The Gravity of the Classical Klein-Gordon field

The work shows that the evolution of the field of the free Klein-Gordon equation (KGE), in the hydrodynamic representation, can be represented by the motion of a mass density subject to the Bohm-type quantum potential, whose equation can be derived by a minimum action principle. Once the quantum hydrodynamic motion equations have been covariantly extended to the curved space-time, the gravity equation (GE), determining the geometry of the space-time, is obtained by minimizing the overall action comprehending the gravitational field. The derived Einstein-like gravity for the KGE field shows an energy-impulse tensor density (EITD) that is a function of the field with the spontaneous emergence of the cosmological pressure tensor density (CPTD) that in the classical limit leads to the cosmological constant(CC). The energy-impulse tensor of the theory shows analogies with the modified Brans-Dick gravity with an effective gravity constant G divided by the field squared. Even if the classical cosmological constant is set to zero, the model shows the emergence of a theory-derived quantum CPTD that, in principle, allows to have a stable quantum vacuum (out of the collapsed branched polymer phase) without postulating a non-zero classical CC. In the classical macroscopic limit, the gravity equation of the KGE field leads to the Einstein equation. Moreover, if the boson field of the photon is considered, the EITD correctly leads to its electromagnetic energy-impulse tensor density. The outputs of the theory show that the expectation value of the CPTD is independent by the zero-point vacuum energy density and that it tends to zero as the space-time approaches to the flat vacuum, leading to an overall cosmological effect on the motion of the galaxies that may possibly be compatible with the astronomical observations.


Introduction
One of the serious problems the nowadays gravity physics [1] refers to the connection between the quantum fields theory (QFT) and the GE. The problem has come to a partial solution in the semi-classical approximation where the energy-impulse tensor density is substituted by its expectation value [2][3][4][5][6].
Even if unable to give answers in a fully quantum regime, the semiclassical approximation has brought to successful results such as the explanation of the Hawking radiation and BH evaporation [7].
The difficulties about the integration of QFT and the GE become really evident in the so called cosmological constant problem, a term that Einstein added to its equation to give stability to the solution of universe evolution that in the general relativity would lead to its final collapse. The introduction, by hand, of the cosmological constant was then refused by Einstein himself that defined it as the biggest mistake of my life [8]. Actually, the CC has been introduced [1] to explain the astronomical observations about the motion of galaxies [9] and to give stability to the quantum vacuum bringing it out from the unphysical collapsed branched polymer phase [10], being the physical vacuum of the strong gravity phase related to a positive, not null, CC [11]. Moreover, the EITD for classical bodies in the GE owns a point-dependence by the mass density (i.e., 2 | |   ), without any analytical complete connection with the field of matter As discussed by Thiemann [12], this connection cannot be build up by simply replacing the EITD by its Minkowskian vacuum expectation value. If we do so, we end with a non-Minkowskian metric tensor solution that has to feed back into the vacuum expectation value and so on with the iteration that does not converge in general. As a consequence of this fault, modifications to the Einstein equation have been proposed both from theoretical point of view, such as the Brans-Dicke modified gravity [13], and by using a semi-empirical approach such as the covariant running G or the slip function QFT [14]; However, the cosmological constant itself in the quantum pure gravity can be considered a modification of the general relativity for defining a GE compatible with the needs of a quantum theory.
Due to the undefined connection between the GE and the particle fields, the integration between the QFT and the GE is still an open question that is object of intense theoretical investigation. Generally speaking, the link between the matter fields and the GE can be obtained by: 1. Defining an adequate GE for matter fields (as, for instance, happens for the photon field); 2. Defining the link between the GE and the QFT by quantizing the action of the new GE. At glance with the first point, the paper shows that it is possible to obtain the GE with analytical connection with the KGE field subject to the non-local quantum potential interaction) is utilized. Then, by using the minimum action principle applied to the hydrodynamic model, the gravity generated by the KGE field  is derived.
The paper is organized as follows: In section 2 the Lagrangean version of the hydrodynamic KGE is developed; In section 3 the gravity equation is derived by the minimum action principle; In section 4, the perturbative approach to the GE-KGE system of evolutionary equations is derived; In section 5, the expectation value of the cosmological constant of the quantum KGE massive field is calculated; In section 6, some features of the GE as well as the check of the theory are discussed.

The hydrodynamic representation of the Klein Gordon equation
In this section we derive the hydrodynamic representation of the Klein-Gordon equation (KGE) in the form of Lagrangean equations that allow to define the minimum action principle for the hydrodynamic formalism.

The Lagrangean form of the KGE
Equation (2.02) in the low velocity limit leads to the Madelung quantum hydrodynamic analogy of Schrodinger equation [19] that in the classical limit (i.e., 0   ) leads to the classical Lagrangean equation of motion [20]. For the purpose of this work, we generalize the Lagrangean formulation to the hydrodynamic KGEs (2.02-3).
Since in curved space-time under the gravitational field we may have discrete energy values, in the following we use the discrete formalism (i.e., ) that is also useful in the numerical approach.
For the generic superposition of eigenstates  (2.1.4) and where j, k run from k min and k max ; (2.1.8) Thence, by using the identities ( k ) Moreover, given that, for stationary states of time-independent systems (i.e., eigenstates) [19] the hydrodynamic Lagrangean function (2.1.5) does not explicitly depend by time, it holds so that, for the eigenstates, the system of Lagrangean hydrodynamic motion equations read where, by using (2.0.13), it follows that and that (2.1.20) For sake of accuracy, it must be observed that the solutions of (2.1.20) have to be submitted to quantization (given by the irrotational property [17,19]) and to the current conservation condition (2.0.3) As shown in [19] the stationary states of (2. that making the summation over k leads to (2.1.22) and to

The hydrodynamic energy-impulse tensor
By using the hydrodynamic energy-impulse tensor (EIT) leading to the equation of motion where we pose For the generic quantum state (2.1.1), the energy-impulse tensor reads it can be recast as k min   and k max   . Nevertheless, it is useful to derive such Minkowskian expressions that, by In the quantum case, due to the force generated by the quantum potential (e.g., responsible of the ballistic expansion of an isolated Gaussian packet) for the gradient of the EITD, it follows that Finally, it is worth noting that, since in the classical limit (whose resolution length is much bigger than the De Broglie length) due to the quantum decoherence [21] (produced by fluctuations) the superposition of states (2.1.1) undergo collapse to an eigenstate, the classical macroscopic limit is obtained by the limiting procedure where the subscript "dec" stands for decoherence and where

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 : Given that the quantum motion equations for eigenstates (2.1.14-15) satisfy the condition (that explicitly defines ( q ,t )  ) the variation of the action  S for the k-th eigenstates reads that is not null since it takes contribution from the quantum potential contained into the hydrodynamic Thence, for the quantum hydrodynamic evolution it follows that that, since in the classical limit, for is recovered.
Moreover, generally speaking, by using (2.1.21), for the general superposition of state (2.1.1) the variation of the action reads is due to the quantum mixing of superposition of states. Thence, by using (2.2.3), the condition (2.3.8) can be generalized to in the classical limit leads to

The hydrodynamic KGE in the curvilinear space-time
As far as it concerns the motion equations (2.1. [8][9], there is no way to univocally define them in the non-Minkowskian space-time without a postulate that fixes the criterion of generalization.
In the classical general relativity this criterion is given by the equivalence of inertial and gravitational mass that, by fact, is equivalent to postulate that the classical equation of motion is covariant in general relativity [19] .
Analogously, assuming the covariance of the KGE [19] that reads also the hydrodynamic motion equations (2.0.2-3) own a covariant form [19] and they read where the quantum potential reads where  g is the metric tensor and where Moreover, given the covariance of (2.4.2-3) also the motion equation ( are covariant.
Once the quantum equations are defined in the non-Minkowskian space-time, their meaning is fully determined when the metric of the space-time is defined by the gravity equation (GE) based on additional condition (e.g., on the hydrodynamic action covariantly generalized). Moreover, it is useful to note that, due to the biunique relation between the quantum hydrodynamic equations ( where the last identity has been obtained by inverting (2.0.13).
It is worth mentioning that the KGE-GE system of evolutionary equations owns the advantage to contain only the irrotational states that satisfy the current conservation condition.

The minimum action in curved space-time and the gravity equation for the hydrodynamic KGE
In order to follow an analytical procedure we derive the gravity equation, by applying the minimum action principle to the quantum hydrodynamic matter evolution associated to the field  .
Given that the quantum hydrodynamic equations in the Minkowskian space-time [19], satisfies the minimum action principle (2.3.11), 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 (2.3.11), we define the condition that leads to the definition of the GE.
By considering the variation of the action due to the curvilinear coordinates [22] and the functional dependence by  , it follows that If we postulate that the variation of action of the gravitational field offsets that one produced by the KGE field so that we obtain the gravitational equation Moreover, given that [22] (3.20) Moreover, by posing is recovered in the classical limit

The GE for the KGE eigenstates
Moreover, by using the identity

The GE of the general KGE field
Finally, by using (3.6-3.10) the GE as a function of the general KGE field (2. where both . Actually, the hydrodynamic approach has been used as a "Trojan horse" to find the GE where the nonphysical states are implicitly excluded by writing it as a function of the KGE field (i.e., we do not need to impose the irrotational condition). Finally, it must be noted that equation (3.1.5) represents the decoherent limit of (3.2.1).

Perturbative approach to the GE-KGE system
For particles very far from the Planckian mass density by a perturbative iteration  is the solution of the zero order KGE, '  the solution of the first order KGE where the Christoffel symbol reads [22]   (4.10) leading to [22]   Moreover, by using the zero-order relations  (4.13) and, being (4.16) By making the macroscopic limit of (4.16) (with    ) we obtain from which we can readily see that the weak gravity limit, on macroscopic scale at the first-order, leads to the Newtonian potential of gravity. The first contribution to the cosmological constant comes from the second order of approximation    and the EITD is calculated by using '  obtained by (4.4), where where, at first order, and where If the macroscopic GE (4.17) and the Einstein equation of the general relativity coincide themselves at first order and the Newtonian gravity is purely classic, the second order GE (4.20) contains contributions (among those the cosmological isotropic pressure  actually is the macroscopic quantum-mechanical contribution to the Newtonian gravity.
It is worth mentioning that the macroscopic GE shows the additional contribution It is noteworthy that the EIT stress component stress 


, that leads to a non-zero slip function [23][24], is specific of the (microscopic) quantum-coherent curved space-time but it decays to

The CPTD expectation-value of the quantum KGE field
Generally speaking, when the KGE field is quantized, the EITD on the right side of the GE becomes a quantum operator and thence also the Ricci's tensor (as well as the metric tensor) of the GE, on the left side, become quantum operators. As it can be easily shown for the pure gravity [25], the commutating rules for the KGE field quantization fixes the commuting relations for the metric tensor.
At zero order, the GE equation leads to a Minkowskian KGE field and, hence, when the KGE field is quantized, the standard QFT outputs are obtained.
If at zero order the GE is decoupled by the field of the massive KGE, at higher order it is not.
The quantization of the GE-KGE system of equations is not the goal of this work, nevertheless, it is interesting to evaluate the CPTD expectation value of the vacuum in order to evaluate if it can lead to the lowering of the theoretical value of the CC on cosmological scale and can help to solve the problem of the disagreement of the QFT with the experimental observations.
In order to evaluate the macroscopic cosmological constant of the quantum KGE field (i.e., at the zero order Minkowskian limit of the GE-KGE system of equations for the ordinary QFT) we need to calculate the expectation value To this end, we need to express the quantum potential as a function of the annihilation and creation operators ( k ) can be established with notation in (2.
it follows that the matter itself stabilizes the vacuum in the physical strong gravity phase [10] as we perceive it.
On the other hand, under this hypothesis, a perfect Minkowskian vacuum (i.e., without matter) will make transition to the unphysical collapsed branched polymer phase with no sensible continuum limit [10][11], leading to no-space and no-time as we experience.

Check of the hydrodynamic GE
If in the general relativity the energy-impulse tensor density for classical bodies [30] is defined only with a point-dependence by the mass density, for the electromagnetic (EM) field, the EITD is defined as a function of the EM field itself [22].
On this basis, since the photon is a boson (obeying the a KGE), we can have a direct check of the theory by comparing the known EITD EM expression with the EITD (3.6) for a boson field given by the quantum hydrodynamic gravity. In fact, given the plane wave of the vector potential, in the Minkowskian case, for the photon (e.g. linearly polarized)   (4.20) must be primarily detectable in the motion of stars around the big black holes at the center of the galaxies., in the rotation of twin neutron stars and in the inter-galactic interaction .. Finally, it is worth mentioning that the interferometric detection of the gravitational waves represents an experimental technique whose angular and frequency-dependent response functions can discriminate among the existing theories of gravity [32].

Conclusion
The quantum hydrodynamic representation of the Klein-Gordon equation, describing the evolution of the mass density 2 | |  owing the hydrodynamic moment S p      and subject to the quantum potential, has been used to derive the correspondent gravity equation, defining the geometry of the space-time, by using the minimum action principle. The gravity equation associated to the KGE field takes into account the gravitational effects of the energy of the non-local quantum potential. The hydrodynamic approach owns three main properties: 1. The energy-impulse tensor of the GE is written as a function of the KGE field; 2. In the classical limit, the GE leads to the Einstein one; 3. If we apply the EITD of the GE to the photon field (that is a boson described by the KGE) we obtain the EITD of the EM theory; The self-generation of the CPTD   Q Tr  leads to the attractive hypothesis that the matter itself generates the physical stable vacuum phase in which is embedded.
The paper shows that the macroscopic CPTD The depletion of the CPTD in the vacuum, far from material bodies, lowers its mean value on cosmological scale so that it can possibly agree with the astronomical observations on the motion of the galaxies.
The GE of the classical KGE field shows that the CPTD ( k ) Q   and other out-diagonal components of the EITD can be considered as "decoherent quantum-mechanical" gravitational effects generated in highly curved space-time near dense matter such as black holes and neutron stars. The hydrodynamic gravity model defines a coupling between the boson field of the free KGE and the GE in a form that, at some extend, mimics the Brans-Dicke gravity leading to an effective gravitational constant inversely proportional to the field squared.

Equivalence between the field and the hydrodynamic solutions of a quantum equation
In addition to the field solution of a quantum equation, it is possible to express it in the hydrodynamic form. This approach was firstly proposed by Madelung [13] and confirmed by the Aharonov-Bohm effect [12]. In order to obtain the hydrodynamic form of a quantum equation, for the complex field (A.0.1), it is split into two equations regarding its real and imaginary part, leading to two differential equations as a function of With the help of the so called quantum pseudo potential [11][12][13][14], the hydrodynamic approach describes the evolution of the particles density and subject to the quantum potential [11][12][13][14]. Actually, the hydrodynamic system of equations broaden the solutions of the starting quantum equation since not all momenta  p , solutions of the hydrodynamic description, are solutions of the field equation [11]. This because not all momenta deriving from hydrodynamic equations can be obtained as the gradient of a function representing the action S of the exponential argument of the field. The restriction of the hydrodynamic momenta to those ones of the quantum problem derives by imposing the existence of the action S. The integrability of the impulse requires [11]  are possible, so that the action results quantized. With the "irrotational" condition the hydrodynamic approach becomes equivalent to the field one. Even this approach is fully quantum [11][12][13][14] and provides outputs that completely overlap those ones of the standard quantum treatment [28], its use is curtailed to the semiclassical approximation and to one-particle problem.

Implementation of quantization in the hydrodynamic equations
The quantization condition, that reduces the hydrodynamic solutions to those of the field of the quantum equation, as formulated by (A.0.4) is external to the hydrodynamic quantum equations (HQE). Actually, under inspection, it is implicitly included into the HQE through the quantum potential. In fact, in the hydrodynamic description, the eigenstates are identified by their property of stationarity that is given by the "equilibrium" condition that (for unidimensional systems) reads This equilibrium, in bounded systems, happens when the force generated by the quantum potential exactly counterbalances that one of the Hamiltonian potential (see, further on, relation (A.1.10)). For 3-D systems, the equilibrium condition also refers to the rotational "equilibrium "(i.e., angular momentum conservation) in a way that the overall "torque" due both to the Hamiltonian and to the quantum potential is null. The initial condition Moreover, since the quantum potential is not fixed but changes with the state of the system, more than one stationary state (each one with its own n qu V ) may exist. In this case, we have multiple quantized action Thence, given that the stationary states, generated by the quantum potential, are the field eigenstates, the quantization is implicitly-contained in the quantum hydrodynamic equations.

A.1.Simple examples
In this section, in order to elucidate how the quantum potential defines the eigenstates, some simple cases are reported. The skilled reader may skip these paragraphs.
Here we consider the low velocity limit of the KGE submitted to the potential ( q ) V at first order in | q | c  (i.e., the Schrödinger equation) for which it holds