Discrete phase space and continuous time relativistic quantum mechanics II: Peano circles, hyper-tori phase cells, and fiber bundles

2021 ◽  
Vol 36 (35) ◽  
Author(s):  
Anadijiban Das ◽  
Rupak Chatterjee
Keyword(s):  
Phase Space ◽  
State Space ◽  
Continuous Time ◽  
Gordon Equation ◽  
Relativistic Quantum ◽  
Relativistic Quantum Mechanics ◽  
Fiber Bundles ◽  
Phase Cell ◽  
Klein Gordon ◽  
Discrete Phase

The discrete phase space and continuous time representation of relativistic quantum mechanics are further investigated here as a continuation of paper I.1 The main mathematical construct used here will be that of an area filling Peano curve. We show that the limit of a sequence of a class of Peano curves is a Peano circle denoted as [Formula: see text], a circle of radius [Formula: see text] where [Formula: see text]. We interpret this two-dimensional (2D) Peano circle in our framework as a phase cell inside our 2D discrete phase plane. We postulate that a first quantized Planck oscillator, being very light, and small beyond current experimental detection, occupies this phase cell [Formula: see text]. The time evolution of this Peano circle sweeps out a 2D vertical cylinder analogous to the worldsheet of string theory. Extending this to 3D space, we introduce a [Formula: see text]-dimensional phase space hyper-tori [Formula: see text] as the appropriate phase cell in the physical dimensional discrete phase space. A geometric interpretation of this structure in state space is given in terms of product fiber bundles. We also study free scalar Bosons in the background [Formula: see text]-dimensional discrete phase space and continuous time state space using the relativistic partial difference-differential Klein–Gordon equation. The second quantized field quanta of this system can cohabit with the tiny Planck oscillators inside the [Formula: see text] phase cells for eternity. Finally, a generalized free second quantized Klein–Gordon equation in a higher [Formula: see text]-dimensional discrete state space is explored. The resulting discrete phase space dimension is compared to the significant spatial dimensions of some of the popular models of string theory.

scholarly journals Discrete phase space - II: The second quantization of free relativistic wave fields

2010 ◽  
Vol 88 (2) ◽  
pp. 93-109 ◽  
Author(s):  
A. Das
Keyword(s):  
Phase Space ◽  
Continuous Time ◽  
Gordon Equation ◽  
Partial Difference Equations ◽  
Partial Difference ◽  
Commutation Relations ◽  
Quantized Fields ◽  
Klein Gordon ◽  
Discrete Phase ◽  
Time Continuum

The Klein–Gordon equation, the Maxwell equation, and the Dirac equation are presented as partial difference equations in the eight-dimensional covariant discrete phase space. These equations are also furnished as difference-differential equations in the arena of discrete phase space and continuous time. The scalar field and electromagnetic fields are quantized with commutation relations. The spin-1/2 field is quantized with anti-commutation relations. Moreover, the total momentum, energy and charge of these free relativisitic quantized fields in the discrete phase space and continuous time are computed exactly. The results agree completely with those computed from the relativisitic fields defined on the space-time continuum.

BILINEAR FORM AND SOLITON SOLUTIONS FOR THE COUPLED NONLINEAR KLEIN–GORDON EQUATIONS

2012 ◽  
Vol 26 (15) ◽  
pp. 1250057
Author(s):  
HE LI ◽  
XIANG-HUA MENG ◽  
BO TIAN
Keyword(s):  
Field Theory ◽  
Bilinear Form ◽  
Gordon Equation ◽  
Relativistic Quantum ◽  
Soliton Solutions ◽  
Relativistic Quantum Mechanics ◽  
Klein Gordon ◽  
Truncated Painlevé Expansion ◽  
Klein Gordon Equation ◽  
Singular Manifolds

With the coupling of a scalar field, a generalization of the nonlinear Klein–Gordon equation which arises in the relativistic quantum mechanics and field theory, i.e., the coupled nonlinear Klein–Gordon equations, is investigated via the Hirota method. With the truncated Painlevé expansion at the constant level term with two singular manifolds, the coupled nonlinear Klein–Gordon equations are transformed to a bilinear form. Starting from the bilinear form, with symbolic computation, we obtain the N-soliton solutions for the coupled nonlinear Klein–Gordon equations.

scholarly journals Bohmian mechanics of Klein–Gordon equation via quantum metric and mass

2019 ◽  
Vol 34 (33) ◽  
pp. 1950270
Author(s):  
Shahram Jalalzadeh ◽  
A. J. S. Capistrano
Keyword(s):  
Quantum Mechanics ◽  
Relativistic Theory ◽  
Gordon Equation ◽  
Relativistic Quantum ◽  
Bohmian Mechanics ◽  
Relativistic Quantum Mechanics ◽  
Relativistic Limit ◽  
Klein Gordon ◽  
Klein Gordon Equation ◽  
De Broglie Bohm

The causal stochastic interpretation of relativistic quantum mechanics has the problems of superluminal velocities, motion backward in time and the incorrect non-relativistic limit. In this paper, according to the original ideas of de Broglie, Bohm and Takabayasi, we introduce simultaneously a quantum mass and a quantum metric of a curved spacetime to obtain a correct relativistic theory free of mentioned problems.

scholarly journals High-Order Energy and Linear Momentum Conserving Methods for the Klein-Gordon Equation

Mathematics ◽  
2018 ◽  
Vol 6 (10) ◽  
pp. 200 ◽  
Author(s):  
He Yang
Keyword(s):  
Numerical Methods ◽  
Gordon Equation ◽  
Free Particle ◽  
High Order ◽  
Linear Momentum ◽  
Relativistic Quantum Mechanics ◽  
Optimal Error Estimates ◽  
Numerical Schemes ◽  
Klein Gordon ◽  
Klein Gordon Equation

The Klein-Gordon equation is a model for free particle wave function in relativistic quantum mechanics. Many numerical methods have been proposed to solve the Klein-Gordon equation. However, efficient high-order numerical methods that preserve energy and linear momentum of the equation have not been considered. In this paper, we propose high-order numerical methods to solve the Klein-Gordon equation, present the energy and linear momentum conservation properties of our numerical schemes, and show the optimal error estimates and superconvergence property. We also verify the performance of our numerical schemes by some numerical examples.

SCHRÖDINGER EQUATION FOR QUANTUM FRACTAL SPACE–TIME OF ORDER n VIA THE COMPLEX-VALUED FRACTIONAL BROWNIAN MOTION

2001 ◽  
Vol 16 (31) ◽  
pp. 5061-5084 ◽  
Author(s):  
GUY JUMARIE
Keyword(s):  
Brownian Motion ◽  
Schrödinger Equation ◽  
Fractional Order ◽  
Schrodinger Equation ◽  
Relativistic Quantum ◽  
Relativistic Quantum Mechanics ◽  
Dimensional System ◽  
One Dimensional ◽  
Klein Gordon ◽  
Complex Valued

First remark: Feynman's discovery in accordance of which quantum trajectories are of fractal nature (continuous everywhere but nowhere differentiable) suggests describing the dynamics of such systems by explicitly introducing the Brownian motion of fractional order in their equations. The second remark is that, apparently, it is only in the complex plane that the Brownian motion of fractional order with independent increments can be generated, by using random walks defined with the complex roots of the unity; in such a manner that, as a result, the use of complex variables would be compulsory to describe quantum systems. Here one proposes a very simple set of axioms in order to expand the consequences of these remarks. Loosely speaking, a one-dimensional system with real-valued coordinate is in fact the average observation of a one-dimensional system with complex-valued coordinate: It is a strip modeling. Assuming that the system is governed by a stochastic differential equation driven by a complex valued fractional Brownian of order n, one can then obtain the explicit expression of the corresponding covariant stochastic derivative with respect to time, whereby we switch to the extension of Lagrangian mechanics. One can then derive a Schrödinger equation of order n in quite a direct way. The extension to relativistic quantum mechanics is outlined, and a generalized Klein–Gordon equation of order n is obtained. As a by-product, one so obtains a new proof of the Schrödinger equation.

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