The Hydrodynamic Representation of the Klein-Gordon Equation with Self-Interacting Field

The interest in obtaining such a description lies in the fact that such type of KGE can describe the states of bosons, such as mesons. The goal of the paper is to obtain the energy-impulse tensor density of such particles that can be useful in the coupling the field of a meson with the Einstein equation [11]. The paper is organized as follows: in the section 2 the hydrodynamic KGE with a self-interaction term is derived for an uncharged scalar particle as well as the Lagrangean motion equations for the eigenstates and the associated energy impulse tensor density.. In the subsection 2.2 the theory is developed for a charged field. In section 3. the formulas are generalized to a non-Euclidean space-time.


Introduction
Since the introduction of the quantum wave equation by Schrödinger, the quantum hydrodynamic approach (QHA) was presented by Madelung [1]. In this quantum representation, developed by Madelung and then by Bhom, the evolution of a complex variable ψ ψ =  i | | exp S is solved as a function of the two real variables, | |ψ and S [2][3][4][5]. As shown by Weiner et al. [6], the outputs of the quantum hydrodynamic model agree with the outputs of the Schrödinger problem and, more recently, as shown by Koide and Kodama [7], it agrees with the outputs of the stochastic variational method.
Recently, the author has shown that the hydrodynamic approach is strictly correlated to the properties of vacuum on small scale [8].
Moreover, as shown by Bohm and Hiley [9,10] the hydrodynamic approach can be generalized for the description of the quantum fields.
The present work develops the quantum hydrodynamic form of the Klein-Gordon equation (KGE) containing an additional self-interaction term.
The interest in obtaining such a description lies in the fact that such type of KGE can describe the states of bosons, such as mesons. The goal of the paper is to obtain the energy-impulse tensor density of such particles that can be useful in the coupling the field of a meson with the Einstein equation [11]. The paper is organized as follows: in the section 2 the hydrodynamic KGE with a self-interaction term is derived for an uncharged scalar particle as well as the Lagrangean motion equations for the eigenstates and the associated energy impulse tensor density.. In the subsection 2.2 the theory is developed for a charged field. In section 3. the formulas are generalized to a non-Euclidean space-time.

The Hydrodynamic KGE with Self-Interacting Field
In this section, the Euclidean hydrodynamic representation of the KGE is derived for a scalar uncharged particle with a self-interaction term that reads  Following the procedure given in reference [11,12] (for the ordinary KGE) the hydrodynamic equations of motion are given by the Hamilton-Jacobi type equation coupled to the current equation [ and where ( ) and where P 2 = Pi Pi is the modulus of the spatial momentum.
As shown in reference [11], given the hydrodynamic Lagrangean function equation (2) can be expressed by the following system of Lagrangean equations of motion that for the eigenstates read Generally speaking, for eigenstates, for which it holds E=E n =const it follows that: from where it follows that (where the minus sign stands for antiparticles) and, by using (17), that Following the hydrodynamic protocol [11], the eigenstates are represented by the stationary solutions of the hydrodynamic equations of motion obtained by deriving from (14) and then inserting it into (15) that leads to where, for eigenstates, the quantum energy-impulse tensor (QEIT) ν µ n T reads [11,12], leading to the quantum energy impulse tensor density (QIETD) [11,12], is the (hydrodynamic) Lagrangian density and L is the hydrodynamic Lagrangian function. Moreover, by using the identity The QIETD (23) can be written as a function of the wave function as following:

Charged field
In the case of a charged boson field, equations (1-3) read, respectively, where the 4-current ì J reads and (where µ π is the mechanical momentum) [11,13] and where Moreover, analogously to (9,(17)(18)(19), from (27) it follows that that, by using (24,29,34) we can express as a function of the wave function as The above equations are coupled to the Maxwell one is the potential 4-vector,

Non-Euclidean Generalization
The quartic self-interaction is introduced in the KGE in order to describe the states of charged ( 1 ± ) bosons (e.g., mesons) [15]. The importance of having the hydrodynamic description of bosons [11] lies in the fact that it allows to derive its quantum energy-impulse tensor that can couple them to the Einstein quantum-gravitational equation [11].
The generalization of the quantum hydrodynamic formalism to the non-Euclidean space-time can be obtained by using the General Physics Covariance postulate [11,16]. By using it, it is possible to derive the non-Euclidean expression of the hydrodynamic model of the KGE Equations (2-3) in a non-Euclidean space read, respectively, Moreover, by using the definition of the Lagrangean function where is the total covariant derivative respect the time and where µ µ q are the Christoffel symbols.
Equations (45-46) leads to the motion equation where, n L reads where the stationary condition 0 µ = du dt , that determines the balance between the "force" of gravity and that one of the quantum potential, leads to the stationary equation for the Eigen states  where the quantum energy impulse tensor density reads and where the cosmological energy-impulse density Λ [11], for Eigen states, reads Finally, it is worth noting that, as a function of the quantum field, the quantum energy impulse tensor density reads