The Spinor-Tensor Gravity of the Classical Dirac Field

In this work, with the help of the quantum hydrodynamic formalism, the gravitational equation associated to the Dirac field is derived. The hydrodynamic representation of the Dirac equation have been generalizaed to the curved space-time in the covariant form. Thence, the metric of the spacetime has been defined by imposing the minimum action principle. The derived gravity shows the spontaneous emergence of the cosmological gravity tensor (CGT) as a part of the energy-impulse tensor density (EITD) that in the classical limit leads to the cosmological constant (CC). Even if the classical cosmological constant is set to zero, the CGT is non zero, allowing to have a stable quantum vacuum (out of the collapsed branched polymer phase). The theory shows that in the classical limit, the gravity equation leads to the general relativity equation. In the perturbative approach, the CGT leads to a second order correction to the Newtonian gravity that takes contribution from the space where the mass is localized (and the spacetime is curvilinear) while tends to zero as the spacetime approaches to the flat vacuum leading, as a mean, to an overall cosmological cosmological constant that may possibly be compatible with the astronomical observations. The Dirac field gravity shows analogies with the modified Brans-Dicke gravity where each spinor term brings an effective gravity constant G divided by its field squared.

, that move with momenta

Introduction
One of the problems of nowadays physics is to describe how the quantum mechanical properties of the space-time (ST) and the second quantization affect the gravity. Even if the general relativity (GR) has opened some understanding about the cosmological dynamics [1][2][3][4][5][6] the complete explanation of generation of matter and its distribution in the universe need the integration of the cosmological physics with the quantum one. To this end the quantum gravity (QG) represents the goal of the century for the theoretical research [7][8][9]. Nevertheless, difficulties arise when one attempts to apply the quantum field theory (QFT) to the force of gravity [10][11]. The difficulties about the integration of QFT and the GR become really evident in the so called cosmological constant (CC) that Einstein introduced into its equation to give stability to the solution of universe evolution but that then he refused [6]. Actually, the renewed interest for the CC is basically stated by semi-empirical arguments to explain the motion of the galaxies [10][11]. The great difficulty of the QFT to give a correct value of the CC, relies in the fact that the energy-impulse tensor density (EITD) in the Einstein equation [12] (for classical bodies) owns a point-dependence by the mass density, but does not have any analytical connection with the fields of matter described by the Dirac or by the Klein-Gordon equation (KGE). If we place matter into a classic ST, each infinitesimal element of mass can be put freely in an infinitesimal volume without any interference with the neighboring ones. In the quantum mechanical ST, the things are different and it is more and more difficult to add more and more matter in the same place (e.g., forbidding punctual mass distribution). Obviously, these two kinds of space-times are different and leads to distinct types of gravity. The incompleteness of general relativity in describing the gravity of the physical quantum-ST is recognized by all and modified gravity models (on semi-empirical basis) have been proposed. One example is the scalar-tensor gravity about which the Brans-Dicke [13-15 one is the most known example. If, on one hand, the conceptual generalization about the possible sources of gravity has been fruitful showing how problems such as the cosmological constant, the universe inflation and the dark matter can be solved [16][17][18], on the other hand, it has brought to the appearance of a scalar field of obscure physical reading. The derivation of the quantum-mechanical gravity of a mass distribution obeying to the Klein-Gordon equation [19], by using the classical-like hydrodynamic quantum representation, that leads to terms that are identical to those contained into the Brans-Dicke model, shows that the scalar field of the BD theory, by fact, reproduces the effects of the quantum-mechanical property of the ST upon the gravity. Taking in account the contribution of the quantum potential energy (i.e., the energy connected to the pilot wave) in the definition of the space-time curvature, the author has shown that the quantization of a scalar uncharged field may lead to a mean cosmological constant value (on the universe scale) compatible with the astronomical observation with the energy cut-off defined by the Planck wave length [20]. The objective of this work is to generalize the gravitational equation (GE) of the scalar KGE field [17] to the field of half integer spin charged particles obeying to the Dirac-Fock-Weyl equation (DFWE).
The work is carried out by utilizing the quantum hydrodynamic representation where the bispinor fields is described by the evolution of four mass densities subject to the non-local interaction of the theory-defined quantum potential [20][21][22][23][24] whose energy, following the basic principle of the general relativity, contributes to the definition of the space-time curvature. The gravity generated by the bi-spinor fields  is derived by using the minimum action principle applied to the Lagrangean form of the quantum hydrodynamic motion equations [21]. Thence, the perturbative approach to the gravity-DFWE coupled system of equations is developed in curved space-time. In section five, the expectation value of the cosmological constant generated by the quantum Dirac field, is derived for flat and near-Minkowskian space-time. Finally, in section six, some aspects of the gravity of the Dirac field are discussed: i. The Dirac field gravity and the cosmological constant in the pure quantum gravity; ii.
Analogies with the Brans-Dicke model; iii.
Quantum mechanical gravity and the foundations of the quantum theory; iv.
Experimental tests.

The hydrodynamic form of the DFWE in the Minkowskian space-time
In reference [21] the Dirac equation , substituting (2.0.5) in (2.0.6) it follows that where F  is the electromagnetic (EM) tensor [22]. Furthermore, by using the hydrodynamic notation and equating the real and imaginary part of equation (2.0.8), both the quantum hydrodynamic Hamilton-Jacobi equation (HHJE) of motion and the current conservation equation [21] of the spinors i   follow.
The HHJE reads (see ref. [ , where i=1,2 is the spinor italic index and where the quantum potential reads that by using (2.0.5-6), leads to [21] the known relation the Hamilton-Jacobi equation (2.0.10), as a function of the spinors components, for the k-th eigenstate reads Moreover, by posing and by using the notation where the superscript ( )  stands for positive and negative solutions of (2.1.5), respectively, for the k eigenstate it follows that is given by (2.0.11) and where the Lagrangean for negative-energy states reads Moreover, it is useful to recast (2.1.7, 2.1.9) as follows, respectively, where for the k -eigenstates Thence, by using (2.2.5, 2.2.9) it follows that Moreover, given that, for the k  -eigenstates and that the hydrodynamic Lagrangean function of th k  -eigenstate  (2.2.13) and that, for the eigenstates, the Lagrangean hydrodynamic motion equations read leading to the motion equation of spinors It must be observed that, in principle, the solutions of (2.2.16) have to be submitted to the irrotational condition [22]) and to the current conservation condition (2.0.12).
On the other hand, as shown in [23] the stationary states (i.e., the eigenstates), of (2.2.16) obey to the current conservation (2.0.12) and are irrotational solutions of the DFWE. Equation (2.2.16) (whose non-relativistic limit (i.e., the Pauli equation) is given in ref. [22]) is not solved here since we are not interested in the quantum hydrodynamic description but in deriving the Lagrangean quantum hydrodynamic equation of motion from a minimum action principle. Moreover, generally speaking, by using the identities (2.2.5, 2.2.9) the motion equation where, in the following, it is useful to pose

The hydrodynamic energy-impulse tensor of the Dirac field
In this section we derive the expression of the hydrodynamic energy-impulse tensor as a function of the Dirac field.
Given the EITD that it is useful to recast in the form with the help of the identities (see (2.1.1, 2.1.7, 2.2.1) and the following one it is possible to obtain the expression of the EITD (2.3.1) as a function of the Dirac field, independently by the hydrodynamic formalism, as follows It is useful to note that expression (2.3.6) excludes the hydrodynamic solutions that do not satisfy the irrotational condition. Moreover, since for So far, we have not introduced the explicit form of the Minkowskian DFWE field since we are interested in covariantly generalizing the theory to the curvilinear space-time

The macroscopic limit
Since in the macroscopic classical limit (i.e., when the action values are much bigger than the Planck constant and when the physical length is much bigger than the De Broglie length [24]) the superposition of states (2.2.1) undergo collapse to an eigenstate (due to the quantum decoherence produced by fluctuations [23][24][25][26]), the macroscopic GE of the general relativity cannot be obtained just by applying the limiting condition where the decoherence effect on the quantum state reads leading to where k  is the eigenstates to which the wave-function collapses and where is the solution of the following equation Moreover, by introducing (2.4.6-7) into (2.4.2) the reversibility of the two limits follows.
The detailed stochastic hydrodynamic derivation of the macroscopic limit (2.4.1) is given in ref. [24][25][26], but it is suffice to say here that the quantum superposition of state can be maintained beyond the De Broglie length in the case of strong interactions [23][24][25][26] such as in the strong gravitational field of a black hole. Therefore, we assume the existence of the macroscopic limit in the curved space-time taking care that the macroscopic scale may go much beyond the De Broglie length in very high curvature space-time.

The minimum action in the hydrodynamic formalism
Since the hydrodynamic Lagrangean depends also by the quantum potential and hence by , the problem of defining the equation of motion can be generally carried out by using the set of variables : Thence, the variation of the hydrodynamic action dVdt      (between the fixed starting and ending points, Given that the quantum motion equations for eigenstates satisfy the condition the variation of the action that is not null for the dependence of the hydrodynamic Lagrangean Thence, for the quantum hydrodynamic evolution the variational principle reads that in the classical limit (i.e., for leads to the classical extremal principle Moreover, generally speaking, by using (2.2.17), for the general superposition of state (2.2.1) the variation of the action reads is due to the mixing of the quantum superposition of states and where Thence, by from (2.5.8), the minimum action principle can be generalized to that, for the decoherence (decay to an eigenstate) at the macroscopic scale leads to that converges to the extremal classical principle.

The hydrodynamic DFWE in the curved space-time
In order to derive the gravitational equation for the Dirac field, we assume that the hydrodynamic representation of the Dirac bi-spinor field, depicted by four mass densities moving with under the action of the quantum potential (2.0.11), has its own gravitational effect by producing the curvature of the space-time that makes stationary the overall action comprehending the gravitational field [19]). This assumption is supported the by the fact the hydrodynamic approach leads to the derivation of the Einstein equation with the correct electromagnetic energy-impulse tensor when applied to the boson field of the photon [19] (see appendix A).
Also the experimental demonstration of the physical existence of the quantum potential energy by the Bohm-Aharonov effect supports this approach.
One possible generalization of the Dirac equation to the curved space-time can be obtained by assuming the physics covariance of the equation. In reference [27] it has been shown that the physics covariance condition plays the same role as the inertial to gravitational mass equivalence postulate in the classical general relativity.
The covariant Dirac equation in curved space-time, respect affine and spin connections, reads [28] 0 4 ab ab and , and b f  , are the vielbein and the inverse vielbein, respectively.
By assuming the covariant derivative for affine and spinor connections leading, by following the procedure in section 2.0, to and of equation ( (2.6.14) As far as it concerns the current (2.0.12) the covariant form reads that by using (2.6.7-8), leads to † † † † in agreement with the standard results for the DFWE that show that the connection acting on the current J  is the Levi-Civita connection where the condition for its conservation reads is the co-variant derivative. For a direct check, equation (2.6.18) can be derived by equating the imaginary part of equation (2.6.11) (the demonstration is not given here).

7. The hydrodynamic Hamilton-Jacobi equation for eigenstates in curved space-time
By using (2.6.13) for the k-th eigenstates it follows that As far as it concerns the EITD (2.7.9) Finally, by using the identities (2.7.1) the identity (2.7.7) can be recast as follows that, as a function of the DFW field, reads represents the eigenstates of the DFWE in curved space-time that for positive (2.7.14)

The electromagnetic field
Since we are considering fermions that bring the electric charge, the overall Einstein equation also contains the coupling with electromagnetic field ) . Moreover, since the equations for the EM field are the same both in classical and quantum case, so that the hydrodynamic description coincides with the classical one (i.e., the quantum potential is null), it follows that EITD for the EM field f T  equals the classical form and reads [19,29] (see Appendix A).

The minimum action in curved space-time and the gravity equation for the DFWE
In this section we derive the gravity equation, by applying the minimum action principle to the quantum hydrodynamic evolution associated to the fields Given that the quantum hydrodynamic equations in the Minkowskian space-time, satisfies the minimum action principle (2.5.11), when we consider the covariant formulation in the curved space-time, such variation takes a contribution from the variability of the metric tensor. When we consider the gravity and we assume that the geometry of the space-time is that one which makes null the overall variation of the action (2.5.11), we have a condition that leads to the definition of the GE. By considering the variation of the action due both to the curvilinear coordinates and to the functional dependence by i   [19], it follows that and, from (2.5.11), If we postulate that the variation of action of the gravitational field [30] offsets that one produced both by the DFWE field, D  , and by the EM field [30] 1 em em we obtain the gravitational equation for the DFWE field where by posing it follows that and that  (3.20) for the infinitesimal transformation of coordinates (3.21) in the Minkowskian case (3.5) leads to   (3.22) and thence, for the arbitrariness of   , to Moreover, since in the decoherent macroscopic Minkowskian limit it holds: Furthermore, given that in the Minkowskian decoherent macroscopic limit it holds (see (C.11) in Appendix C)     (3.27) and, by defining  such as (3.28) finally, that where the classical CC  is independent by k.

The GE for the DFWE eigenstates
By using (2.7.5-6, 3.8-9) for the k eigenstate, it follows that Equation ( where from (2.26-27) it has been used the identity

The GE of the DFWE field
Finally, in the matrix form, the GE can be written as a function of the general DFWE field (2.
Actually, the hydrodynamic approach has been used as a "Trojan horse" to find the GE (3.

The GE-DFWE-EM system of quantum evolution
Once the system of evolutionary equations for the charged classical fermion field is defined in curved space-time where the cosmological gravity tensor (CGT) From the general point of view, we observe that the quantization of the Dirac field in curved spacetime, with the metric defined by the GE (4.1), owns the following properties: 1. In the limit of flat space-time, the standard quantum electrodynamics (QED) is recovered (see section 5.).
2. Since all ordinary elementary particles (low energy vacuum states) own by far mass densities very much smaller than the Planck mass in a sphere of Planck length, the first order of approximation of the gravitational evolutionary system of equations can lead to precise results by using the perturbative approach.
3. For fermions in very high curved space-time or of Planckian mass, the superposition of the Dirac fields eigenstates is quite different by the Fourier superposition of the Minkowskian case and the full treatment (e.g., high energy QFT) strongly diverges by the traditional results.

Cosmological tensor density of the quantum Dirac field
In the classical treatment, except for the photon, the Einstein equation is not coupled to the particles fields, but just to the energy impulse tensor of classical bodies. The GE (4.1) is analytically coupled to the classical fermion field and it takes into account the energy of the (non-local) quantum potential for determining the geometry of the space-time. The most important effect of the quantum potential is the generation of the CGT.
If we put to zero the quantum potential in the decoherent GE (3.1.7) (as it happens in the classical limit (i.e., Thence, the gravitational effect of the quantum potential leads to a not null CGT that allows the quantum vacuum to be in the stable physical phase [9] even if the classical cosmological constant is null. Given that the contribution of the quantum potential modifies both the classical metric tensor of general relativity and the quantization rules (see Appendix E), it also affects the field quantization. In this section, we will derive the CPTD of the quantum fermion field in in the quasi Minkowskian space-time in order to evaluate its mean value on cosmological scale.
Since the EM field owns a null CGT (the classical theory coincides with the quantum hydrodynamic description (i.e., 0 qu V  for the photon) and the gravitational coupling is the same of the classical general relativity), for sake of simplicity we derive the CPTD by considering the case of uncharged fermions in the quasi-Minkowskian limit whose Fourier decomposition [31] : : : : : : : (5.18) it follows that : In appendix F it is shown that the Newtonian gravity leads to a null gravitational perturbation to the quantum Dirac field. This fact agrees with the experimental outputs of the Minkowskian QED that leads to the prediction of the anomalous magnetic moment with six significant digits [31]. The gravitational corrections coming from higher order terms can be a test for the theory by predicting the anomalous magnetic moment with higher precision. Moreover, (5.1.4) may give an important contribution in very heavy fermion particles.

Discussion
The hydrodynamic representation of the DFWE field makes it equivalent to mass distributions submitted to the non-local quantum potential (2.6.14) obeying to the GE (4.1). In agreement with the basic principle of the general relativity, the assumption of the theory is that the energy of the quantum potential (expression of the quantum-mechanical properties of the vacuum) contributes to the curvature of the space-time.
On the basis of this postulate, the quantum-mechanical non-local effects (e.g., the uncertainty principle) come into the gravity leading to the theoretical appearance of the CGT in the GE and to the removal of the point singularity of black holes in general relativity [32]. In the classical treatment, the Einstein equation for massive particles is not coupled to any matter field, but just to the energy impulse tensor of the classical bodies and does not contain any information about how it couples with their quantum-mechanical fields (i.e., their phases). On the other hand, the GE-DFWE-EM (4.1-3) system owns a complete gravitational coupling and can be further quantized leading to a quantum gravity based on the description of the physical vacuum that includes its quantum-mechanical properties (e.g the quantum non-local potential that generates repulsion against the mass density concentration in smaller and smaller volume).

The GE and the quantum gravity
Even if the quantization of the DFWE field in the curved ST is not treated in this work, the inspection of some features of the quantum gravity to the light of the GE (4.1) deserves a mention. Since the action of the GE (4.1) is basically given by the standard Einstein-Hilbert action plus terms stemming by the energy of the non-local quantum potential of massive DFWE, the outputs of the quantum "pure gravity" practically remains valid for the GE presented in this work. As shown in [9], one interesting aspect of the quantum pure gravity is that the vacuum does not make a transition to the collapsed branched polymer phase, if and only if there exists a vanishing small cosmological constant (i.e., 0   ).
Therefore the presence of the term Q  in the GE (4.1), in principle, allows to pose 0   (as strongly supported by Einstein [6]) with the GE that reads.  it follows that the matter itself stabilizes the vacuum in the physical strong gravity phase [9]. On the other hand, a perfect Minkowskian vacuum (i.e., without matter) will make transition to the non-physical collapsed branched polymer phase with no sensible continuum limit [9], leading to nospace and no-time as we know.

Analogy with the Brans-Dicke gravity
The output the work highlights an interesting analogy with the Brans-Dicke [13][14][15] gravity that solves the problem of the cosmological constant [16] as well as those of the inflation [17] and dark energy [18].
that is the summation over the spinor terms, each one owing the form of the scalar Brans-Dicke gravity [13] in absence of external potentials (for sake of simplicity we have considered the case of uncharged fermions).
If is worth noting that the DWFE gravity, endorses the concept of the effective gravitational constant of the Brans-Dicke model that in the spinor-tensor form (6.1.2) leads to effective gravitational constants for each spinor

Quantum-mechanical gravity and the foundations of the quantum theory
The output (2.5.12) shows that in order to have the macroscopic behaviour in addition to the smallness of the Planck constants, the decoherence process induced by uncorrelated fluctuations is necessary: The decoherence process of the quantum states is necessary to obtain the classical extremal principle and the axiomatic description of the macroscopic classical physics.
In the quantum hydrodynamic model (Madelung -De Broglie-Bohm pilot wave approach) the passage to the classic equation of motion is obtained just by the condition of assuming the Planck's constant vanishing compared to the action scales of the problem [22]. Actually, in the quantum hydrodynamic model, the decoherence remains undefined since it shows itself in the open-system stochastic generalization of quantum systems [22][23][24][25][26]. The decoherence process does not appear in the quantum hydrodynamic theory since fluctuations are absent. Actually, in the classical limit for 0   there is the change of the "mathematical nature" of the quantum hydrodynamic motion equations, produced by the disappearance of the quantum potential that generates the eigenstates and the coherent evolution of their superposition [22][23][24].
In presence of a noisy environment (when the quantum potential energy is much smaller than the energy amplitude of fluctuations) the coherent superposition of states cannot be maintained in time and only the eigenstates configurations are stable [24,33] The approach based on the minimum action condition (2.5.11) shows that the Madelung-De Broglie-Bohm pilot wave approach needs to be completed by the existence of the decoherence process

Experimental tests
The result (5.23) basically shows that in the Minkowskian space-time (i.e., the vacuum very far from particles) the CPTD expectation value is vanishing regardless its zero-point energy density. The contribution to the CPTD appears at second order in the perturbative development of the GE. The macroscopic CPTD expectation value is vanishing in the region of space-time with (Newtonian) weak gravity and it increases at higher gravity as quantum-mechanical corrections. In fact, by observing that, due to the high density of mass distribution in black holes (e.g., qu i V  acquires a value comparable to the mass energy 2 mc (e.g., ) i( k a    [32]) it turns out that the largest contribution to the measured cosmological constant of the universe comes from the high-gravity regions of space (e.g., black holes, neutron stars and super massive BHs (SMBHs) at the center of the galaxies). On the other hand, from the ordinary baryonic mass, a very small bit of contribution to the CPTD may come from the space very close to the center of ordinary elementary particles, where the gravity reaches the largest value. Thence, the quantum-mechanical properties of the vacuum lead to a great lowering of the expectation value of the CPTD on cosmological scale with an order of magnitude that can be in agreement with the astronomical observations [20]. From the experimental point of view, the quantum-mechanical corrections to the Newtonian gravity, correspondent gravity equation by using the minimum action principle. The gravity equation associated to the DFWE field takes into account the gravitational effects of the energy of the nonlocal theory-defined quantum potential.
The self-generation of the CPTD Q  leads to the attractive hypothesis that the matter itself generates the physical stable vacuum phase in which it is embedded. The GE of the fermion field shows that the measured cosmological constant is the effect of the cosmological pressure tensor density that emerges as 2 nd order quantum-mechanical correction to the Newton gravity.
The macroscopic CPTD ( k ) Q  it is not null if, and only if, massive particles are present but it tends to zero in the flat vacuum so that its spatial mean, on the cosmological scale, have a much smaller order of magnitude than that one deriving tout court by accounting the zero-point vacuum energy density, and possibly agrees with the measured cosmological constant .
The GE of the classical DFWE field shows that the CPTD ( k ) Q  , as well as the quantum CGT, are 2 nd order macroscopic "quantum-mechanical" gravitational effects that become more and more relevant higher and higher the space-time curvature. This output shows that the gravity near super massive black holes at the center of the galaxies (e.g., in the bulge), between galaxies on cosmological scale and between twin black holes can be deeply influenced by the quantummechanical effects of the gravity. The spinor-tensor gravitational coupling of the fermion field owns (for each spinor) terms typical of the Brans-Dicke modified gravity in absence of external potentials. The gravitational corrections to the quantum fermion field can be tested by predicting the anomalous magnetic moment with higher precision and in determining the states of very heavy fermions in high-energy QED.
and, by dividing each component by