2.1.3. Covariant Field Equations

When in 1916 Einstein wanted to construct the theory of gravitation, the idea he wished to follow was inspired geometrically, based on the principle of equivalence.

The principle of equivalence states the equivalence at a local level between inertia and gravitation, in the sense that locally inertial and gravitational forces can simulate one another so well that, when both present, their effects can be made to cancel: it can be stated by saying that one can always find a system of coordinates in which locally the accelerations due to gravitation are negligible.

On the other hand, one can demonstrate a theorem originally due to Weyl whose statement sounds analogous: it states that one can always find a system of coordinates in which in a point the symmetric part of the connection vanishes.

In the previous sections, we have discussed in what way the condition of complete antisymmetry of torsion gives rise to a unique symmetric part of the connection, thus removing any possible ambiguity in the implementation of Weyl theorem: hence, for a completely antisymmetric torsion, the Weyl theorem is the mathematical implementation of the principle of equivalence insofar as the acceleration due to gravitation is encoded within the symmetric connection. A unique symmetric connection corresponds to a uniquely defined gravitational field as our physical intuition would suggest. Furthermore, the single symmetric connection is entirely written in terms of the derivatives of the metric, and therefore, if the gravitational field is encoded within the symmetric connection, then the gravitational potential is encoded within the metric tensor.

The metric tensor is a tensor, but it cannot vanish and none of its derived scalar is non-trivial, and the connection is not a tensor, so they will always depend on the choice of coordinates: hence, the information about gravity will always be intertwined with inertial information, which is not a surprise, since after all we know, they are locally indistinguishable. On the other hand, we wish to have a way to tell gravity apart from inertial information, and, to do that, it is necessary to take a less local level, then considering the Riemann curvature tensor: if gravity is contained in the metric tensor as well as in the connection, then it is contained in the Riemann curvature tensor too, but the Riemann curvature tensor is a tensor from which non-trivial scalars can be derived or which can be vanished, and this is what makes it able to discriminate gravity from inertial forces. If the metric is Minkowskian and the connection is zero, we cannot know whether this is because gravity is absent or compensated by inertial forces, and, similarly, if the metric is not Minkowskian and the connection is not zero, we cannot know whether this is because gravity is present or simulated by inertial forces as above. However, if the Riemann curvature tensor is zero, we know it is because gravity is absent, and, if the Riemann curvature tensor is not zero, we know gravity is present in general terms. This has to be so, as there can not be any compensation due to inertial forces since there can be no inertial forces, within the Riemann curvature tensor.

Therefore, the principle of equivalence is the manifestation of the interpretative principle telling that gravitation is geometrized, and this is so as a consequence of the fact that gravity alone is contained in the Riemann curvature.

This statement has to be taken into account together with the parallel fact that, in Einstein relativity, the mass is a form of energy, as it is very well known indeed.

Putting the two things together, it becomes clear that the gravitational field equations that were given in terms of a second-order differential operator of the gravitational potential proportional to the mass density have to be considered as an approximated form of a more general set of gravitational field equations given by a certain linear combination of the curvature proportional to the energy.

The energy density is a tensor having two indices and therefore the curvature we are looking for must have two indices as well, which tells that we need the contraction of the Riemann curvature given by the Ricci curvature.

In 1916, all matter forms that were known consisted of macroscopic fluids, scalars, and electro-dynamic fields, all of which have an energy density symmetric in the two indices. This may be a problem as the Ricci curvature is not symmetric.

In addition, this is where Einstein assumption of the vanishing torsion came about: assuming torsion to be equal to zero meant that a specific linear combination of the Ricci curvatures were symmetric, and thus proportional to the energy.

To see this, consider identity (29) in the case in which torsion vanishes. Its full contraction gives, in the most general case, the following identity:

$$\nabla\_{\mu} \left( \mathcal{R}^{\mu \nu} - \frac{1}{2} \mathcal{g}^{\mu \nu} \mathcal{R} - \mathcal{g}^{\mu \nu} \Lambda \right) = 0$$

where the object in parenthesis is symmetric indeed, and so it can be taken to be proportional to the energy density.

Now, Einstein geometrical insight is expressed by the gravitational field equations

$$R^{\mu\nu} - \frac{1}{2}g^{\mu\nu}R - g^{\mu\nu}\Lambda = \frac{1}{2}kE^{\mu\nu} \tag{120}$$

called Einstein field equations: from them, it follows that the energy density verifies *E µν* =*E νµ* and <sup>∇</sup>*µ<sup>E</sup> µν* =0 as is well known.

Therefore, Einstein field equations are the most general linear combination of curvatures for which geometric identities imply the validity of the symmetry and conservation law for the energy of matter. In this sense, the field equations are established on the bases of their conservation laws, themselves obtained from geometric identities, and this is what represents the Einstein spirit of geometrization—at least in the most general case without torsion.

Then, one might wonder what happens if torsion were not neglected.

The first thing we would have to keep in mind is that, in this case, geometry would provide both a curvature and a torsion tensor. The second point to be retained is that the Einsteinian gravitational theory is based on the fact that the curvature tensor is sourced by the energy. Putting things together, we should expect in the presence of torsion that there be another conserved quantity in parallel to the energy and another field equation coupling such a conserved quantity to the torsion tensor itself.

Such a quantity, however, is already at hand.

In 1928, Dirac was the first to describe a system of matter fields, named spinors, which possessed an energy together with a spin, and this is the quantity we are seeking.

In a torsional completion of the theory of gravitation, matter fields described by both an energy and a spin can naturally find a place when the spin is coupled to torsion much in the same way in which the energy is coupled to curvature. For such a theory, the full system of field equations is given by the spin–torsion field equations, which simply spell the proportionality between torsion and spin, called Sciama–Kibble field equations, alongside the curvature–energy field equations, which are formally the same as in Einstein gravity, and therefore still called Einstein field equations. Altogether, they are known under the name of Einstein–Sciama–Kibble ESK field equations [18–20].

However, contrary to what is believed, the ESK field equations are actually not the most general either because, while torsion and gravitation are independent, their field equations have the same coupling constant, and this accounts for an arbitrary restriction.

If we want independent fields to have independent coupling to their independent sources, we must find a way to obtain the ESK field equations generalized so that the two coupling constants are different.

We will not spend time on the mathematical details of this generalization, but the interested reader can find such generalized system of field equations in the case of two different coupling constants in [21].

However, then again, this is not still the most general system of field equations because the torsion tensor enters algebraically in its coupling to the spin density tensor.

As mentioned, the above system of field equations has the feature that torsion and spin are algebraically related and this constitutes a conceptual problem because in the case in which the spin density were to vanish, then torsion would vanish too, with no possibility to propagate, and hence the torsion tensor would be unphysical.

That the torsion–spin coupling is algebraic might not be seen as a problem because also the curvature–energy coupling is algebraic, but there are reasons for this situation not be to entirely analogous: the most important is that the torsion that enters in the field equations is the general Cartan torsion, with the consequence that, if the spin density were to be vanishing everywhere the Cartan torsion would be vanishing as well, but the curvature that enters the field equations is the Ricci curvature and not the Riemann curvature, with the consequence that, even if the energy density were to be vanishing everywhere, the Ricci curvature would also be vanishing, but this would not imply that the Riemann curvature would be equal to zero, and gravity may still be present.

In addition, the curvature has an internal structure given in terms of first-order derivatives of the connection and thus in terms of second-order derivatives of the metric tensor, so that there exists a dynamics for the gravitational field, unlike for torsion.

If we desire that the torsion dynamics be implemented in the theory, then we have to look for dynamical terms in the torsion–spin field equations, and also for torsional contribution in all of the other field equations as well.

We specify that our main goal is following the Einstein spirit of geometrization, and, in order to do so, we are going to obtain the field equations for the theory in a genuinely geometric way by finding the most general form of the field equations that is compatible with the constraints given by underlying geometric identities.

In order to construct the most general system of field equations, we are going to start by distinguishing them into two different types: the field equations for the geometry– matter coupling, which shall be written in the form of second-order derivatives of the metric and torsion and also gauge potentials equal to sources given by the energy and spin and also the current of fields; and the matter field equations, which will be written in the form of a first-order differential operator containing metric and torsion and gauge potentials acting on the spinor field and equalling the spinor field itself. This discrimination

comes from the fact that, on the one hand, it is possible to employ spinors to construct sources for the tensor and gauge field equations, but, on the other hand, it is not possible to use tensor and gauge fields to build sources of the spinorial field equations. In the spinorial field equations, the derivatives of the spinor field must be proportional to the spinor field itself. This discrimination between the form of geometric and matter field equations is therefore intrinsic to the structure of the fields we use.

We start by considering the fact that field equations for the metric have to be in the form of some derivative of the metric equal to some source: because the covariant derivative of the metric tensor vanishes identically, then any dynamics of the metric can only be described in terms of the partial derivatives of the metric, or, equivalently, by the metric connection (15). Again, the metric connection is not a tensor, and the only way we have from the symmetric connection to form a tensor is to take another partial derivative, therefore forming the metric curvature tensor as given by (22). As Equation (27) shows, the metric curvature tensor is one peculiar combination of second-order partial derivatives of the metric, that is, arguments of symmetry under the most general coordinate transformations force at least second-order derivatives of the metric in the differential field equations. Then, arguments of simplicity would require that we do not take any further differential structure. In the following, we will see that second-order derivatives in the metric field equations endow them with a character that no other field equation will have, rendering them somewhat peculiar indeed.

For the moment, what we have established is that the metric field equations will have to be given in the form of some combination of the metric curvature tensor, and to see what combination, we start from considering that, if the leading term were to be given by the Riemann metric curvature tensor *R ατσν*, then the vacuum equations would reduce to the condition of vanishing of Riemann metric curvature tensor, so that they would imply that there only be the trivial metric. Hence, if we want non-trivial metrics to be possible in vacuum, then the Riemann metric curvature tensor must appear contracted as the Ricci metric curvature tensor *R αµ* for leading term, and of course we may have contractions such as the Ricci metric curvature scalar *Rgαµ* or even Λ*gαµ* as sub-leading terms in general: as we have already seen above, the most general form of linear combination of curvatures in the field equations is given by (120), in which the only constant Λ is still undetermined, and it will remain undetermined since there is no way to fix it on geometrical grounds. Thus, we might well think of it as a generic integration constant, which can always be added and whose value cannot be fixed.

We now turn our attention to the other field equations, for which the covariant derivatives of the fields will not be identically zero.

The field equations for the torsion have to be in the form of covariant derivatives of the torsion axial-vector equal to some source: taking covariant derivatives of the torsion axial-vector implies that we will have to write the field equation in the form of the covariant divergence of the torsion axial-vector equal to a source constituted by a pseudo-scalar field, but the temporal derivative will be specified for the temporal component of the torsion axial-vector solely. In addition, thus, we must take two covariant derivatives of the torsion axial-vector as a leading term.

To assess what are the most general field equations for the torsion axial-vector, we consider that the leading term given in the form of two covariant derivatives of the torsion axial-vector ∇*σ*∇*αW<sup>ρ</sup>* is to be such that one of the indices of the derivatives has to be contracted yielding the two forms <sup>∇</sup>*σ*∇*σW<sup>ρ</sup>* and <sup>∇</sup>*ρ*∇*σW<sup>σ</sup>* as leading terms: sub-leading terms may be added eventually and so we may establish the most general form of field equations as

$$\begin{aligned} 2\Pi \nabla\_{\sigma} \nabla^{\sigma} W^{\eta} - 2H \nabla^{\eta} \nabla\_{\rho} W^{\rho} - \\ -V \nabla\_{\kappa} W\_{\nu} W\_{\rho} \varepsilon^{\alpha \nu \rho \eta} - \mathcal{U} W^{\kappa} W\_{\kappa} W^{\eta} - \\ -2LR^{\eta \rho} W\_{\rho} + 2NRW^{\eta} + PW^{\eta} = \kappa S^{\eta} \end{aligned}$$

where *S <sup>α</sup>* will have to be fixed on general grounds.

This general field equation can be restricted with the Velo–Zwanziger method [22,23]. Thus, taking its divergence

$$\begin{aligned} 2(\Pi - H)\nabla\_{\eta}\nabla^{\eta}\nabla\_{\rho}W^{\rho} &+ \\ &+ V\nabla\_{\eta}\nabla\_{\kappa}W\_{\nu}W\_{\rho}\varepsilon^{\eta\alpha\nu\rho} + \\ &+ V\nabla\_{\kappa}W\_{\nu}\nabla\_{\eta}W\_{\rho}\varepsilon^{\eta\alpha\nu\rho} - \\ &- 2\left[LIW^{\rho}W^{\eta} + (L - \Pi)R^{\eta\rho}\right]\nabla\_{\eta}W\_{\rho} + \\ &+ \left(2N - L + \Pi\right)\nabla\_{\eta}RW^{\eta} - \\ &- \left(\Pi W^{\alpha}W\_{\alpha} - 2NR - P\right)\nabla\_{\eta}W^{\eta} = \kappa\nabla\_{\eta}S^{\eta}\,,\end{aligned}$$

it becomes possible to see that there appears a third-order time derivative for the temporal component of the torsion axial-vector implying that the constraint obtained from the field equations would actually determine the time evolution of some components of the torsion axial-vector field. Since this would spoil a balance between the number of independent field equations and the amount of degrees of freedom of a given field, then no higher-order derivative terms must be produced in the constraints and thus we set Π = *H* identically. Once this is done, there is no second-order derivative in time for any components of the field in the constraint, which is thus a true constraint, which substituted back into the field equations gives

$$\begin{aligned} 2H\nabla\_{\sigma}\nabla^{\sigma}W^{\eta} - 2H(LW^{\alpha}W\_{\alpha} - 2NR - P)^{-1} \cdot \\ \cdot \nabla^{\eta}[V\nabla\_{\tau}\nabla\_{\alpha}W\_{\nu}W\_{\rho}\varepsilon^{\tau\alpha\nu\rho} + V\nabla\_{\alpha}W\_{\nu}\nabla\_{\tau}W\_{\rho}\varepsilon^{\tau\alpha\nu\rho} - \\ -2[LW^{\rho}W^{\tau} + (L - H)R^{\tau\rho}]\nabla\_{\tau}W\_{\rho} + \\ + (2N - L + H)\nabla\_{\tau}RW^{\tau} - \kappa\nabla\_{\tau}S^{\tau}] + \\ + 2H\nabla^{\eta}\left(LW^{\alpha}W\_{\alpha} - 2NR\right)(LW^{\alpha}W\_{\alpha} - 2NR - P)^{-2} \cdot \\ -[V\nabla\_{\tau}\nabla\_{\alpha}W\_{\nu}W\_{\rho}\varepsilon^{\tau\alpha\nu\rho} + V\nabla\_{\alpha}W\_{\nu}\nabla\_{\tau}W\_{\rho}\varepsilon^{\tau\alpha\nu\rho} - \\ -2[LW^{\rho}W^{\tau} + (L - H)R^{\tau\rho}]\nabla\_{\tau}W\_{\rho} + \\ + (2N - L + H)\nabla\_{\tau}RW^{\tau} - \kappa\nabla\_{\tau}S^{\tau}] - \\ -V\nabla\_{\alpha}W\_{\nu}W\_{\rho}\varepsilon^{\alpha\nu\rho\eta} - &LW^{\alpha}W\_{\alpha}W^{\eta} - \\ -2LR^{\eta\rho}W\_{\rho} + 2NRW^{\eta} + PW^{\eta} &= \kappa S^{\eta} \end{aligned}$$

which contains second-order time derivatives of all components of the torsion axial-vector, and therefore this is a true field equation. To check the propagation properties of the field, we consider its characteristic determinant

$$\left[ (\mathcal{U}W^{\mu}W\_{\alpha} - 2NR - P)n^{2} + 2[\mathcal{U}W^{\tau}W^{\nu} + (L - H)R^{\tau\nu}]n\_{\tau}n\_{\nu} = 0 \right]$$

and, by following the general discussion of Velo and Zwanziger, one can see that, in general, acausality may be possible unless we have *L* = *H* and *N* = *U* = 0 identically, in which case *n* <sup>2</sup> =0 and thus causality is ensured. Notice that there are no constraints on *V*, which remains a free parameter.

Placing all constraints together gives field equations

$$4\nabla\_{\rho}(\partial \mathcal{W})^{\rho \eta} - V \mathcal{W}\_{\rho}(\partial \mathcal{W})\_{\alpha \nu} \varepsilon^{\rho \alpha \nu \eta} + 2P \mathcal{W}^{\eta} = 2\kappa S^{\eta}$$

because *H* can be reabsorbed within a redefinition of all the other constants.

To proceed, we notice that, for the metric field equations, the source contribution from the torsion axial-vector field has to be built with no quartic torsion term because they would correspond to what in the torsion field equations are cubic torsion terms, which are absent, and no second derivatives of torsion because they would give rise to curvatures, which cannot be present since they are already addressed. Thus, it is possible to come to the most general form of this contribution as the one given by

$$\begin{split} E^{\mu\nu} &= a\mathcal{W}^{\mu}\mathcal{W}^{\nu} + b\mathcal{W}^{2}\mathcal{g}^{\mu\nu} + z(\mathcal{W}^{\nu}\mathcal{W}\_{\rho}(\partial\mathcal{W})\_{a\sigma}\varepsilon^{\rho a\sigma\mu} + \\ &+ \mathcal{W}^{\mu}\mathcal{W}\_{\rho}(\partial\mathcal{W})\_{a\sigma}\varepsilon^{\rho a\sigma\nu}) + \mathcal{y}(\nabla\_{\sigma}\mathcal{W}^{\mu}(\partial\mathcal{W})^{\sigma\nu} + \nabla\_{\sigma}\mathcal{W}^{\nu}(\partial\mathcal{W})^{\sigma\mu}) + \ge \nabla^{\mu}\mathcal{W}\_{\sigma}\nabla^{\nu}\mathcal{W}^{\sigma} + \\ &+ \mathcal{w}\nabla\_{\sigma}\mathcal{W}^{\mu}\nabla^{\sigma}\mathcal{W}^{\nu} + \nu\nabla\_{a}\mathcal{W}\_{\sigma}\nabla^{a}\mathcal{W}^{\sigma}\mathcal{g}^{\mu\nu} + \nu(\partial\mathcal{W})^{\nu\sigma}(\partial\mathcal{W})^{\mu}{}\_{\sigma} + t(\partial\mathcal{W})^{2}\mathcal{g}^{\mu\nu} \end{split}$$

in terms of ten constants: because we know that ∇*νE νµ* = 0 and because in vacuum the divergence of the torsion field equations gives

$$4P\nabla \cdot \mathcal{W} + V(\partial \mathcal{W})\_{\eta\rho}(\partial \mathcal{W})\_{\alpha\nu}\varepsilon^{\eta\rho\alpha\nu} = 0,$$

then one can easily see that it must be *V* =*z*=*v*=0 with *x* =*y*=−*w* and *x*+*u*=−4*t* and together with *a*=−2*b*=2*tP* which must hold identically.

We also notice that we must have *P* =2*M*<sup>2</sup> because this is just the mass term of the torsion axial-vector field as it is well known.

The field equations for the gauge field are also in the form of covariant derivatives of the gauge potential equal to some source: nevertheless, taking derivatives of the gauge potential means that that we have to consider the gauge strength because this is the only term that is differential in the potential and which is still gauge invariant, but, since this is irreducible, any contraction of the gauge strength vanishes and therefore these terms alone cannot be not enough. Hence, we have to take one more covariant derivative of the gauge strength as a leading term.

The most general field equations for the gauge fields have a leading term in the form ∇*σFαρ* and, after contraction, we get ∇*σF σρ* as the leading term: then, we get

$$\nabla\_{\sigma}F^{\sigma\eta} - \frac{1}{\Pi^2}BF\_{\alpha\nu}W\_{\rho}\varepsilon^{\alpha\nu\rho\eta} = qJ^{\eta\rho}$$

in which the source *J <sup>α</sup>* will have to be fixed as well.

The contribution from the gauge field is similarly built in terms of squares of the gauge strength strength, since any other term would violate gauge symmetry. Thus,

$$E^{\mu\nu} = \alpha F^{\mu\rho} F^{\nu}\_{~\rho} + \beta F^{\kappa\pi} F\_{\alpha\pi} \mathcal{g}^{\mu\nu}$$

in terms of two constants: again, because ∇*νE νµ* =0 and using the form of the electrodynamic field equations, we can see that *B*=0 and *α*=−4*β* identically.

In the metric field equation, the contributions due to torsion and gauge fields are analogous, and torsion and gauge fields are independent, so we may normalize torsion and gauge fields with no loss of generality in order to have the two constants *t* and *β* with the same value, and it is still without losing generality that they can be reabsorbed in the *k* constant. We notice that, in reabsorbing within a renaming of the constant *k* the values of the constants *t* and *β*, we did not lose any generality in their absolute value, but, in order not to lose any generality also for the sign, all constants would have to be positive, and this in general may not be the case: the reason why we did it anyway is that those constants are in front of torsion and gauge fields' energy contributions, which are positive defined. Of course, we might have assumed those constants to possess a generic sign, but, in the final form of the field equations, we would have discovered that those signs were positive, and thus we can assume this immediately with no loss of generality.

To proceed with the inclusion of matter fields, it is fundamental to notice that spinor fields are defined in terms of gamma matrices that can also be used in building fundamental quantities, whose employment allows for lowering the order of derivatives in all such quantities because every time covariance demands for a single covariant index to be present, and one gamma matrix can be used instead of one spinorial derivative.

The most general field equations for the spinor field have a leading term containing ∇*µψ* so that, after multiplying by the matrix *γ ν* , it is possible to contract the indices getting *γ <sup>µ</sup>*∇*µ<sup>ψ</sup>* as a leading term: therefore, we may establish the most general form of field equations as

$$i\gamma^{\mu}\nabla\_{\mu}\psi - X\mathcal{W}\_{\sigma}\gamma^{\sigma}\pi\psi - m\psi = 0$$

where the imaginary unit has been placed because, in free cases, *iγ <sup>µ</sup>*∇*µψ*−*mψ*=0 so that taking the square of the derivative gives <sup>∇</sup>2*ψ*+*m*2*ψ*=0, and *<sup>m</sup>* can be interpreted as the mass term, which is what is expected. That is, the imaginary unit has to be interpreted as what ensures the mass of the field will behave as to provide non-imaginary contribution to the dynamics of the free field equations.

Then, we have to write the general form of their contribution in the metric field equations, and this can be constructed by employing no more than one spinorial derivative of the spinor field, since gamma matrices can be used to saturate indices: eventually,

$$\begin{split} \mathcal{E}^{\rho\sigma} &= \mathfrak{zeta} \left[ \nabla^{\rho} (\overline{\psi} \gamma^{\sigma} \psi) + \nabla^{\sigma} (\overline{\psi} \gamma^{\rho} \psi) \right] + \\ &+ i \mathfrak{j} (\overline{\psi} \gamma^{\rho} \nabla^{\sigma} \psi - \nabla^{\sigma} \overline{\psi} \gamma^{\rho} \psi + \overline{\psi} \gamma^{\sigma} \nabla^{\rho} \psi - \nabla^{\rho} \overline{\psi} \gamma^{\sigma} \psi) + \\ &+ \chi \nabla\_{\alpha} (\overline{\psi} \gamma^{\alpha} \psi) g^{\rho\sigma} + \lambda i (\overline{\psi} \gamma^{\alpha} \nabla\_{a} \psi - \nabla\_{\alpha} \overline{\psi} \gamma^{\alpha} \psi) g^{\rho\sigma} + \\ &+ \tau (\mathcal{W}^{\sigma} \overline{\psi} \gamma^{\rho} \pi \psi + \mathcal{W}^{\rho} \overline{\psi} \gamma^{\sigma} \pi \psi) + \upsilon \mathcal{W}\_{a} \overline{\psi} \gamma^{a} \pi \psi g^{\sigma\rho} + \mu \overline{\psi} \psi g^{\rho\sigma} \end{split}$$

in general; the contribution as a source of the torsion field equations is the spin density of the material field, and it can be taken without any spinorial derivative at all when gamma matrices are considered, therefore obtaining that

$$S^{\mu} = \omega \overline{\psi} \gamma^{\mu} \pi \psi$$

also in general; the contribution as source of the gauge field equations is the current density of the material field, and similarly it is given according to

$$J^{\rho} = p\overline{\psi}\gamma r^{\rho}\psi$$

again in the most general case: by considering again the divergences of all field equations and with the same reasoning as before, one can eventually see that *ζ* = 0 as well as *µ*=−2*λm* and *p*=4*ξ* with *τ*=−2*ξX* and *υ*=−2*λX* and also *κω*=2*ξX* identically.

Finally, we notice that, without affecting the metric or the torsion or the gauge fields, the spinor field may be renormalized in such a way that, without losing generality, we can always set 4*ξ* = 1 and, as a consequence, it is possible to see that the full system of field equations has been completely determined.

It is constituted by the metric field equations given according to the expression

$$\begin{split} &R^{\rho\sigma} - \frac{1}{2} \mathcal{R} \mathcal{g}^{\rho\sigma} - \frac{k}{2} \left[ \frac{1}{4} (\partial \mathcal{W})^2 \mathcal{g}^{\rho\sigma} - (\partial \mathcal{W})^{\sigma\alpha} (\partial \mathcal{W})^{\rho}\_{\;\;\alpha} \right] - \\ & - \frac{k}{2} \left( \frac{1}{4} F^2 \mathcal{g}^{\rho\sigma} - F^{\rho\alpha} F^{\sigma}{}\_{\;\;\alpha} \right) - \frac{k}{2} M^2 (\mathcal{W}^{\rho} \mathcal{W}^{\sigma} - \frac{1}{2} \mathcal{W}^2 \mathcal{g}^{\rho\sigma}) - \\ & - \Lambda \mathcal{g}^{\rho\sigma} = \frac{1}{2} k \left[ \frac{i}{4} \left( \overline{\Psi} \gamma^{\rho} \nabla^{\sigma} \psi - \nabla^{\sigma} \overline{\Psi} \gamma^{\rho} \psi + \overline{\Psi} \gamma^{\sigma} \nabla^{\rho} \psi - \nabla^{\rho} \overline{\Psi} \gamma^{\sigma} \psi \right) - 1 \right. \\ & \left. - \frac{1}{2} X (\mathcal{W}^{\sigma} \overline{\Psi} \gamma^{\rho} \sigma \tau \psi + \mathcal{W}^{\rho} \overline{\Psi} \gamma^{\sigma} \pi \psi) \right] \end{split}$$

and the torsion field equations given according to

$$\nabla\_{\rho}(\partial \mathcal{W})^{\rho \mu} + \mathcal{M}^2 \mathcal{W}^{\mu} = \mathcal{X} \overline{\psi} \gamma^{\mu} \pi \psi$$

with gauge field equations given by

$$\nabla\_{\sigma} F^{\sigma \mu} = q \overline{\psi} \gamma^{\mu} \psi$$

as the form that is usually known, while the matter field equations are

$$i\gamma^{\mu}\nabla\_{\mu}\psi - \mathcal{X}\mathcal{W}\_{\sigma}\gamma^{\sigma}\pi\psi - m\psi = 0$$

with parameters Λ and *M* and also *m* describing intrinsic properties of metric and torsion and also spinor fields, while parameters *k*, *X*, and *q* are the constants that measure the strength with which metric, torsion, and gauge fields couple to energy, spin, and current.

It is possible to write the above system of coupled field equations into the system of coupled field equations with respect to which all the torsionless derivatives and curvatures are the torsionful derivatives and curvatures.

Thus, we can give the full system of field equations as the torsion–spin and the curvature–energy field equations as

$$D\_{[\rho}D^{\sigma}Q\_{\mu\nu]\sigma} + Q\_{\eta[\mu\nu}G\_{\rho]\sigma}\mathcal{g}^{\sigma\eta} - G\_{\sigma[\rho}Q\_{\mu\nu]\eta}\mathcal{g}^{\sigma\eta} + M^2Q\_{\rho\mu\nu} = \frac{1}{12}S\_{\rho\mu\nu} \tag{121}$$

and

$$\begin{split} &G^{\rho\tau} - \frac{1}{2} G \varrho^{\rho\tau} - 18k [\frac{1}{3} D\_a D^{[a} D\_\tau Q^{\rho\sigma] \pi} - \frac{1}{3} D\_a D\_\eta Q^{\eta\pi [a} Q^{\rho\sigma] \nu} g\_{\nu\tau} - \\ &- \frac{1}{3} Q^{\rho\eta\rho} D^{[\sigma} D\_\pi Q^{\eta\rho] \pi} - \frac{1}{3} Q^{\tau\eta\rho} D^{[\rho} D\_\pi Q^{\eta\rho] \pi} + \frac{1}{2} D^\pi D\_\tau Q^{\tau\mu\nu} Q\_{\pi\mu\nu} g^{\rho\sigma} + \\ &+ \frac{1}{4} D\_\pi Q^{\tau\mu\nu} D^\tau Q\_{\tau\mu\rho} g^{\rho\sigma} - D\_\pi Q^{\tau\mu\rho} D^\tau Q\_{\tau\mu} \frac{\sigma}{\tau} - \frac{1}{3} D\_\eta Q^{\eta\tau\alpha} D\_\alpha Q^{\rho\sigma} \frac{\sigma}{\pi} + \\ &+ \frac{1}{3} (Q^{\rho\eta\rho} D\_\tau Q^{\tau\mu\sigma} + Q^{\tau\eta\rho} D\_\tau Q^{\tau\mu\rho}) Q^{\eta\rho}{}\_\pi \left] - \frac{1}{2} k (\frac{1}{4} F^2 \mathcal{G}^{\rho\sigma} - F^{\rho\alpha} F^{\sigma}\_a) - \\ &- (12 k M^2 + 1) (\frac{1}{2} D\_\theta Q^{\mu\rho\sigma} - \frac{1}{4} Q^{\rho\alpha\tau} Q^{\sigma}{}\_{a\pi} + \frac{1}{8} Q^2 \mathcal{G}^{\rho\sigma}) - \Lambda g^{\rho\sigma} = \frac{1}{2} k T^{\rho\sigma} \end{split} \tag{122}$$

called *Sciama–Kibble field equations* and *Einstein field equations* and they come alongside the gauge-current field equations

$$D\_{\sigma}F^{\sigma\mu} + \frac{1}{2}F\_{\alpha\nu}Q^{\alpha\nu\mu} = \mathcal{J}^{\mu} \tag{123}$$

called *Maxwell field equations*, where the sources are given by the spin and the energy

$$S^{\rho\mu\nu} = -8X^{\frac{i}{4}}\overline{\psi}\{\gamma^{\rho}, \sigma^{\mu\nu}\}\psi\tag{124}$$

and

$$\begin{split} T^{\rho\tau} &= \frac{i}{2} (\overline{\Psi}\gamma^{\rho}\mathbf{D}^{\sigma}\psi - \mathbf{D}^{\sigma}\overline{\Psi}\gamma^{\rho}\psi) + (8X+1)D\_{a}(\frac{i}{4}\overline{\Psi}\{\gamma^{\mu},\sigma^{\rho\sigma}\}\psi) + \\ &+ \frac{1}{2}(8X+1)Q^{\rho\mu\upsilon}\frac{i}{4}\overline{\Psi}\{\gamma^{\sigma},\sigma\_{\mu\upsilon}\}\psi - (8X+1)Q^{\sigma\mu\upsilon}\frac{i}{4}\overline{\Psi}\{\gamma^{\rho},\sigma\_{\mu\upsilon}\}\psi \end{split} \tag{125}$$

alongside the current

$$
\overline{\gamma}^{\mu} = q \overline{\psi} \gamma^{\mu} \psi \tag{126}
$$

given in terms of the matter field. They come alongside the spinorial field equation

*J*

$$i\gamma^{\mu}\mathbf{D}\_{\mu}\psi - i(X + \frac{1}{8})Q\_{\nu\tau\mathfrak{a}}\gamma^{\nu}\gamma^{\tau}\gamma^{\mathfrak{a}}\psi - m\psi = 0\tag{127}$$

called *Dirac spinorial field equations*, which decompose according to

$$\frac{1}{2}(\overline{\psi}\gamma^{\mu}\mathcal{D}\_{\mu}\psi - \mathcal{D}\_{\mu}\overline{\psi}\gamma^{\mu}\psi) - (X + \frac{1}{8})Q^{\pi\tau\eta}S^{\sigma}\varepsilon\_{\pi\tau\eta\sigma} - m\Phi = 0\tag{128}$$

$$D\_{\mu}U^{\mu} = 0\tag{129}$$

$$\frac{1}{2}(\overline{\psi}\gamma^{\mu}\pi\mathbf{D}\_{\mu}\psi - \mathbf{D}\_{\mu}\overline{\psi}\gamma^{\mu}\pi\psi) - (X + \frac{1}{8})Q^{\pi\tau\eta}L^{\sigma}\varepsilon\_{\pi\tau\eta\sigma} = 0\tag{130}$$

$$D\_{\mu}S^{\mu} - 2m\Theta = 0\tag{131}$$

$$i(\overline{\psi}\mathcal{D}^a\psi - \mathcal{D}^a\overline{\psi}\psi) - D\_\mu M^{\mu a} + (2X + \frac{1}{4})\varepsilon\_{\pi\tau\eta\sigma}Q^{\pi\tau\eta}\Sigma^{\sigma a} - 2m\mathcal{U}^a = 0\tag{132}$$

$$D\_{\mathfrak{a}}\Phi - 2(\overline{\psi}\sigma\_{\mu\mathfrak{a}}\mathbf{D}^{\mu}\psi - \mathbf{D}^{\mu}\overline{\psi}\sigma\_{\mu\mathfrak{a}}\psi) + (2X + \frac{1}{4})\Theta Q^{\pi\tau\eta}\varepsilon\_{\pi\tau\eta\mathfrak{a}} = 0\tag{133}$$

$$D\_{\nu} \Theta - 2i(\overline{\psi} \sigma\_{\mu \nu} \pi \mathbf{D}^{\mu} \psi - \mathbf{D}^{\mu} \overline{\psi} \sigma\_{\mu \nu} \pi \psi) - (2X + \frac{1}{4}) \Phi Q^{\pi \tau \eta} \varepsilon\_{\pi \tau \eta \nu} + 2mS\_{\nu} = 0\tag{134}$$

$$\partial\_t \left( \mathbf{D}\_a \overline{\psi} \pi \psi - \overline{\psi} \pi \mathbf{D}\_a \psi \right) + D^\mu \Sigma\_{\mu a} + \left( 2X + \frac{1}{4} \right) \varepsilon^{\pi \tau \eta \mu} Q\_{\pi \tau \eta} M\_{\mu a} = 0 \tag{135}$$

$$(D^{\mu}S^{\rho}\varepsilon\_{\mu\rho\alpha\upsilon} + i(\overline{\psi}\gamma\_{[a}\mathbf{D}\_{\upsilon]}\psi - \mathbf{D}\_{[\nu}\overline{\psi}\gamma\_{a]}\psi) + (2X + \frac{1}{4})Q^{\pi\tau\eta}\varepsilon\_{\pi\tau\eta[a}S\_{\nu]} = 0\tag{136}$$

$$-D^{[a}\mathcal{U}^{\nu]} - i\varepsilon^{av\mu\rho}(\mathbf{D}\_{\mu}\overline{\psi}\gamma\_{\rho}\pi\psi - \overline{\psi}\gamma\_{\rho}\pi\mathbf{D}\_{\mu}\psi) - (12X + \frac{3}{2})Q^{av\rho}\mathcal{U}\_{\rho} - 2mM^{av} = 0\tag{137}$$

which are altogether equivalent to the Dirac spinor field equations and called *Gordon decompositions*. Spin, energy, and current verify

$$D\_{\rho} \mathbb{S}^{\rho \mu \nu} + \frac{1}{2} T^{[\mu \nu]} \equiv 0 \tag{138}$$

and

$$D\_{\mu}T^{\mu\nu} + T\_{\rho\oint}Q^{\rho\oint} - S\_{\mu\rho\oint}G^{\mu\rho\beta\nu} + I\_{\rho}F^{\rho\nu} \equiv 0\tag{139}$$

alongside

$$(D\_{\rho})^{\rho} = 0\tag{140}$$

satisfied in the most general case.

Intriguingly, we notice that, in the spinor field equations, the mass appears linearly, and thus it may be positive as well as negative, and therefore it is possible to have two different types of spinor field equations. Such possibility is clear because, if *m* → −*m* is accompanied by the discrete transformation *ψ*→*πψ*, then the system of field equations is invariant, and, consequently, any solutions of the first are also a solution of the second. The fact that we may have two different spinor field equations is translated into the fact that we may have two different solutions linked by *ψ*→*πψ* in general.

The full system of field equations is invariant under the transformation of parity reflection [24], and it is the most general under the restriction of being at the least-order differential form [25]. What this means is that arguments of compatibility with covariance, generality, and having field equations at their least-order derivative are enough to lead to the above physical field equations. In addition, this is true regardless of the principle of equivalence. The principle of equivalence might have been a guide for Einstein from a historical perspective, but mathematically there is no need for it. Its role is reduced to that of an interpretative principle telling us that the metric is what encodes the information about gravitation. Moreover, it is common knowledge of Einsteinian gravity that, when Einstein field equations are linearized and taken in the static case and for small velocities, they reduce to Newton equations, in which the time–time component of the metric is witnessed to be the Newtonian gravitational potential. Henceforth, the principle of equivalence can be abandoned. Certainly, this principle may give important insights, but it can be equally well disregarded, as the interpretation of gravity within the metric tensor naturally emerges from specific limits of the Einstein field equations and these come from arguments of simplicity, generality, and compatibility with identities proper to the underlying geometric structure.

### **3. Torsion-Spin Interactions**

In this second section, we will consider the above physical field equations in order to investigate their properties: the idea will be to write them in an equivalent but somewhat clearer manner. We will end with general remarks about the interaction of geometry with its material content.

### *3.1. Torsion and Spinor Decomposition*

To have the physical field equations converted in more manageable forms, we will decompose all quantities that can be decomposed into more fundamental ones: we will separate torsion from all torsionless quantities in all the covariant derivatives and curvature tensors. Finally, the spinor field will also be decomposed into its two irreducible chiral projections and elementary degrees of freedom.

### 3.1.1. Torsion as Axial-Vector Massive Field

Among all geometric fields, torsion has a special property indeed. The gauge potential is a gauge field for phase transformations, and the metric tensor can be considered a gauge field for coordinate transformations, so both are always depending on the phase or the coordinate system, while torsion is a tensor that does not have any relation with such properties. Thus, torsion can be split from gauge and metric connections, with all the covariant derivatives and curvatures being written as covariant derivatives and curvatures with no torsion but with all the torsion terms appearing as independent.

To have the most general connection decomposed into the simplest symmetric connection plus torsion terms, we only need to substitute (16) in (37), and this in (114).

Thus, the system of field equations reduces to

$$\nabla\_{\rho}(\overline{\partial}\mathcal{W})^{\rho\mu} + M^2 \mathcal{W}^{\mu} = X \overline{\psi} \gamma^{\mu} \pi \tau \psi \tag{141}$$

and

$$\begin{aligned} R^{\rho\sigma} - \frac{1}{2} R g^{\rho\sigma} - \Lambda g^{\rho\sigma} &= \frac{k}{2} \left[ \frac{1}{4} F^2 g^{\rho\sigma} - F^{\rho\sigma} F^{\sigma}{}\_{\alpha} + \\ + \frac{1}{4} (\partial \mathcal{W})^2 g^{\rho\sigma} - (\partial \mathcal{W})^{\sigma\alpha} (\partial \mathcal{W})^{\rho}{}\_{\alpha} + M^2 (\mathcal{W}^{\rho} \mathcal{W}^{\sigma} - \frac{1}{2} \mathcal{W}^2 g^{\rho\sigma}) + \\ + \frac{i}{4} (\overline{\psi} \gamma^{\rho} \nabla^{\sigma} \psi - \nabla^{\sigma} \overline{\psi} \gamma^{\rho} \psi + \overline{\psi} \gamma^{\sigma} \nabla^{\rho} \psi - \nabla^{\rho} \overline{\psi} \gamma^{\sigma} \psi) - \\ - \frac{1}{2} X (\mathcal{W}^{\sigma} \overline{\psi} \gamma^{\rho} \pi \psi + \mathcal{W}^{\rho} \overline{\psi} \gamma^{\sigma} \pi \psi)) \end{aligned} \tag{142}$$

for the torsion–spin and curvature–energy coupling, and

$$
\nabla\_{\sigma} \mathbf{F}^{\sigma \mu} = q \overline{\psi} \gamma^{\mu} \psi \tag{143}
$$

for the gauge–current coupling. Then, we also have that

$$i\gamma^{\mu}\nabla\_{\mu}\psi - \mathbf{X}\mathcal{W}\_{\sigma}\gamma^{\sigma}\pi\psi - m\psi = 0\tag{144}$$

for the spinor field equations.

If we take the divergence of (141) and contract (142), we obtain the constraints

$$M^2 \nabla\_{\mu} W^{\mu} = 2Xmi \overline{\psi} \pi \pi \psi \tag{145}$$

and

$$
\epsilon\_{\overline{k}}^2 \mathcal{R} + \frac{8}{k} \Lambda - M^2 \mathcal{W}^2 = -m\overline{\psi}\psi \tag{146}
$$

where (144) has been used.

It is now possible to interpret torsion: just a quick look at the torsion–spin and curvature–energy field equations simply reveals that *torsion is an axial-vector massive field* verifying Proca field equations with corresponding energy and torsional-spin coupling within the gravitational field equations [26]. With this insight, one might now wonder if there really was the necessity to go through the trouble of insisting on the presence of torsion if all comes to the presence of an axial-vector massive field, asking why we could not simply impose torsion equal to zero and then allowing an axial-vector massive field to be included into the theory. The answer is that, although mathematically it is equivalent to follow both approaches, conceptually the former approach is the most straightforward construction in which all quantities are defined and all relationships are built in the most general manner, while, on the other hand, the latter approach would be afflicted by a number of arbitrary assumptions. If this latter approach were the one to be followed, we would have to justify why torsion albeit in general present should be removed, why, among all fields that could be included, we pick precisely a vector field with pseudotensorial properties, and why it would have to be massive, and hence resulting into an approach having three unjustified assumptions in alternative to the other approach in which assumptions are either justified or not assumed at all. In order to avoid this high degree of arbitrariness, we prefer to follow the approach that we actually followed here. This leads after all to the presence of an axial-vector massive field. Then, if in some part of the theory, there were to appear new physics that could somehow be reconducted to an axial-vector massive field, we would know that these effects would emerge from the existence of torsion. In fact, such effects might be something that we have already observed, even if we ignored that they could come from the torsion tensor.

### Spinors as Sum of Chiral Parts

Analogously to the covariant decomposition of torsion, there is also a perfectly covariant split of the spinor field into its two chiral parts according to (80) and (81) and therefore in its degrees of freedom as expressed by (105).

When (105) is plugged into the Gordon decompositions, we obtain the polar forms of the Gordon decompositions, among which we find the following two equations:

$$-X\mathcal{W}\_{\mu} - \frac{1}{4}g\_{\mu\nu}\varepsilon^{\nu\rho\sigma a}\partial\_{\rho}\xi^{k}\_{\sigma}\xi^{j}\_{a}\eta\_{j\bar{k}} - (\nabla\mathfrak{a} - qA)^{\iota}u\_{[\iota}s\_{\mu]} + s\_{\mu}m\cos\theta + \frac{1}{2}\nabla\_{\mu}\beta = 0$$

and

$$s\_{\mu}m\sin\beta - (\nabla a - qA)^{\rho}u^{\nu}s^{a}\varepsilon\_{\mu\rho\nu a} + \frac{1}{2}|\mathring{\xi}|^{-1}\mathring{\xi}\_{\mu}^{k}\partial\_{\mu}(|\mathring{\xi}|\mathring{\xi}\_{k}^{a}) + \nabla\_{\mu}\ln\phi = 0$$

which are very special since we can show that these two expressions imply the spinor field Equations (144): in fact, by employing the above pair of equations, we have that

$$\begin{aligned} & \qquad i\gamma^{\mu}\nabla\_{\mu}\psi - \mathcal{X}\mathcal{W}\_{\sigma}\gamma^{\sigma}\pi\psi - m\psi = \\ & = (\nabla\alpha - qA)^{\iota}(i\gamma^{\mu}u^{\nu}s^{\mu}\varepsilon\_{\mu\nu\alpha} + \iota\_{[\iota}s\_{\mu]}\gamma^{\mu}\pi + \gamma\_{\iota})\psi - \\ & \qquad - m(is\_{\mu}\gamma^{\mu}\sin\beta + s\_{\mu}\gamma^{\mu}\pi\cos\beta + \mathbb{I})\psi \end{aligned}$$

and then, using (90, 91), we get

$$i\gamma^{\mu}\nabla\_{\mu}\psi - X\mathcal{W}\_{\sigma}\gamma^{\sigma}\pi\psi - m\psi = 0$$

which is the Dirac spinor field Equation (144) as expected.

Therefore, we may summarize by saying that the Dirac spinorial field equation are equivalent to the equations

$$2\nabla\_{\mu}\mathfrak{E} - 2X\mathcal{W}\_{\mu} - \frac{1}{2}\mathcal{g}\_{\mu\nu}\varepsilon^{\nu\rho\sigma\mathfrak{a}}\partial\_{\rho}\mathfrak{X}^{k}\_{\sigma}\mathfrak{X}^{j}\_{\mathfrak{a}}\eta\_{j\bar{k}} - 2(\nabla\mathfrak{a} - qA)^{\iota}u\_{[\![\![s}\!]\_{\mu]} + 2ms\_{\mu}\cos\beta = 0 \tag{147}$$

and

$$\nabla\_{\mu} \ln \phi^2 + |\mathfrak{f}|^{-1} \mathfrak{J}\_{\mu}^k \partial\_{\mathfrak{a}} (|\mathfrak{f}| \mathfrak{J}\_k^a) - 2(\nabla a - qA)^{\rho} u^{\nu} s^a \varepsilon\_{\mu \rho \nu \mathfrak{a}} + 2ms\_{\mu} \sin \beta = 0 \tag{148}$$

in the most general case that is possible.

Thus, we interpret spinor fields in this way: despite being fundamental, *spinor fields are reducible, constituted from two chiral parts. Their independence is measured by the Yvon– Takabayashi angle describing internal dynamics of the spinor field. The module describes the overall matter distribution*. These are the two degrees of freedom of the spinor field, with all space–time derivatives of these two degrees of freedom specified by (147) and (148) [27].

### *3.2. Torsion–Spinor Interactions*

We now have all elements to deepen the investigation about the interaction between geometry and matter.

### 3.2.1. Torsion–Spinor Binding

In the recent parts, we have seen that (145) and (146) provide very simply links between geometrical structures and the bi-linear spinorial scalars. In addition, (147) and (148) constitute some form of dynamical conditions upon such spinorial scalars.

We have in fact that (145) and (146) can be written as

$$M^2 \nabla\_{\mu} W^{\mu} = 4Xm\phi^2 \sin \beta \tag{149}$$

and

$$\frac{2}{k}R + \frac{8}{k}\Lambda - M^2W^2 = -2m\phi^2\cos\beta\tag{150}$$

linking torsion and curvature to Yvon–Takabayashi angle and module, these last being subject to

$$\nabla\_{\mu}\boldsymbol{\beta} - 2X\boldsymbol{W}\_{\mu} - \frac{1}{2}g\_{\mu\nu}\varepsilon^{\nu\rho\sigma\imath a}\partial\_{\rho}\xi^{k}\_{\sigma}\xi^{j}\_{a}\eta\_{jk} - 2(\nabla\boldsymbol{u} - q\boldsymbol{A})^{\iota}u\_{[\boldsymbol{u}}s\_{\mu]} + 2ms\_{\mu}\cos\beta = 0\tag{151}$$

and

$$\nabla\_{\mu} \ln \phi^2 + |\mathfrak{f}|^{-1} \mathfrak{J}\_{\mu}^k \partial\_{\mathfrak{a}} (|\mathfrak{f}| \mathfrak{J}\_{\mathbb{K}}^a) - 2(\nabla a - qA)^\rho u^\nu s^a \varepsilon\_{\mu\rho\nu\mathfrak{a}} + 2ms\_{\mu} \sin \beta = 0 \tag{152}$$

as dynamical conditions. Therefore, by solving these last equations, *we can always integrate spinor fields as all their degrees of freedom can be tied to torsion and curvature*.

This is remarkable because it shows that formally the spinorial degrees of freedom can always be replaced by quantities related to the underlying geometric structure.

To conclude this part, we will give a few more results starting from the introduction of the potentials

$$K\_{\mu} = 2X\mathcal{W}\_{\mu} + \frac{1}{2}g\_{\mu\nu}\varepsilon^{\nu\rho\sigma\alpha}\partial\_{\rho}\mathfrak{J}^{k}\_{\sigma}\mathfrak{J}^{j}\_{\alpha}\eta\_{j\dot{k}} + 2(\nabla\alpha - qA)^{\iota}u\_{[\iota}s\_{\mu]}\tag{153}$$

and

$$G\_{\mu} = -|\mathfrak{f}|^{-1} \mathfrak{f}\_{\mu}^{k} \partial\_{\mathfrak{a}} (|\mathfrak{f}| \mathfrak{f}\_{k}^{a}) + 2(\nabla \mathfrak{a} - qA)^{\rho} \mathfrak{u}^{\nu} \mathfrak{s}^{a} \varepsilon\_{\mu\rho\nu\omega} \tag{154}$$

in terms of which we have

$$\nabla\_{\mu}\mathcal{S} - \mathbf{K}\_{\mu} + \mathbf{s}\_{\mu}\mathbf{2}m\cos\beta = 0\tag{155}$$

and

$$\nabla\_{\mu} \ln \phi^2 - G\_{\mu} + s\_{\mu} \mathcal{D}m \sin \beta = 0 \tag{156}$$

as the Dirac equations in polar form. From these, we get

$$\left|\nabla\frac{\rho}{2}\right|^2 - m^2 - \phi^{-1}\nabla^2\phi + \frac{1}{2}(\nabla G + \frac{1}{2}G^2 - \frac{1}{2}K^2) = 0\tag{157}$$

and

$$\nabla\_{\mu}(\phi^2 \nabla^{\mu} \frac{\oint}{\Sigma}) - \frac{1}{2}(\nabla \mathcal{K} + \mathcal{K} \mathcal{G})\phi^2 = 0 \tag{158}$$

as a Hamilton–Jacobi equation and a continuity equation.

Alternatively, we may define

$$\frac{1}{2}(\nabla\_{\mu}\beta - 2X\mathsf{W}\_{\mu} - \frac{1}{2}\mathcal{g}\_{\mu\nu}\varepsilon^{\nu\rho\sigma\mu}\partial\_{\rho}\mathfrak{J}^{k}\_{\sigma}\mathfrak{J}^{j}\_{a}\mathfrak{\eta}\_{jk}) = \mathcal{Y}\_{\mu} \tag{159}$$

and

$$-\frac{1}{2} [\nabla\_{\mu} \ln \phi^2 + |\xi|^{-1} \mathfrak{F}\_{\mu}^k \partial\_{\mu} (|\xi| \mathfrak{F}\_k^a)] = Z\_{\mu} \tag{160}$$

in terms of which

$$(Y\_{\mu} - (\nabla \alpha - qA)^{\iota} u\_{[\iota} s\_{\mu]} + m s\_{\mu} \cos \beta = 0\tag{161}$$

and

$$Z\_{\mu} - (\nabla \mathfrak{a} - qA)^{\rho} u^{\nu} s^{\mathfrak{a}} \varepsilon\_{\mu \rho \nu \mathfrak{a}} + m s\_{\mu} \sin \beta = 0 \tag{162}$$

as Dirac equations in polar form. Then, defining

$$P\_{\nu} = \nabla\_{\nu} \mathfrak{a} - qA\_{\nu} \tag{163}$$

as the momentum, we have that

$$P^{\nu} = m \cos \beta \mu^{\nu} + \mathcal{Y}\_{\mu} \mu^{[\mu} \mathbf{s}^{\nu]} + \mathcal{Z}\_{\mu} \mathbf{s}\_{\rho} \mu\_{\sigma} \varepsilon^{\mu \rho \sigma \nu} \tag{164}$$

giving its explicit form in terms of mass and velocity but also in terms of the Yvon– Takabayashi angle and spin as well as the potentials given in the (159) and (160) above.

There is a very important point to be clarified regarding the spinorial active transformations acting on spinorial fields. Consider the rotations around the third axis and the spinors in polar form (105): despite the fact that these spinors are aligned along the axis around which we perform the above rotation, that rotation does not leave them unchanged (as we have for vectors). This might already sound problematic, but, in addition, we also have that, when such a rotation is given for an angle *<sup>θ</sup>* <sup>=</sup> <sup>2</sup>*π*, it is **<sup>Λ</sup>** <sup>=</sup> <sup>−</sup><sup>I</sup> implying that the spinor would not go back to the initial configuration (as we have when we perform a passive rotation). This too sounds peculiar. Thus, we might ask, is there any intuitive way to see things under which these odd behaviors would look natural? First of all, we have to take into account the fact that the rotation is an active rotation, and therefore an operation that, keeping fixed the space–time, moves the spinor. Then, we have to keep in mind that spinors are more sensitive than vectors to the structure of the space–time, as if anchored instead of being free to slide in it. Thus, for a given active rotation around a certain axis, a vector behaves like a pole, and, if aligned to the axis of rotation, it would be left unchanged as a whole. For the same active rotation, however, spinors would behave as a pole with a flag, so, even if aligned to the axis of rotation, they would be left unchanged almost fully but not quite entirely. They would indeed behave as if the rotation was taking place on a Möbius band. One way to picture them would be that of the belt trick, or the spinning plates, as described in [28].

We have shown that there is a duplicity in the spinorial structure made clear from the fact that spinors were defined up to the *ψ*→*πψ* discrete transformation, or from the fact that the combined *ψ*→*πψ* and *m*→ −*m* is one symmetry of the physical field equations. Such duplicity may suggest a form of matter/antimatter duality [29,30].

Physical effects and phenomenological implications provided by a torsion tensor with a dynamical axial-vector field have also been recently presented in [31].

### **4. Limiting Situations**

In this third section, we will consider the above theory in some specific cases so to deepen their examination: we will first consider what happens as a consequence of the fact that torsion is an axial-vector field with mass and in addition we will discuss what happens as a consequence of the fact that also the spinor field is massive. Eventually, we will see what happens in the complementary situation in which masslessness will allow another symmetry.
