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Journal of Nonlinear Mathematical Physics

A generalized Schrödinger equation via a complex Lagrangian of electrodynamics

Nicoleta Aldea, Gheorghe Munteanu

Published Jun 3, 2015• 1 citation
Publisher

Abstract

In this paper we give a generalized form of the Schrödinger equation in the relativistic case, which contains a generalization of the Klein-Gordon equation.By complex Legendre transformation, the complex Lagrangian of electrodynamics produces a complex relativistic Hamiltonian H of electrodynamics, on the holomorphic cotangent bundle T * M. By a special quantization process, a relativistic time dependent Schrödinger equation, in the adapted frames of (T * M, H) is obtained.This generalized Schrödinger equation can be expressed with respect to the Laplace operator of the complex Hamilton space (T * M, H).Finally, under some additional conditions on the proper time s of the complex space-time M and the time parameter t along the quantum state, by the method of separation of variables, we obtain two classes of solutions for the Schrödinger equation, one for the weakly gravitational complex curved space M, and the second in the complex space-time with Schwarzschild metric.

DOI
10.1080/14029251.2015.1056619

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In this paper we give a generalized form of the Schrödinger equation in the relativistic case, which contains a generalization of the Klein-Gordon equation.By complex Legendre transformation, the complex Lagrangian of electrodynamics produces a complex relativistic Hamiltonian H of electrodynamics, on the holomorphic cotangent bundle T * M. By a special quantization process, a relativistic time dependent Schrödinger equation, in the adapted frames of (T * M, H) is obtained.This generalized Schrödinger equation can be expressed with respect to the Laplace operator of the complex Hamilton space (T * M, H).Finally, under some additional conditions on the proper time s of the complex space-time M and the time parameter t along the quantum state, by the method of separation of variables, we obtain two classes of solutions for the Schrödinger equation, one for the weakly gravitational complex curved space M, and the second in the complex space-time with Schwarzschild metric.
In a previous paper, we showed that the Hamilton equations of motion of a quantum particle are obtained as group velocities of the wave packet describing this particle only if the time dependent phase of a particle wave is proportional to the Lagrangian, not to the Hamiltonian as in a solution of the conventional Schrodinger equation. When a Lagrangian of relativistic form is considered, the wave packet of a quantum particle takes a physical form, with a finite spectrum of a cut off velocity (c). Based on a relativistic quantum principle, asserting the invariance of the time dependent phase for an arbitrary change of coordinates, we obtained the relativistic kinematics and dynamics, the electromagnetic field equations, the spin and the electromagnetic and gravitational interactions. When the Lagrangian is considered as a function of the Hamiltonian, we obtain a Schrodinger type equation with an additional term depending on the velocity and the momentum operator. 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On the other hand, one could find that the simplest way to define a quantum particle is by a wave packet, with momenta conjugated to coordinates and energy conjugated to time Consider a quantum particle as a wave packet, and find that the group velocities in the coordinate and momentum spaces are in agreement with the Hamilton equations only when the Lagrangian is considered in the time dependent phases, instead of the Hamiltonian in the conventional forms of these waves as solutions of the Schrodinger equation. We define a relativistic quantum principle, and derive a wave equation for a relativistic quantum particle, the relativistic kinematics and dynamics of the particle waves, the Maxwell equations and the Lorentz force of a field interacting with the particle waves, the relativistic transform of such a field, and the spin as a characteristic of the particle waves. 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The starting point of this research is a representation of a quantum particle according to the Schrodinger equation of the conventional quantum mechanics. In this representation, a quantum particle is described by a wave packet in the coordinate space and the conjugated wave packet in the momentum space. The problem is that while in the coordinate space the group velocity is in agreement with one of the Hamilton equations, the group velocity in the momentum space is in contradiction with the other Hamilton equation - a minus sign is missing. Group velocities in agreement with the Hamilton equations are obtained only when in the time dependent phase of a quantum particle wave packet, instead of the Hamiltonian coming from the conventional Schrodinger equation, the Lagrangian is considered. This suggests us to consider the relativistic Lagrangian in the time dependent phase. In this way, the conventional relativistic principle of invariance of the time-space interval gets the more physically understandable form of the invariance of the time dependent phase of a quantum particle - the time dependent phase of a quantum particle is the same in any system of coordinates. Based on the relativistic time dependent phase invariance of a quantum particle, from the group velocities of this particle the relativistic kinematics and dynamics are obtained. The interaction with an electromagnetic field is described by a modification of the time dependent phase with a scalar potential conjugated to time and a vector potential conjugated to the space coordinates. In this paper, we find that, according to the formalism of the general relativity, any matter element in a field of forces is accelerated only perpendicularly to its velocity. This means that the matter propagation of a quantum particle can be conceived in planes perpendicular to velocity, while the mater distribution can be considered in a Fourier representation - quantum waves. We study the quantum particle dynamics in gravitational waves and derive the graviton spin. We consider a quantum particle in electromagnetic field and obtain a Schrödinger-Dirac-type equation, with additional terms describing the velocity dependence of the particle dynamics.
This paper deals with computational derivation of the Klein-Gordon quantum relativist equation and the Schrödinger quantum equation with forward and backward space-time shifts. The first part introduces forward and backward derivatives for discrete and continuous systems. Generalized complex discrete and continuous derivatives are deduced. In the second part, we apply these forward and backward derivatives to the harmonic oscillator. The third part deduces the Klein-Gordon equation from the space-time complex continuous derivatives. These derivatives take into account forward-backward space-time shifts related to a phase velocity u. Without time shift, the Schrödinger equation is deduced in part four. This quantum equation is interpreted as a Burgers fluid with a complex momentum in relation to the Bohm-Hiley quantum potential concept. Finally a time-dependent Schrödinger equation for a free particle in a constant potential is simulated. The quantum particle looks like an oscillator with two frequencies.
We present a consistent and comprehensive treatise on the foundations of the extended Hamilton–Lagrange formalism — where the dynamical system is parametrized along a general system evolution parameter s, and the time t is treated as a dependent variable t(s) on equal footing with all other configuration space variables qi(s). In the action principle, the conventional classical action L1d t is then replaced by the generalized action L1d s, with L and L1denoting the conventional and the extended Lagrangian, respectively. It is shown that a unique correlation of L1and L exists if we refrain from performing simultaneously a transformation of the dynamical variables. With the appropriate correlation of L1and L in place, the extension of the formalism preserves its canonical form. In the extended formalism, the dynamical system is described as a constrained motion within an extended space. We show that the value of the constraint and the parameter s constitutes an additional pair of canonically conjugate variables. In the corresponding quantum system, we thus encounter an additional uncertainty relation. As a consequence of the formal similarity of conventional and extended Hamilton–Lagrange formalisms, Feynman's nonrelativistic path integral approach can be converted on a general level into a form appropriate for relativistic quantum physics. In the emerging parametrized quantum description, the additional uncertainty relation serves as the means to incorporate the constraint and hence to finally eliminate the parametrization. We derive the extended Lagrangian L1of a classical relativistic point particle in an external electromagnetic field and show that the generalized path integral approach yields the Klein–Gordon equation as the corresponding quantum description. We furthermore derive the space–time propagator for a free relativistic particle from its extended Lagrangian L1. These results can be regarded as the proof of principle of the relativistic generalization of Feynman's path integral approach to quantum physics.
The Schrödinger operators on the Newtonian spacetime are defined in a way which make them independent of the class of inertial observers. In this picture the Schrödinger operators act not on functions on the spacetime but on sections of a certain one-dimensional complex vector bundle—the Schrödinger line bundle . This line bundle has trivializations indexed by inertial observers and is associated with an U (1)-principal bundle with an analogous list of trivializations—the Schrödinger principal bundle . If an inertial frame is fixed, the Schrödinger bundle can be identified with the trivial bundle over spacetime, but as there is no canonical trivialization (inertial frame), these sections interpreted as 'wavefunctions' cannot be viewed as actual functions on the spacetime. In this approach, the change of an observer results not only in the change of actual coordinates in the spacetime but also in a change of the phase of wavefunctions. For the Schrödinger principal bundle, a natural differential calculus for 'wave forms' is developed that leads to a natural generalization of the concept of the Laplace–Beltrami operator associated with a pseudo-Riemannian metric. The free Schrödinger operator turns out to be the Laplace–Beltrami operator associated with a naturally distinguished invariant pseudo-Riemannian metric on the Schrödinger principal bundle. The presented framework does not involve any ad hoc or axiomatically introduced geometrical structures. It is based on the traditional understanding of the Schrödinger operator in a given reference frame—which is supported by producing right physics predictions—and it is proven to be strictly related to the frame-independent formulation of analytical Newtonian mechanics and Hamilton–Jacobi equations that makes a bridge between the classical and quantum theory.
This paper is a continuation of our preceding paper dealing with computational derivation of the Klein–Gordon quantum relativist equation and the Schrödinger quantum equation with forward and backward space-time shifts. The first part introduces forward and backward derivatives for discrete and continuous systems. Generalized complex discrete and continuous derivatives are deduced. The second part deduces the Klein–Gordon equation from the space-time complex continuous derivatives. These derivatives take into account forward-backward space-time shifts related to an internal phase velocity u. The internal group velocity v is related to the speed of light u.v=c2 and to the external group and phase velocities u.v=vg.vp. Without time shift, the Schrödinger equation is deduced, with a supplementary term, which could represent a reference potential. The third part deduces the Quantum Relativist Klein–Gordon equation for a particle in an electromagnetic field.