The Non-Euclidean Hydrodynamic Klein-Gordon Equation with Perturbative Self-Interacting Field

In this paper the quantum hydrodynamic approach for the KGE owning a perturbative self-interaction term is developed. The generalized model to non-Euclidean space-time allows to determine the quantum energy impulse tensor density of mesons for the gravitational equation of quantum mechanical systems.

Moreover, given that the uncertainty principle forbids a black hole (BH) collapse [39][40][41][42][43][44] and that the quantization of a black hole requires the existence of a fundamental state with a minimum mass, the BH cannot have a mass smaller than that of Planck and comparable with those of elementary particles [45].
If we try to define the QG equation from the minimum action principle (MAP), the quantizing action is not self-contained under renormalization, since loop diagrams generate terms not present in the initial expression [55][56][57][58][59].
In the present work the author makes use of the hydrodynamic quantum description (HQD) [60][61][62][63][64][65][66][67][68] that has the advantage of being embedded in a classical-like formalism, where the quantization is implicitly contained into the equation of evolution [68], for deriving a gravity equation for quantum mechanical systems.
The quantum hydrodynamic approach, firstly developed by Madelung [62], describes the evolution of the complex wavefunction as a function of the two real variables, | | and S [60,62,63,[69][70][71][72][73]. The model gives rise to classical-like analogy describing the motion of particles' density As shown by Weiner et al. [71], the outputs of the quantum hydrodynamic model agree with the outputs of the Schrödinger problem, but not only for the semi-classical limit or for a single particle [62,63,69]. More recently, Koide and Kodama [72], showed that it agrees with the outputs of the stochastic variational method.
Equation (2) can be expressed by the following system of Lagrangean equations of motion that for the eigenstates read where the Lagrangean for positive and negative-energy states n L  and n L  , respectively reads from where it follows that (where the minus sign stands for antiparticles) and, by using (17), that Following the hydrodynamic protocol [68], the eigenstates are represented by the stationary solutions of the hydrodynamic equations of motion obtained by deriving (14) and then inserting it into (15) that leads to and to where for eigenstates, the quantum energy-impulse tensor (QEIT) T n   reads [45,68] leading to the quantum energy impulse tensor density (QEITD) [45,68] 2 2 nn n n nn n n n n the QEITD (24) can be written as a function of the wave function as following  

Charged Boson[M1]
In the case of a charged boson, Equations (1)-(3) read, respectively, Moreover, analogously to (9) and (17)- (19), from (28) it follows that that leads to and to that by using (25), as a function of  and  A , reads For 0   the Lagrangean (37) acquires the known classical expression Moreover, with the help of (25), (30), (33)-(35) it follows that that by using (25), (30) and (35) we can be expressed as a function of the wave function as The above equations are coupled to the Maxwell equation: where [75]     , , and where is the potential 4-vector.

Non-Euclidean Space-Time
The quartic self-interaction is introduced in the KGE in order to describe the states of charged ( 1  ) bosons (e.g., mesons) [76]. The importance of having the hydrodynamic description of bosons [68] lies in the fact that it becomes possible to derive its quantum energy-impulse tensor that can couple them to the Einstein gravitational equation for quantum mechanical systems [68]. By using the General Physics Covariance postulate [3,68], it is possible to derive the non-Euclidean expression of the hydrodynamic model of the KGE: Equations (2) and (3) in a non-Euclidean space-time read, respectively, Moreover, by using the Lagrangean function n n n dS L g p q dt the covariant form of the motion Equations (14) and (15) reads where t D p p p q is the total covariant derivative respect the time and where    are the Christoffel symbols.
Equations (47) and (48) leads to the motion equation where n L reads From (50) where the stationary condition 0 du dt   , that determines the balance between the "force" of gravity and that one of the quantum potential, defines the stationary equation for the eigenstates  Given the classical-like hydrodynamic model of quantum mechanics, the energy-impulse tensor for the Einstein gravitational equation of quantum mechanical systems is straightforwardly derived. The availability of such a description for mesons facilitates the coupling of them to the GE.