Relativistic Generalized Uncertainty Principle

The Generalized Uncertainty Principle and the related minimum length are normally considered in non-relativistic Quantum Mechanics. Extending it to relativistic theories is important for having a Lorentz invariant minimum length and for testing the modified Heisenberg principle at high energies.In this paper, we formulate a relativistic Generalized Uncertainty Principle. We then use this to write the modified Klein-Gordon, Schr\"odinger and Dirac equations, and compute quantum gravity corrections to the relativistic hydrogen atom, particle in a box, and the linear harmonic oscillator.Comment: 6 pages, Revte

Relativistic generalized uncertainty principle and its implications

The fundamental physical description of the Universe is based on two theories:Quantum Mechanics and General Relativity. Unified theory of Quantum Gravity (QG) is an open problem. Quantum Gravity Phenomenology (QGP) studies QG effects in low-energy systems. The basis of one such phenomenological model is the Generalized Uncertainty Principle (GUP), which is a modified Heisenberg uncertainty relation and predicts a deformed position-momentum commutator. Relativistic Generalized Uncertainty Principle (RGUP) is proposed in this thesis,which gives a Loerentz invariant minimum length and resolves the composition law problem. RGUP modified Klein–Gordon, Schrödinger and Dirac equations with QG corrections to several systems are presented. The Lagrangians of Quantum Electro-dynamics for the gauge, scalar, and spinor fields are obtained. The RGUP corrections to scattering amplitudes are then calculated. The results are applied to high energy scattering experiments providing much needed window for testing minimum length and QG theories in the laboratory.This work was supported by the Natural Sciences and Engineering Research Council of Canada, and University of Lethbridge. This research was additionally supported by Quantum Major Innovation Fund Project, funded by the Government of Alberta

Effective field theory from Relativistic Generalized Uncertainty

Theories of Quantum Gravity predict a minimum measurable length and a corresponding modification of the Heisenberg Uncertainty Principle to the so-called Generalized Uncertainty Principle (GUP). However, this modification is usually formulated in non-relativistic language, making it unclear whether the minimum length is Lorentz invariant. We have formulated a Relativistic Generalized Uncertainty Principle, resulting in a Lorentz invariant minimum measurable length and the resolution of the composition law problem. This proved to be an important step in the formulation of Quantum Field Theory with minimum length. We derived the Lagrangians consistent with the existence of minimal length and describing the behaviour of scalar, spinor, and U(1) gauge fields. We calculated the Feynman rules (propagators and vertices) associated with these Lagrangians. Furthermore, we calculated the Quantum Gravity corrected scattering cross-sections for a lepton-lepton scattering. Finally, we compared our results with current experiments, which allowed us to improve the bounds on the scale at which quantum gravity phenomena will become relevant.Comment: Sixteenth Marcel Grossmann Meeting - MG16 2021, 19 pages, 2 table

Experimental test of fair three-sided coins

A simple model for a fair 'three-sided coin' is proposed and tested. Describing the coin as a cylinder with a given height and basis radius, this model efficiently characterizes the problem, constraining the size of the coin. A statistical analysis of the data collected from actual realizations of such coins has been performed, supporting the proposed model. Besides studying the case of a fair three-sided coin, this work represents a model for an explicit application of the scientific method, in which all parts (problem characterization, statement of a hypothesis, experiment, analysis, description, conclusions) have clearly directed its development. Thus, it represents an useful illustration of such method for undergraduate students