The Behavior of Quantum Particles at Very Low Scale
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The Behavior of Quantum Particles at Very Low Scale A. A. Abutaleb1 · E. Ahmed1 · A. I. Elmahdy1 Received: 1 June 2020 / Accepted: 21 October 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we propose that the behavior of quantum particles assuming discreteness of the spacetime at very low scale is similar to the behavior of Anyons.
1 Introduction The nature of spacetime at different scales is essentially important in formulating physical theories. In general relativity, the spacetime is assumed to be continuous, but quantum field theory indicates that spacetime should be discrete at some very small scales [1–3]. Some physicists believe that it is often useful to define physical theories in continuous spacetime as the continuum limit of that theories in discrete spacetime [3]. One of the most important problems of physics today is to understand how we pass from a continuous to a discrete description of spacetime. Is there suddenly change or is there gradual transition and where does the change occur? In order to solve this problem, Kempf [1] suggested that spacetime could be simultaneously continuous and discrete!, in the same way that information can be. The transformation rules between continuous and discrete representations of information are described by Shannon’s sampling theory [4]. Here, we assume that the structure of the spacetime is inherited by a mathematical function called the graininess function, which allow one to assume that the structure of the spacetime is highly dependent on the scale measurement of the observer (i.e. the observer will see the structure of the spacetime according to his/her measurement scale). In 1988, Hilger [5] had established a new branch of mathematics called Time-Scale calculus in order to unify continuous and discrete analysis. The general idea of Time-Scale calculus is to study the dynamic equations where the domain of the functions is an arbitrary closed subset of the real numbers called Time-Scale T [6–10]. When T = R (the real numbers), we recover our usual frame work of calculus, and when T = Z (the integer numbers), we arrive at the usual version of difference equations. For example, consider a function F : D → R, and suppose that there are some points of discontinuity x ∈ D. In ordinary calculus, one cannot speak about the differentiability of the function F at x, but in the Time-Scale calculus, one can differentiate the function F even it was discontinuous (in
A. A. Abutaleb
[email protected] 1
Department of Mathematics, Faculty of Science, University of Mansoura, Mansoura, Egypt
International Journal of Theoretical Physics
ordinary sense) at every point of its domain. Moreover, one can study the dynamic equations even the domain D has a fractal nature. (Notice that in a fractal-like curve, the concept of neighborhood in Euclidean Topology is destroyed, because the distance of any two points on this curve is infinity) [11]. Some results of physical literature expect a kind of discretization nature
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