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

ISSN: 2329-6542

Journal of Astrophysics & Aerospace Technology

  • Research Article   
  • J Astrophys Aerospace Technol 2017, Vol 5(2): 148
  • DOI: 10.4172/2329-6542.1000148

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

Piero Chiarelli*
National Council of Research of Italy, Interdepartmental Center “E. Piaggio”, University of Pisa, Area of Pisa, Pisa, Moruzzi 1, Italy
*Corresponding Author: Piero Chiarelli, National Council of Research of Italy, Interdepartmental Center “E. Piaggio”, University of Pisa, Italy, Tel: +39-050-315- 2359, Fax: +39-050-315-2166 , Email: [email protected]

Received Date: Jul 31, 2017 / Accepted Date: Aug 09, 2017 / Published Date: Aug 14, 2017

Abstract

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

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 Equation is solved as a function of the two real variables,Equation and S [2-5]. 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

Equation (1)

where Equation and where, for instance, we assume the quartic renormalizable interaction Equation

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

Equation (6)

coupled to the current equation [2]

Equation (3)

where

Equation (4)

and where

Equation (5)

Moreover, being the 4-impulse in the hydrodynamic analogy

Equation (6)

it follows that

Equation (7)

where

Equation (8)

Moreover, by using (6), equation (2) can be rewritten as

Equation (9)

where

Equation (10)

and where P2 = Pi Pi is the modulus of the spatial momentum.

As shown in reference [11], given the hydrodynamic Lagrangean function

Equation

Equation (11)

Equation

equation (2) can be expressed by the following system of Lagrangean equations of motion

Equation(12)

Equation (13)

that for the eigenstates read

Equation (14)

Equation (15)

where

Equation (16)

Generally speaking, for eigenstates, for which it holds E=En=const it follows that:

Equation (17)

from where it follows that

Equation (18)

Equation

(where the minus sign stands for antiparticles) and, by using (17), that

Equation (19)

Following the hydrodynamic protocol [11], the eigenstates are represented by the stationary solutions of the hydrodynamic equations of motion obtained by deriving Equation from (14) and then inserting it into (15) that leads to

Equation (20)

where

Equation

and to

Equation (21)

where, for eigenstates, the quantum energy-impulse tensor (QEIT) Equation reads [11,12],

Equation (22)

leading to the quantum energy impulse tensor density (QIETD) [11,12],

Equation (23)

Equation

where Equation is the (hydrodynamic) Lagrangian density and L is the hydrodynamic Lagrangian function. Moreover, by using the identity

Equation (24)

The QIETD (23) can be written as a function of the wave function as following:

Equation (25)

Equation

Charged field

In the case of a charged boson field, equations (1-3) read, respectively,

Equation (26)

Equation

Equation (27)

Equation (28)

where the 4-current Equation reads

Equation (29)

Equation

and

Equation (30)

(where Equation is the mechanical momentum) [11,13] and where

Equation (31)

Moreover, analogously to (9,17-19), from (27) it follows that

Equation (32)

Equation

that leads to

Equation (33)

to

Equation (34)

and to

Equation (35)

that, by using (24), as a function of Equation and Equation reads

Equation

Moreover, with the help of (24,29,32-34) it follows that

Equation (36)

that, by using (24,29,34) we can express as a function of the wave function as

Equation (37)

The above equations are coupled to the Maxwell one

Equation (38)

(where Equation is the current of the charged particles) where [14]

Equation (39)

and where

Equation (40)

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

Equation (41)

Equations (2-3) in a non- Euclidean space read, respectively,

Equation

Equation (42)

where

Equation (43)

Moreover, by using the definition of the Lagrangean function

Equation (44)

the covariant form of the motion equations (14-15) reads

Equation (45)

Equation (46)

where

Equation (47)

is the total covariant derivative respect the time and where Equation are the Christoffel symbols.

Equations (45-46) leads to the motion equation

Equation (48)

where, Ln reads

Equation

Equation (49)

From (48) it follows that the motion equation reads

Equation (50)

where the stationary condition Equation 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

Equation (51)

where Equation whereEquation is Jacobean of the transformation of the Galilean co-ordinates to non-Euclidean ones and where Equation is the metric tensor defined by the quantum gravitational equation [11]

Equation (52)

where the quantum energy impulse tensor density reads

Equation (53)

Equation

and where the cosmological energy-impulse density Equation[11], for Eigen states, reads

Equation (54)

where, for scalar uncharged particles leads to

Equation (55)

Finally, it is worth noting that, as a function of the quantum field, the quantum energy impulse tensor density reads

Equation (56)

Equation

Charged boson in non-Euclidean space-time

The KGE in non-Euclidean space-time for electromagnetic charged boson

Equation (57)

leads to the hydrodynamic system of equations

Equation (58)

Equation (59)

where

Equation (60)

Moreover, the Lagrangean motion equations read

Equation (61)

Equation (62)

where

Equation (63)

and to the QIETD

Equation (64)

Conclusion

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.

References

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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