The Hydrodynamic Representation of the Klein-Gordon Equation with Self- Interacting Field
Received Date: Jul 31, 2017 / Accepted Date: Aug 09, 2017 / Published Date: Aug 14, 2017
In this paper the quantum hydrodynamic approach for the KGE owing a self-interaction term is developed both for scalar and charged boson. The model allows to determine the quantum energy impulse tensor density of massive bosons such as the mesons. The generalization of the hydrodynamic Klein-Gordon equation to the non-Euclidean space-time is also derived for a quantum relativity approach.
Keywords: Quantum hydrodynamic representation; Bhom-Madelung approach; Self-interacting field; Non-Euclidean quantumhydrodynamics
Since the introduction of the quantum wave equation by Schrödinger, the quantum hydrodynamic approach (QHA) was presented by Madelung . In this quantum representation, developed by Madelung and then by Bhom, the evolution of a complex variable is solved as a function of the two real variables, and S [2-5]. As shown by Weiner et al. , the outputs of the quantum hydrodynamic model agree with the outputs of the Schrödinger problem and, more recently, as shown by Koide and Kodama , 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 .
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 . 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
where and where, for instance, we assume the quartic renormalizable interaction
coupled to the current equation 
Moreover, being the 4-impulse in the hydrodynamic analogy
it follows that
Moreover, by using (6), equation (2) can be rewritten as
and where P2 = Pi Pi is the modulus of the spatial momentum.
As shown in reference , 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=En=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 , 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 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:
In the case of a charged boson field, equations (1-3) read, respectively,
where the 4-current reads
Moreover, analogously to (9,17-19), from (27) it follows that
that leads to
that, by using (24), as a function of and reads
Moreover, with the help of (24,29,32-34) 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
(where is the current of the charged particles) where 
is the potential 4-vector,
The quartic self-interaction is introduced in the KGE in order to describe the states of charged (±1) bosons (e.g., mesons) . The importance of having the hydrodynamic description of bosons  lies in the fact that it allows to derive its quantum energy-impulse tensor that can couple them to the Einstein quantum-gravitational equation .
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
the covariant form of the motion equations (14-15) reads
is the total covariant derivative respect the time and where are the Christoffel symbols.
Equations (45-46) leads to the motion equation
where, Ln reads
From (48) it follows that the motion equation reads
where the stationary condition 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 where is Jacobean of the transformation of the Galilean co-ordinates to non-Euclidean ones and where is the metric tensor defined by the quantum gravitational equation 
where the quantum energy impulse tensor density reads
and where the cosmological energy-impulse density , for Eigen states, reads
where, for scalar uncharged particles leads to
Finally, it is worth noting that, as a function of the quantum field, the quantum energy impulse tensor density reads
Charged boson in non-Euclidean space-time
The KGE in non-Euclidean space-time for electromagnetic charged boson
leads to the hydrodynamic system of equations
Moreover, the Lagrangean motion equations read
and to the QIETD
The hydrodynamic approach allows obtaining the quantum energy-impulse tensor density as a function the field of the particle.
The biunique correspondence between the standard quantum mechanics and the hydrodynamic representation [1-6,17] warrants that the quantum energy-impulse tensor density can be independently defined by the used formalism.
In this work the quantum energy-impulse tensor, for massive bosons described by a KGE with self-interacting field is derived for defining the coupling with the quantum gravitational equation.
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Citation: Chiarelli P (2017) The Hydrodynamic Representation of the Klein-Gordon Equation with Self-Interacting Field. J Astrophys Aerospace Technol 5: 148. Doi: 10.4172/2329-6542.1000148
Copyright: © 2017 Chiarelli P. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricteduse, distribution, and reproduction in any medium, provided the original author and source are credited.
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