditions granted by its hypotheses. In a classical theory of fields, Lorentz covariance and causality are ensured, but positive norms and energies are not. In fact, we have seen that negative energies are not only possible but also needed to ensure the mechanism to avoid the formation of singularities.

It so appears that the exclusion principle and singularity avoidance can not both be implemented in the same framework. In addition, usually, the common behavior is that of implementing the spin-statistic theorem and leaving the singularity formation unsolved. However, one can instead consider the complementary position of ensuring singularity avoidance and leave the Pauli exclusion open.

However, in a theory where spinors interact with torsion, we have seen that, in effective approximation, the torsionally-induced spin–contact interactions of the spinor give rise to self-interactions for the spinor field. These nonlinear contributions in the matter field equations are enough to ensure that no superposition of two identical solutions can also be a solution. This entails the exclusion principle.

Notice that, in case the two solutions are not identical, that is, if the two solutions correspond to opposite spins, their superposition is allowed by the double-valuedness that characterizes the spinorial fields in general cases.

### **6. Conditions on Energy**

In this second section, we intend to deepen the investigation of the problem of the positive energies. We conclude with comments on the macroscopic approximation.

### *6.1. Positive Energy*

In the development of field theories, it is not uncommon for some properties to be present in a given approximation but not in the full theory. Thus, particles behave in a certain manner in classical mechanics and very differently in quantum mechanics, and quantum particles behave in a given way in quantum mechanics and rather differently in the relativistic version of quantum mechanics.

Following a bottom-up approach in terms of successive generalizations, fewer and fewer properties will be found within the most general theory that is possible. There is, however, a property that does not appear to follow such a pattern, which is the energy. From classical mechanics to quantum mechanics, to relativistic quantum mechanics, to relativistic quantum mechanics of spinning fields, the energy of a particle is always taken to be positive, either because it is proven positive, or because we force it to be positive by correcting the theory in an appropriate way.

Forcing the energy to be positive does have a number of consequences, not only for the interpretation, but also to obtain results like the spin-statistic theorem as discussed above. However, we have seen that the exclusion principle can also be entailed in a different manner, and there should be no surprise in finding a generalization of field the theory in which some energies happen to be negative after all.

Allowing negative energies has considerable advantages too, not only for the fact that the mathematics tells us that they are possible, but also to obtain results like the avoidance of singularities as we had discussed before.

Just the same, even assuming that energies can be negative, we have that they will have to turn out to be positive in those approximations in which we know they are.

To see this is in fact the case, let us consider the energy given as the right-hand side of (142), and that is

$$\begin{aligned} T^{\rho\sigma} &= \frac{1}{4} F^2 \mathcal{g}^{\rho\sigma} - F^{\rho\sigma} F^{\sigma}\_{\;\;\;a} + \\ &+ \frac{1}{4} (\overline{\partial} \mathcal{W})^2 \mathcal{g}^{\rho\sigma} - (\overline{\partial} \mathcal{W})^{\sigma a} (\overline{\partial} \mathcal{W})^{\rho}{}\_{a} + \\ &+ M^2 (\mathcal{W}^{\rho} \mathcal{W}^{\sigma} - \frac{1}{2} \mathcal{W}^2 \mathcal{g}^{\rho\sigma}) + \\ &+ \frac{i}{4} (\overline{\Psi} \gamma^{\rho} \nabla^{\sigma} \psi - \nabla^{\sigma} \overline{\Psi} \gamma^{\rho} \psi + \overline{\psi} \gamma^{\sigma} \nabla^{\rho} \psi - \nabla^{\rho} \overline{\Psi} \gamma^{\sigma} \psi) - \\ &- \frac{1}{2} X (\mathcal{W}^{\sigma} \overline{\Psi} \gamma^{\rho} \sigma \, \psi + \mathcal{W}^{\rho} \overline{\Psi} \gamma^{\sigma} \, \pi \, \psi) \end{aligned} \tag{211}$$

in general. In particular, as the electro-dynamic and torsional contributions are positive, we will consider only

$$\begin{aligned} E^{\rho \sigma} &= \frac{i}{4} (\overline{\Psi} \gamma^{\rho} \nabla^{\sigma} \psi - \nabla^{\sigma} \overline{\Psi} \gamma^{\rho} \psi + \\ &+ \overline{\Psi} \gamma^{\sigma} \nabla^{\rho} \psi - \nabla^{\rho} \overline{\Psi} \gamma^{\sigma} \psi) - \\ &- \frac{1}{2} X (\mathcal{W}^{\sigma} \overline{\Psi} \gamma^{\rho} \pi \psi + \mathcal{W}^{\rho} \overline{\Psi} \gamma^{\sigma} \pi \psi) \end{aligned} \tag{212}$$

as pure spinorial contribution and which is not positive.

To better see this, we go in the frame where the spinor assumes the polar form (105) in which

$$\begin{aligned} E\_{\rho\sigma} &= \phi^2 \left[ P\_{\sigma} \mu\_{\rho} + P\_{\rho} \mu\_{\sigma} + \\ &+ [\frac{1}{4} (\boldsymbol{\Omega}^{ij}{}\_{\sigma} \boldsymbol{\varepsilon}\_{\rho ijk} + \boldsymbol{\Omega}^{ij}{}\_{\rho} \boldsymbol{\varepsilon}\_{\sigma ijk}) \eta^{ka} + \\ &+ \xi^a\_{\rho} (\nabla \boldsymbol{\beta} / 2 - \mathbf{X} \boldsymbol{\mathcal{W}})\_{\sigma} + \xi^a\_{\sigma} (\nabla \boldsymbol{\beta} / 2 - \mathbf{X} \boldsymbol{\mathcal{W}})\_{\rho} \right] \mathbf{s}\_a \end{aligned} \tag{213}$$

whose time-time component is not positive defined as the straightforward check would immediately show.

In it, the momentum is given by (164) as

$$P^{\nu} = m \cos \beta \mu^{\nu} + \mathcal{Y}\_{\mu} \mu^{[\mu} \mathbf{s}^{\nu]} + \mathcal{Z}\_{\mu} \mathbf{s}\_{\rho} \mu\_{\sigma} \varepsilon^{\mu \rho \sigma \upsilon} \tag{214}$$

whose time component is also not positive defined.

However, if we could justify the assumption in terms of which we neglect all contributions coming from the spin, then the energy would reduce to

$$E\_{00} = 2\phi^2 P\_0 \mu\_0 \tag{215}$$

for the time-time component.

The momentum becomes

$$P^0 = m \cos \beta u^0 \tag{216}$$

for the time component.

Therefore, if now the Yvon–Takabayashi angle vanishes, then the energy is ensured to be positive defined.

Summarizing, we can say that, if

$$
\beta \to 0 \tag{217}
$$

$$s\_a \to 0\tag{218}$$

then the energy of spinor fields is necessarily positive [41].

These two conditions together condense a very simple situation, as we are going to discuss in what follows.

### *6.2. Macroscopic Limit*

In the previous part, we have discussed how the energy is positive if *β*→0 and *s<sup>a</sup>* →0 happen to occur.

To understand the meaning of these conditions, let us consider again the field equations for the gravitational field and for electro-dynamics (142) and (143) and compute the divergences: they are respectively given by

$$\begin{aligned} \nabla\_{\rho} \left[ \frac{1}{4} F^2 g^{\rho \sigma} - F^{\rho a} F^{\sigma}{}\_a + \\ + \frac{1}{4} (\partial W)^2 g^{\rho \sigma} - (\partial W)^{\sigma a} (\partial W)^{\rho}{}\_a + \\ + M^2 (W^{\rho} W^{\sigma} - \frac{1}{2} W^2 g^{\rho \sigma}) + \\ + \frac{i}{4} (\overline{\Psi} \gamma^{\rho} \nabla^{\sigma} \psi - \nabla^{\sigma} \overline{\Psi} \gamma^{\rho} \psi + \overline{\Psi} \gamma^{\sigma} \nabla^{\rho} \psi - \nabla^{\rho} \overline{\Psi} \gamma^{\sigma} \psi) - \\ - \frac{1}{2} X (\mathcal{W}^{\sigma} \overline{\Psi} \gamma^{\rho} \pi \psi + \mathcal{W}^{\rho} \overline{\Psi} \gamma^{\sigma} \pi \psi)) = 0 \end{aligned} \tag{219}$$

and

$$\nabla\_{\mu}(\overline{\psi}\gamma^{\mu}\psi) = 0 \tag{220}$$

identically, as we already know from the first chapter.

By substituting the polar form (105) and implementing the above conditions *β* →0 and *s<sup>a</sup>* →0, we get

$$\nabla\_{\rho} \left( \frac{1}{4} F^2 g^{\rho \sigma} - F^{\rho a} F^{\sigma}{}\_{a} + 2m \phi^2 u^{\rho} u^{\sigma} \right) = 0 \tag{221}$$

and

$$\nabla\_{\mu} (2\phi^2 \mu^{\mu}) = 0 \tag{222}$$

as it is straightforward to see.

Evaluating the divergence of the former and employing the latter, we obtain the expression

$$-2\eta \phi^2 \mu^\mu \mathcal{F}^\sigma{}\_\alpha + 2m\phi^2 \mu^\rho \nabla\_\rho \mu^\sigma = 0\tag{223}$$

having used the Maxwell field equations.

After the necessary simplifications, we get

$$
\rho \mathbf{u} \boldsymbol{u}^{\rho} \nabla\_{\rho} \mathbf{u}^{\sigma} = q \mathbf{F}^{\sigma \mathbf{u}} \mathbf{u}\_{\mathfrak{n}} \tag{224}
$$

which is just the Newton law in the presence of Lorentz force. This is what we have in macroscopic approximation.

Thus, we can interpret *β*→0 and *s<sup>a</sup>* →0 as the conditions that implement the known macroscopic approximation.

This is reasonable because vanishing the internal dynamics and all information about internal structures essentially means that we are considering situations where internal contributions are concealed within the spinorial field, which means we are in macroscopic approximation.

Spinor fields have energy density that can be negative as a consequence of all contributions of spin and internal dynamics, and it is only when these are hidden that the positivity of the energy density is also ensured for spinors.

### *6.3. Three: Special Models*

This third and last chapter will be about the application of the above theory for phenomenological cases: we will in fact consider what the effects are of torsion for the two standard models of particles and cosmology.

### **7. Particles and Cosmology**

In the first chapter, we have encountered the theorem of the polar form, which specified that, if both scalars Θ and Φ do not vanish identically, then we can always find special frames where the most general spinor is as in (105).

However, what if <sup>Θ</sup> = <sup>Φ</sup> ≡ 0 everywhere? The answer to this question has already been given in [15], and it is that we could still find a special frame in which the most general spinor can be written in some type of polar form.

Specifically, if <sup>Θ</sup> = <sup>Φ</sup> ≡ 0, then we can always find special frames where the most general spinor is given by

$$\psi = \frac{1}{\sqrt{2}} (\mathbb{I} \cos \frac{a}{2} - \pi \sin \frac{a}{2}) \begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{pmatrix} \tag{225}$$

up to the reversal of the third axis.

Spinor fields undergoing these constraints are called flag-dipoles, and they contain two special cases: one with constraint *S <sup>a</sup>* =0 and called flagpoles, written as

$$
\psi = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{pmatrix} \tag{226}
$$

up to the reversal of the third axis and extinguishing the class of Majorana spinors; the other with a constraint given by *Mab* =0 and called dipoles, written as

$$
\psi = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix} \tag{227}
$$

up to the reversal of the third axis and the switch between chiral parts and accounting for the Weyl spinors.

Therefore, as it can be seen quite clearly, Majorana as well as Weyl spinors can always be Lorentz transformed into a frame in which they remain with a fixed structure, and, consequently, they have no degree of freedom at all.

This may sound surprising, and thus we are going to give a direct proof of this statement for the Weyl spinors.

To do that, consider a general Weyl spinor, for instance left-handed, in the form

$$
\psi = \begin{pmatrix} ae^{i\alpha} \\ be^{i\beta} \\ 0 \\ 0 \end{pmatrix}
$$

where the two complex components have been written in polar form. Consider now as Lorentz transformation the rotation of angle *θ* around the second axis given by

$$\mathbf{A}\_{R2} = \begin{pmatrix} \cos\theta/2 & -\sin\theta/2 & 0 & 0\\ \sin\theta/2 & \cos\theta/2 & 0 & 0\\ 0 & 0 & \cos\theta/2 & -\sin\theta/2\\ 0 & 0 & \sin\theta/2 & \cos\theta/2 \end{pmatrix}.$$

followed by the rotation of angle *ϕ* around the third axis

$$\mathbf{A}\_{R3} = \begin{pmatrix} e^{i\boldsymbol{\wp}/2} & 0 & 0 & 0 \\ 0 & e^{-i\boldsymbol{\wp}/2} & 0 & 0 \\ 0 & 0 & e^{i\boldsymbol{\wp}/2} & 0 \\ 0 & 0 & 0 & e^{-i\boldsymbol{\wp}/2} \end{pmatrix}.$$

applied to the spinor. The results are given by expression

$$\begin{aligned} \psi' &= \begin{pmatrix} \cos\theta/2 & -\sin\theta/2 & 0 & 0\\ \sin\theta/2 & \cos\theta/2 & 0 & 0\\ 0 & 0 & \cos\theta/2 & -\sin\theta/2\\ 0 & 0 & \sin\theta/2 & \cos\theta/2 \end{pmatrix},\\ \begin{pmatrix} e^{i\varphi/2} & 0 & 0 & 0\\ 0 & e^{-i\varphi/2} & 0 & 0\\ 0 & 0 & e^{i\varphi/2} & 0\\ 0 & 0 & 0 & e^{-i\varphi/2} \end{pmatrix} \begin{pmatrix} ae^{i\theta} \\ be^{i\theta} \\ 0 \\ 0 \end{pmatrix} \end{aligned}$$

and that is

$$\psi' = \begin{pmatrix} a\cos\theta/2e^{i\varphi/2}e^{i\alpha} - b\sin\theta/2e^{-i\varphi/2}e^{i\beta} \\ a\sin\theta/2e^{i\varphi/2}e^{i\alpha} + b\cos\theta/2e^{-i\varphi/2}e^{i\beta} \\ 0 \\ 0 \end{pmatrix}$$

after multiplication. The spin-down component is zero if

$$a\sin\theta / 2e^{i\varphi/2}e^{ia} + b\cos\theta / 2e^{-i\varphi/2}e^{i\beta} = 0$$

which can be worked out to be

$$\frac{a}{b}e^{i(\alpha-\beta)} = -e^{-i\varphi}\cot\theta/2$$

splitting into

$$
\cot \theta / 2 = -a / b$$

$$
\varphi = \beta - \alpha$$

for the two angles. Thus, we can always find a combination of two rotations that brings the spin-down component to vanish identically. When this is done, we have

$$
\psi' = \sqrt{a^2 + b^2} e^{i(\beta + \alpha)/2} \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix},
$$

for spin-up Weyl spinors. With another rotation of angle *ζ* = *β*+*α* around the third axis given as the above

$$\mathbf{A}\_{R3} = \begin{pmatrix} e^{-i\zeta/2} & 0 & 0 & 0 \\ 0 & e^{i\zeta/2} & 0 & 0 \\ 0 & 0 & e^{-i\zeta/2} & 0 \\ 0 & 0 & 0 & e^{i\zeta/2} \end{pmatrix}.$$

the phase can also be vanished. Thus, we have

$$
\psi'' = \sqrt{a^2 + b^2} \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix},
$$

and employing a boost of rapidity *η* =ln |*a* <sup>2</sup> + *b* 2 | along the third axis given by

$$
\mathbf{A}\_{B3} = \begin{pmatrix}
 e^{-\eta/2} & 0 & 0 & 0 \\
 0 & e^{\eta/2} & 0 & 0 \\
 0 & 0 & e^{\eta/2} & 0 \\
 0 & 0 & 0 & e^{-\eta/2}
\end{pmatrix},
$$

the module is also removed, and we get

$$
\psi^{\prime \prime \prime} = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix},
$$

for the final form of the Weyl spinor. Obviously, the same would be true if we intended to keep only the spin-down component. In addition, of course, the same remains true for the right-handed case. This result for Weyl spinors is general.

Although more calculations would be needed, it would still be straightforward to see that, by employing exactly the same method, we would obtain exactly the same result also if we were to consider the Majorana spinors.

Such a result may be surprising, but it is a mathematical consequence of the definition of Majorana and Weyl spinors alone and therefore it is true in full generality.

Thus, these spinors do not have degrees of freedom.

If we take this to conclude that these spinors cannot be physical, then we are bound to accept that such spinors cannot form the matter content of any theory, in particular, the standard model of particle physics as we know.

This leaves us with a remarkable consequence: if these spinors, and in particular Weyl spinors, cannot be used in physics, and in particular in the standard model of particle physics, then we cannot employ neutrinos as defined at the moment. Neutrinos need be right-handed too, and, after the symmetry breaking, they must get a Dirac mass.

Because the charge count of the standard model cannot change, neutrinos are sterile. We will next try to see what happens when sterile neutrinos with a Dirac mass term are then included. Of course, the first application is neutrino oscillations.

We now try to see what the effects of torsion can be. However, in order to do so, we have first to make one little digression in order to generalize the theory.

Throughout the entire presentation, we have been considering single spinor fields, but clearly the treatment of two spinor fields, or even more spinor fields, is doable, and it is achieved by replicating the spinor field Lagrangian as many times as the number of independent spinor fields.

For instance, in the case of two spinor fields, we have

$$\begin{split} \mathcal{L} = & -\frac{1}{4} (\overline{\partial} \mathcal{W})^2 + \frac{1}{2} M^2 \mathcal{W}^2 - \frac{1}{k} \mathcal{R} - \frac{2}{k} \Lambda - \frac{1}{4} F^2 + \\ & + i \overline{\psi}\_1 \gamma^\mu \nabla\_\mu \psi\_1 + i \overline{\psi}\_2 \gamma^\mu \nabla\_\mu \psi\_2 + \\ & - X\_1 \overline{\psi}\_1 \gamma^\mu \pi \psi\_1 \mathcal{W}\_\mu - X\_2 \overline{\psi}\_2 \gamma^\mu \pi \psi\_2 \mathcal{W}\_\mu + \\ & - m\_1 \overline{\psi}\_1 \psi\_1 - m\_2 \overline{\psi}\_2 \psi\_2 \end{split} \tag{228}$$

as it is reasonable to expect.

Taking the variation with respect to torsion gives

$$\begin{aligned} \nabla\_{\nu} (\partial \mathcal{W})^{\nu \mu} + M^2 \mathcal{W}^{\mu} &= X\_1 \overline{\psi}\_1 \gamma^{\mu} \pi \pi \psi\_1 + \\ &+ X\_2 \overline{\psi}\_2 \gamma^{\mu} \pi \pi \psi\_2 \end{aligned} \tag{229}$$

as the torsion field equations with two sources.

In effective approximation, we obtain expressions

$$M^2 \mathcal{W}^\mu \approx X\_1 \overline{\psi}\_1 \gamma^\mu \pi \psi\_1 + X\_2 \overline{\psi}\_2 \gamma^\mu \pi \psi\_2 \tag{250}$$

which can be plugged back into the Lagrangian giving

$$\begin{split} \mathcal{L}^{\rho} &= -\frac{1}{k}R - \frac{2}{k}\Lambda - \frac{1}{4}F^{2} + i\overline{\psi}\_{1}\gamma^{\mu}\nabla\_{\mu}\psi\_{1} + i\overline{\psi}\_{2}\gamma^{\mu}\nabla\_{\mu}\psi\_{2} + \\ &+ \frac{1}{2}\left|\frac{X\_{1}}{M}\right|^{2}\overline{\psi}\_{1}\gamma^{\mu}\psi\_{1}\overline{\psi}\_{1}\gamma\_{\mu}\psi\_{1} + \frac{1}{2}\left|\frac{X\_{2}}{M}\right|^{2}\overline{\psi}\_{2}\gamma^{\mu}\psi\_{2}\overline{\psi}\_{2}\gamma\_{\mu}\psi\_{2} - \\ &- \frac{X\_{1}}{M}\frac{X\_{2}}{M}\overline{\psi}\_{1}\gamma^{\mu}\pi\psi\_{1}\overline{\psi}\_{2}\gamma\_{\mu}\pi\psi\_{2} - m\_{1}\overline{\psi}\_{1}\psi\_{1} - m\_{2}\overline{\psi}\_{2}\psi\_{2} \end{split} \tag{231}$$

in which each spinor has self-interaction and between spinors there is mutual interaction.

The extension to three spinor fields, or *n* spinor fields, is similar: there are *n* selfinteractions, always attractive, and <sup>1</sup> 2 *n*(*n*−1) mutual interactions, being either attractive or repulsive according to *XiX<sup>j</sup>* being positive or negative.

This extension is interesting for *n*=3 because this is the situation we have for neutrinos. By neglecting all the interactions apart from the effective interactions, and in them neglecting the self-interaction so to have only the mutual interactions, one may calculate the Hamiltonian

$$\mathcal{AC} = \sum\_{\text{ij}} \overline{\nu}\_i (\mathbf{U}\_{\text{ij}} - \mathbf{X}\_i \mathbf{X}\_{\text{j}} \gamma^\mu \pi \mathbf{v}\_i \overline{\mathbf{v}}\_{\text{j}} \pi \gamma \chi\_{\mu}) \mathbf{v}\_{\text{j}} \tag{232}$$

where the Latin indices run over the three labels associated with the three different flavors of neutrinos. Hence, the matrix *Uij*−*XiXjγ <sup>µ</sup>πνiνjπγ<sup>µ</sup>* is the combination of the constant matrix *Uij* describing kinematic phases that arise from the mass terms, as usual, plus the field-dependent matrix *XiXjγ <sup>µ</sup>πνiνjπγ<sup>µ</sup>* describing the dynamical phases that arise from the torsionally-induced nonlinear potentials, those of the present theory.

Dealing with the nonlinear potentials is problematic, but, in reference [42], this problem is solved by taking neutrinos dense enough to make the torsion field background homogeneous and thus constant. The phase difference is

$$
\Delta\Phi \approx \left(\frac{\Delta m^2}{2E} + \frac{1}{4}|\mathcal{W}^0 - \mathcal{W}^3|\right)L\tag{233}
$$

having assumed *W*<sup>1</sup> =*W*<sup>2</sup> =0 and where *L* is the length of the oscillations. In [43], it was seen that (233) in the case in which the neutrino mass difference is small becomes

$$
\Delta\Phi \approx \left(\Delta m^2 + m \frac{X\_{\rm eff}^2}{4M^2} |\overline{\nu}\gamma\_\mu\nu\overline{\nu}\gamma^\mu\nu|^{\frac{1}{2}}\right) \frac{L}{2E} \tag{234}
$$

where *m* is the value of the nearly-equal masses of neutrinos while *X* 2 eff is a combination of the coupling constants and with the dependence *L*/*E* as the ratio between length and energy of the oscillations, as it is expected.

The phase difference due to the oscillation has the kinematic contribution, as a difference of the squared masses, plus a dynamic contribution, proportional to the neutrino mass density distribution. The novelty torsion introduced is that, even in the case in which neutrino masses were to be non-zero but with insufficient non-degeneracy in mass spectrum, we might still have oscillations, and therefore an ampler margin of freedom before having some tension. Notice also that both *m* and *X* 2 eff depend on the masses and coupling constants of the two neutrinos involved so that they would be different for another pair of neutrinos, making it clear how the parameters of the oscillation depend on the specific pair of neutrinos, as it should be.

This is an immediate and clear effect that the neutrinos with Dirac mass term and interacting in terms of torsion give to us for some new physical insight beyond what is commonly expected from the standard model of particles.

What about the standard model of cosmology? To give an answer to this question, we move on to examine some consequences torsion may have for Dark Matter.

To begin with, we specify that, although we still do not exactly know what dark matter is, nevertheless it has to be a form of matter: albeit many models may fit galactic rotation curves, only dark matter as a real form of matter fits all galactic behaviors [44].

Hence, given dark matter as a form of matter, massive and weakly interacting, we will additionally take it to be described by <sup>1</sup> 2 -spin spinor fields. This makes it possible to have the effects due to torsional interactions.

In reference [45], the torsional effects have been studied in a classical context to see how galactic dynamics could be modified by torsion, and, in [46], we have applied those results to the case in which torsion was coupled to spinors to see how galactic dynamics could be modified by torsion and how torsion could be sourced by dark matter.

Thus, here as before, torsion is not used as an alternative but as a correction over pre-existing physics. Having this in mind, we recall that, in [46], we showed how, if spinors are the source of torsion, the gravitational field in galaxies turns out to be increased: from (142), we see that, in the case of the effective approximation (165), we get

$$R^{\rho\sigma} - \frac{1}{2} R g^{\rho\sigma} - \Lambda g^{\rho\sigma} = \frac{k}{2} (E^{\rho\sigma} - \frac{1}{2} \frac{X^2}{M^2} S^\mu S\_{\mu} g^{\rho\sigma}) \tag{235}$$

showing that the spinor field with the torsionally-induced nonlinear interactions has an effective energy, which is written as the usual term plus a nonlinear contribution.

For this contribution, we have to recall that we are not considering a single spinor field, as we have done when in particle physics, but collective states of spinor fields, as it is natural to assume in cosmology, with the consequence that it is not possible to employ the re-arrangements we used before and thus *S <sup>µ</sup>S<sup>µ</sup>* cannot be reduced. Generally, we do not know how to compute it, but, as the square of a density, it may be positive.

In Ref. [46], we have been discussing precisely what would happen if the spin density square happened to be positive, and we have found that the contribution to the energy would change the gravitational field as to allow for a constant behavior of the rotation curves of galaxies, discussing the value of the torsion–spin coupling constant that is required to fit the galactic observations.

The details of the calculations were based on the fact that in this occurrence and within the approximations of slow rotational velocity and weak gravitational field, the acceleration felt by a point-particle was given by

$$\text{div}\,\vec{a} \approx -m\rho - K^2 \rho^2 \tag{236}$$

in which the Newton gravitational constant has been normalized and where *K* is the effective value of the torsional constant, with constant tangential velocity obtained for densities scaling down as *r* −2 in general. In the standard approach to dark matter, there are only Newtonian source contributions scaling down as *r* <sup>−</sup><sup>1</sup> and so a modification to the density distribution has to be devised, and it is the well known Navarro–Frenk–White profile. In the presence of torsional corrections, the Newtonian profile suffices because, even if the density drops as *r* −1 , it is squared in the torsional correction and the *r* <sup>−</sup><sup>2</sup> drop is obtained. These similarities suggest that the torsion correction might be what gives the Navarro–Frenk–White profile. After all, the NFW profile is obtained in *n*-body dynamics as those assumed here provided that the *n* spinors interact through torsion in terms of some axial-vector simplified model.

Nor is the idea of modeling dark matter, through the NFW profile, unexpected in terms of torsion, since this is precisely what a specific type of effective theories does.

In quite recent years, there has been a shift of approach in looking for physics beyond the standard model, and in particular dark matter. The new way of tackling the issue is based on the idea of studying all types of effective interactions that can be put in a Lagrangian, and, among all of them, there is the axial-vector spin-contact interaction.

However, in even more recent years, this approach has been generalized, shifting the attention from the effective interactions to the mediated interactions, known as simplified models [47]. However, the story does not change, since among all these there is the axial-vector mediated term

$$
\Delta \mathcal{L} = -g \overline{\chi} \gamma^{\mu} \pi \chi B\_{\mu} \tag{237}
$$

where *χ* is the dark matter particle and *B<sup>µ</sup>* is the axial-vector mediator, and where the structure of the interaction is that of the torsion–spin coupling, as it should be quite easily recognizable for the reader at this moment.

When the standard model has been acknowledged to need a complementation, we have been striving to have it placed within a more general model, which should have also contained some new physics, and in particular dark matter. It has been the constant failure in this project that prompted us to reverse the strategy, pushing us to look for simplified models, namely models that can immediately describe dark matter, or in general new physics, and leaving the task of including them, together with the standard model, into a more general model for later, and better, times. If we were, therefore, to see that the dark matter, or generally some new physics, were actually described by one of these simplified models, the following step would be to include it beside the standard model within a more general model, and at this point it should be clear what is our ultimate claim for this entire section.

Our claim is that, if such a simplified model is the one described by the axial-vector mediator, then we will need not look very far, as the general model would be torsion.

We next move to study a more direct effect about a cosmological situation [48,49].

The problem is quite simply the fact that the cosmological constant has a measured value that, in natural units, is about one hundred and twenty orders of magnitude off of the theoretically predicted one. Normally, this would have made physicists reject the theories in which its value is calculated, but those theories are quantum field theory and the standard model, being very successful otherwise.

Philosophers may argue that, in the face of a bad result disproving a theory, there can be no good result that can support it: the history of physics is loaded with examples of good agreements between observations and predictions that were based on theories later seen to be false. In addition, in this specific situation, the bad agreement is not only bad, but it is the worst in all of physics ever. Nowadays, the common behavior would be to claim that this is not really a bad agreement, since new physics might intervene to make the agreement acceptable. It does not take very experienced philosophers to see that this argument could always be invoked to push the problems under the carpet of an even higher energy frontier, and when this frontier will be unreachable, the predictivity of the theory will be annihilated. In this work, we try to embrace a philosophic approach, or merely be reasonable, admitting that such a discrepancy is lethal.

As a consequence, it follows that all theories predicting contributions to the cosmological constant must be dramatically re-adjusted. As we said above, these are the theory of quantum fields, with cosmological constant contributions due to zero-point energy, and the standard model, with cosmological constant contributions given by the same mechanism that gives mass to all fields and that is the spontaneous symmetry breaking.

As for the contribution coming from the general theory of quantum fields in terms of the zero-point energies, we have to recall that the zero-point energies are the result of quantization implemented with commutation relationships. In a normal-ordered quantum theory of fields, or simply in the classical theory of fields, zero-point energy does not appear, and thus no further contribution arises in the cosmological constant.

Leaving us without zero-point energy, it becomes necessary to find a way to compute the Casimir force without using any zero-point energy. It is worth remembering that Casimir forces derived from van der Waals forces was indeed the very first way to describe this phenomenon in the original paper by Casimir and Polder. A more recent account can be found in [50]. See also [51,52].

As for the contribution of the standard model in terms of the mechanism of symmetry breaking, recall that, after the break-down of the symmetry, we have the generation of the masses of all particles interacting with the Higgs field plus that of an effective cosmological constant <sup>1</sup> 2 *λ* 2*v* <sup>4</sup> with a value around 10<sup>120</sup> in natural units. If this term is to disappear, we need to vanish either *λ* or *v*, but, as vanishing the former would imply no symmetry breaking, the only possibility is to vanish *v* so that symmetry breaking can still occur though not spontaneously. We may look for a dynamical symmetry breaking.

To begin our investigation, the very first thing we want to do is remark that, as the reader may have noticed, we never treated the scalar field. The reason was merely to keep an already heavy presentation from being heavier.

Still, it is now time to put in some scalar field. The scalar field complementing the Lagrangian (166) gives

$$\begin{split} \mathcal{L} = & -\frac{1}{4} (\partial \mathcal{W})^2 + \frac{1}{2} M^2 \mathcal{W}^2 - \frac{1}{k} \mathcal{R} - \frac{2}{k} \Lambda - \frac{1}{4} F^2 + \\ & + i \overline{\psi} \gamma^\mu \nabla\_\mu \psi + \nabla^\mu \phi^\dagger \nabla\_\mu \phi - \\ & - X \overline{\psi} \gamma^\mu \tau \psi \mathcal{W} \mu - \frac{1}{2} \Sigma \phi^2 \mathcal{W}^2 - Y \overline{\psi} \psi \phi - \\ & - m \overline{\psi} \psi + \mu^2 \phi^2 - \frac{1}{2} \lambda^2 \phi^4 \end{split} \tag{238}$$

where the *X*, Ξ, *Y* are the constants related to torsion with spinor and scalar interactions.

It is quite interesting to notice that, within this complementation, there is also the term *φ* <sup>2</sup>*W*<sup>2</sup> coupling torsion to the scalar. This may look strange, since torsion is supposed to be sourced by the spin density, which is equal to zero for scalar fields. Therefore, we should expect to have torsion without a pure source of scalar fields, although we will have scalar contributions in torsion field equations.

In fact, upon variation of the Lagrangian, we obtain

$$\nabla\_{\mathfrak{a}}(\partial \mathcal{W})^{a\upsilon} + (M^2 - \Xi \phi^2) \mathcal{W}^{\upsilon} = \mathcal{X} \overline{\psi} \gamma^{\upsilon} \pi \psi \tag{239}$$

in which there is indeed a scalar contribution, although in the form of an interaction giving an effective mass term.

There is, immediately, something rather striking about this expression: in a cosmic scenario, for a universe in an FLRW metric, we would have that the torsion, to respect the same symmetries of isotropy and homogeneity, would have to possess only the temporal component. However, in this case, the dynamical term would disappear leaving

$$(M^2 - \Xi \phi^2)W^\nu = X\overline{\psi}\gamma^\nu \pi\psi\tag{240}$$

as the torsion field equations we would have had in the effective limit, though now the result is exact. The source would have to be the sum of the spin density of all spinors in the universe, and because the spin vector points in all directions, statistically the source vanishes too and

$$(M^2 - \Xi \phi^2) \mathcal{W}^\nu = 0\tag{241}$$

which tells us that, if torsion is present, then

$$M^2 = \Xi \phi^2 \tag{242}$$

and, if Ξ is positive, the scalar acquires the value

$$
\phi^2 = M^2 / \Xi \tag{243}
$$

which is constant throughout the universe.

A constant scalar all over the universe is the condition needed for slow-roll in inflationary scenarios, and in this case there arises an effective cosmological constant

$$
\Lambda\_{\text{effective}} = \Lambda + \frac{1}{2} \left| \frac{\lambda}{2} \left| \frac{M}{\Xi} \right|^2 \right|^2 \tag{244}
$$

in the Lagrangian (238), driving the scale factor of the FLRW metric and therefore driving the inflation itself. Inflation will last, so long as symmetry conditions hold, but as the universe expands and the density of sources decreases, local anisotropies are no longer swamped, and their presence will spoil the symmetries that engaged the above mechanism, bringing inflation to an end [53].

As the universe expands in a non-inflationary scenario, the torsional field equation would no longer lose the dynamic term due to the symmetries, but it might still lose it due to the fact that massive torsion can have an effective approximation. In this case, we would still have

$$(M^2 - \Xi \phi^2) \mathcal{W}^\nu \approx X \overline{\psi} \gamma^\nu \pi \psi \tag{245}$$

although only as an approximated form. We may plug it back into the initial Lagrangian (238) obtaining

$$\begin{split} \mathcal{A}^{\rho} = & -\frac{1}{k}R - \frac{2}{k}\Lambda - \frac{1}{4}F^2 + i\overline{\Psi}\gamma^\mu\nabla\_\mu\psi + \nabla^\mu\phi^\dagger\nabla\_\mu\phi + \\ & + \frac{1}{2}X^2(M^2 - \Xi\phi^2)^{-1}\overline{\Psi}\gamma^\nu\psi\overline{\Psi}\gamma\_\nu\psi - \\ & - Y\overline{\Psi}\psi\phi - m\overline{\Psi}\psi + \mu^2\phi^2 - \frac{1}{2}\Lambda^2\phi^4 \end{split} \tag{246}$$

as the resulting effective Lagrangian. The presence of an effective interaction involving spinors and scalars, having a structure much richer than that of the Yukawa interaction, is obvious. In addition, we observe that, if for vanishingly small scalar this reduces to the above effective interaction for spinors, in the presence of larger values for the scalar, it can even become singular. We might speculate that such a value is the maximum allowed for the scalar as the one at which the above mechanism of inflation takes place.

Consider now the case *µ*=0 in the above Lagrangian.

The scalar potential is minimized by *φ* <sup>2</sup> =*v* 2 such that

$$
\lambda^2 \upsilon^2 = \frac{1}{2} \Xi X^2 (M^2 - \Xi \upsilon^2)^{-2} |\overline{\psi} \gamma^\nu \psi \overline{\psi} \gamma\_\nu \psi|\_\mathbf{v} \tag{247}
$$

linking the square of the Higgs vacuum to the square of the density of the spinor vacuum. Therefore, the dynamical symmetry breaking mechanism occurs eventually.

This break-down of symmetry is a dynamical one because the vacuum is not a constant, but it is the vacuum expectation value of the spinor distribution.

After dynamical symmetry breaking is implemented in the Lagrangian, the effective cosmological constant is still proportional to the Higgs vacuum, but the Higgs vacuum is now proportional to the spinor vacuum. Where material distributions tend to zero, as we would have in cosmology the vacuum for the spinor trivializes, the vacuum for the Higgs trivializes as well and the cosmological constant is no longer generated [54].

The picture that emerges is one for which symmetry breaking is no longer a mechanism that happens throughout the universe but only when spinors are present, with the consequence that, if spinors are not present, the effective cosmological constant is similarly not present. The cosmological constant due to spontaneous symmetry breaking in the standard model is thus avoidable.

No zero-point energy leaves no contribution apart from those due to phase transitions, which can be quenched by a symmetry breaking that is not spontaneous but dynamical, and no effective cosmological constant arises.

In this third chapter, we presented and discussed the possible torsional dynamics in the cosmology and particle physics standard models. Now, it is time to pull together all the loose ends in order to display the general overview.

We have seen and stated repeatedly that torsion can be thought as an axial-vector massive field coupling to the axial-vector bi-linear spinor field according to the term

$$
\Delta \mathcal{L}\_{\text{interaction}}^{\text{Q}-\text{spinor}} = -X \overline{\Psi} \gamma^{\mu} \pi \psi \mathcal{W}\_{\mu} \tag{248}
$$

of which we have one for every spinor. Effective approximations involving two, three, or even more spinor fields have been discussed, with a particular care for the case of neutrino oscillations, for which we have detailed in what way the results of [42] can be generalized in order to have

$$
\Delta\Phi \approx \frac{L}{2\overline{E}} \left( \Delta m^2 + m \frac{X\_{\text{eff}}^2}{4M^2} |\overline{v}\gamma\_\mu v \overline{v}\gamma^\mu v|^{\frac{1}{2}} \right) \tag{249}
$$

describing the phase difference for almost degenerate neutrino masses, as consisting of the *L*/*E* dependence modulating the usual kinetic contribution, plus a new dynamic contribution, so that, even if the neutrino mass spectrum were to be degenerate, torsion would still induce an effective mechanism of oscillation. As these considerations have nothing special about neutrinos, and thus they may as well be extended to all leptons, we then proceeded in studying such extension. However, once the Lagrangian terms of the weak interaction after symmetry breaking and the torsion for an electron and a left-handed neutrino were taken in the effective approximation, we saw that, due to the cleanliness of the scattering and the precision of the measurements, the standard model correction induced by the torsion had to be very small, and, if this occurs because the torsion mass is large, then the effective approximation is no longer viable. We have then re-considered the case without effective approximations, allowing also for sterile right-handed neutrinos in order to maintain the feasibility of the dynamical neutrino oscillations discussed above, therefore reaching the general Lagrangian

$$\begin{split} \Delta \mathcal{L}\_{\text{interaction}}^{\text{Q/weak-spinor}} &= -X\_{\epsilon} \overline{\varepsilon} \gamma^{\mu} \pi \epsilon \mathcal{W}\_{\mu} - X\_{\nu} \overline{\upsilon} \gamma^{\mu} \pi \upsilon \mathcal{W}\_{\mu} + \\ &+ \frac{\mathcal{S}}{\sqrt{2}} \left( \mathcal{W}\_{\mu}^{-} \overline{\upsilon} \gamma^{\mu} e\_{L} + \mathcal{W}\_{\mu}^{+} \overline{\varepsilon}\_{L} \gamma^{\mu} \upsilon \right) + \\ &+ \frac{\mathcal{S}}{\cos \theta} Z\_{\mu} \left[ \frac{1}{2} (\overline{\upsilon} \gamma^{\mu} \upsilon - \overline{\varepsilon}\_{L} \gamma^{\mu} e\_{L}) + |\sin \theta|^{2} \overline{\varepsilon} \gamma^{\mu} e \right] \end{split} \tag{250}$$

showing that, while the sterile right-handed neutrino is by construction insensitive to weak interactions, it is sensitive to the universal torsion interaction, suggesting that, to see torsional interactions, we must pass for neutrino physics.

After having extensively wandered in the microscopic domain of particle physics, we move to see what type of effect torsion might have for a macroscopic application of a yet unseen particle, dark matter, and we have seen that, in the case of effective approximation, the spinor source in the gravitational field equations becomes of the form

$$R^{\rho\sigma} - \frac{1}{2} R g^{\rho\sigma} - \Lambda g^{\rho\sigma} = \frac{k}{2} \left( E^{\rho\sigma} - \frac{1}{2} \frac{X^2}{M^2} S^\mu S\_{\mu} g^{\rho\sigma} \right) \tag{251}$$

showing that, if the spin density square happens to be positive, the contribution to the energy would change the gravitational field as to allow for a constant behavior of the rotation curves of galaxies. We have discussed that this behavior comes from having a matter density scaling according to *r* −2 for large distances. Such behavior, usually, is granted by the Navarro–Frenk–White profile or, here, is due to the presence of torsion, suggesting that the NFW profile is just the manifestation of torsional interactions, and ultimately that dark matter may be described in terms of the axial-vector simplified model, sorting out one privileged type among all possible simplified models now in fashion in particle physics. Then, we proceeded to include into the picture also the scalar fields, getting

$$\begin{split} \mathcal{L} = & -\frac{1}{4} (\partial \mathcal{W})^2 + \frac{1}{2} M^2 \mathcal{W}^2 - \frac{1}{k} R - \frac{2}{k} \Lambda - \frac{1}{4} F^2 + \\ & + i \overline{\psi} \gamma^\mu \nabla\_\mu \psi + \nabla^\mu \phi^\dagger \nabla\_\mu \phi - \\ & - X \overline{\psi} \gamma^\mu \pi \psi \mathcal{W}\_\mu - \frac{1}{2} \Xi \phi^2 \mathcal{W}^2 - Y \overline{\psi} \psi \phi - \\ & - m \overline{\psi} \psi + \mu^2 \phi^2 - \frac{1}{2} \lambda^2 \phi^4 \end{split} \tag{252}$$

showing that, in general, the torsion, besides its coupling to the spinor, may also couple to the scalar, with the scalar behaving as a sort of correction to the mass of torsion and a kind of re-normalization factor in the torsion–spinor effective interactions. We discussed how, within a homogeneous isotropic universe, the torsion field equations grant the condition *M*<sup>2</sup> =Ξ*φ* 2 so that, if Ξ were positive, then the scalar field would acquire a constant value, slow-roll will take place, and inflation could engage. After inflation has ended, torsion contributions to the scalar sector may induce a dynamical symmetry breaking. This may solve the cosmological constant problem in a new manner.

### **8. Conclusions**

In this review, we have constructed the most general geometry with torsion as well as curvature, and, after having also introduced gauge fields in a similarly geometric manner, we have also built a genuinely geometric theory of spinor fields. We have seen how, under the assumption of being at the least-order derivative, the most general fully covariant system of field equations has been found for all physical fields in interaction. Separating torsion from all other fields and splitting spinor fields in their irreducible components allowed us to better see that torsion can be seen as an axial-vector massive field mediating the interaction between chiral projections. A formal integration of the spinorial degrees of freedom has also been discussed in some detail. Studying special situations, we have seen that the torsionally-induced spin-contact interactions are attractive. In addition, we have examined the conditions under which they can be removed with a choice of chiral gauge.

We have then seen that torsional effects for spinor fields can give rise to the conditions for which the gravitational singularities are no longer bound to form. We have hence established a parallel to the instance of the Pauli exclusion principle. We have discussed the problem of positive energies for spinors. In addition, we have determined the conditions under which positivity of the energy can be ensured.

Eventually, we have discussed problems inherent to the standard model of particles, specifically for neutrino oscillations and dark matter. In addition, we have commented on a possible solution to the cosmological constant problem.

In a geometry which, in its most general form, is naturally equipped with torsion, and for a physics which, for the most exhaustive form of coupling, has to couple the spin of matter, the fact that torsion couples to the spin of spinor material field distributions is just as well suited as a coupling can possibly be. In addition, its consequences about the stability of such field distributions are certainly worth receiving further attention.

In the standard model of particles, there are different facts to consider. Assuming the existence of right-handed sterile neutrinos, the torsion–spin coupling gives dynamic corrections to the oscillation pattern that can become important contributions if we were to see that the neutrino masses were too close to one another to fit the observed patterns. By assuming dark matter as constituted by a form of matter with spinorial structure, the torsion field, with its axial-vector massive character, might give rise to the known NFW profile. In cosmology, the most urgent of the problems is that of the cosmological constant, which can be solved, or at least quenched, by a theory in which spontaneous symmetry breaking is replaced by dynamical symmetry breaking, as is the case when torsion is allowed to interact through spinors with the scalar.

In the first two instances, that is, for the case of neutrino oscillations and dark matter, the new contributions are condensed in <sup>∆</sup><sup>L</sup> <sup>=</sup> <sup>−</sup>*Xψ<sup>γ</sup> <sup>µ</sup>πψW<sup>µ</sup>* as axial-vector coupling. In the last instance that is for the dynamical symmetry breaking, new physics are described in terms of the <sup>∆</sup><sup>L</sup> <sup>=</sup>−*Xψ<sup>γ</sup> <sup>µ</sup>πψWµ*−<sup>1</sup> 2 Ξ*φ* <sup>2</sup>*W*<sup>2</sup> contribution as what gives the torsion–spin and scalar interaction. These two potentials for torsion are the only two potentials that can be added into the Lagrangian within the restriction of renormalizability.

It is important to call attention to the fact that, as the general presentation goes, there are two ways to have torsion in differential geometry: the first is the one followed here, and it is the one more mathematical in essence, based on the general argument that torsion is present simply because there is no reason to set it to zero. The second one is more physical, based on the argument that torsion *must* be present since it necessarily arises after gauging translations much in the same way in which curvature is present as it arose after gauging rotations. In this approach, torsion and curvature are the Yang–Mills fields that inevitably emerge because we are considering the gauge theory of the full Poincaré group [55]. The usefulness of this approach has been remarkable in addressing problems related to supersymmetry, and especially supergravity. For more details, we refer the reader to [56,57], and in particular [58]. More recent papers that deal with torsion-gravity as a gauge theory are [59–61]. Still important is the work in [62].

This latter approach of gauging the full Poincaré group is based on the tetradic formalism, on which an overview by Tecchiolli can be found in this Special Issue [63].

Readers might not have failed to notice that there is a great absent in the presentation: field quantization, and there are reasons for it to be so. In spite of all successful predictions and precise measurements, it would not be a proper behavior to deny all mathematical problems the theory of quantum fields still has. From the fact that the equal-time commutator relations may not make sense at all [64] to the non-existence of interaction pictures [65], to mention only the most important of the problems, the rigorous mathematical treatment of quantum fields is yet to be achieved. What can torsion do for this? Honestly, it seems unlikely that any change in the field content can change things for the general structure of the theory from its roots. However, it may still be possible that, after all, torsion could address problems appearing later on in the development of the theory. For example, we have already discussed how torsion could be responsible for avoiding singularities in the case of spinor fields. This tells us that torsion may similarly be responsible for the fact that the elementary particles might not be point-like. If this were the case, then torsion would certainly have something to say about the problem of ultraviolet divergences. Torsion may be what gives a physical meaning to regularization and normalization, with the torsion mass giving the scale of the physical cut-off. There is still quite a way to go in fixing, or at least alleviating, the problems of the quantum field theoretical approach. It does not look unreasonable, however, that torsion might be there to help again.

Then, there are all the possible extensions. To begin, there is the fact that we wrote all field equations under the constraint of being at the least-order derivative possible. This requirement also coincides with renormalizability for all equations except those for gravity. If we wish to have renormalizability for all equations, then the gravitational field equations must be taken at the fourth-order derivative in the metric tensor [66]. This is certainly an opportunity for further research, especially for the effects torsion may have for the problems of singularity formation and positive energies. Another point needing some strengthening is related to the fact that torsion has always been taken completely antisymmetrically. Such a symmetry was duly justified in terms of fundamental arguments, and, for that matter, the completely antisymmetric torsion couples neatly with the completely antisymmetric spin of the Dirac field. However, if higher-spin fields were to be found or more general geometric backgrounds were to be needed, more general torsion would make their appearance in the theory. Studying what may be the role of a trace torsion or of the remaining irreducible torsion component is also an important task.

Regarding the mass generation of spinors through spin–torsion interactions, it is necessary to direct the attention toward [67–69], and recently [70].

Again, a recent work is that of Diether and Christian later in this Special Issue [71].

In particle physics more in general, high-energy experimental constraints on torsion have been placed, especially in [72–76]. Going to cosmology, dark matter has also been studied in the presence of torsion: after the already-mentioned [45], the reader may find it interesting to also have a look at [77–81]. As for the problem of singularity formation during the Big Bang, the following references may be of help [82–86].

More mathematical extensions have been addressed along the years in various manners. In fact, all possible alternatives and extensions of Einstein gravity can also be generalized for the torsional case: for instance, conformal gravity with torsion has been established in [87], while *f*(*R*)-types of torsion-gravity have been studied in [88]. As for the latter, the reader may also find the problem of junction conditions interesting [89].

For a review of such a problem, we invite the reader to the paper by Vignolo that can also be found later in this Special Issue [90].

From a purely mathematical, general point of view, interesting features of the torsional background in the presence of spinors have been investigated in [91,92].

The dynamics of the torsion field may also in principle allow the propagation of parity violating modes, although many constraints have been placed recently [93,94].

Anomalies and constraints on torsion were studied in [95,96].

The possibilities introduced by not neglecting torsion in gravity for Dirac fields can also be more mathematical in essence. Above all, it is paramount to mention all the exact solutions for the coupled system of field equations that may be found. In [97–99], we found exact solutions for the Dirac field in its own gravitational field. Including torsion in gravity and allowing the coupling to the spinor can only increase the interesting features that exact solutions could have. Of course, finding exact solutions for a system of interacting fields is a very difficult enterprise and so we must expect a slow evolution.

It is clearly impossible to draw a complete list of references. Nevertheless, those presented here, and their own references, might be taken as a fair list to help the reader.

We wish to conclude this exposition with one personal note on aesthetics. It is very often stated in philosophical debates that a theory is considered to be beautiful when it has some sense of inevitability built into itself. That is, a sense for which there is nothing that can be modified, or removed, from the theory without looking like a form of unnecessary assumption. We see torsion gravity precisely like that. Such a theory is formed by requiring that some very general principles of symmetry be respected for the four-dimensional continuum space–time. If these hypotheses are put in, they determine the development of the theory without anything else to be postulated. It is at this point therefore that a theory of gravity with no torsion, where we would remove an object that would otherwise be naturally present, has to be regarded as something arbitrary.

The most beautiful, in the sense of the most necessary, theory of gravity is the one in which torsion is allowed to occupy the place that is its own a priori.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** In the last sixteen years, the single constant source of material (and more importantly immaterial) support has been Marie-Hélène Genest, my wife. It is to her that I give my continuous and everlasting thanks for everything.

**Conflicts of Interest:** The author declares no conflict of interest.

### **References**


*Review*

Received: 22 October 2019; Accepted: 3 December 2019; Published: 6 December 2019
