Abstract
Various theories of Quantum Gravity predict modifications of the Heisenberg Uncertainty Principle near the Planck scale to a so-called Generalized Uncertainty Principle (GUP). In some recent papers, we showed that the GUP gives rise to corrections to the Schrödinger equation, which in turn affect all quantum mechanical Hamiltonians. In particular, by applying it to a particle in a one-dimensional box, we showed that the box length must be quantized in terms of a fundamental length (which could be the Planck length), which we interpreted as a signal of fundamental discreteness of space itself. In this Letter, we extend the above results to a relativistic particle in a rectangular as well as a spherical box, by solving the GUP-corrected Klein–Gordon and Dirac equations, and for the latter, to two and three dimensions. We again arrive at quantization of box length, area and volume and an indication of the fundamentally grainy nature of space. We discuss possible implications.
Highlights
Klein–Gordon equation in one dimensionWe see that this is identical to the Schrödinger equation, when one makes the identification: 2mE/h 2 ≡ k2 → E2/h 2c2 − m2c2/h 2
Various theories of Quantum Gravity predict modifications of the Heisenberg Uncertainty Principle near the Planck scale to a so-called Generalized Uncertainty Principle (GUP)
By applying it to a particle in a one-dimensional box, we showed that the box length must be quantized in terms of a fundamental length, which we interpreted as a signal of fundamental discreteness of space itself
Summary
We see that this is identical to the Schrödinger equation, when one makes the identification: 2mE/h 2 ≡ k2 → E2/h 2c2 − m2c2/h 2. The quantization of length, which does not depend on k, continues to hold [1]. In addition to fermions being the most fundamental entities, the 3-dimensional version of KG equation (7), when combined with Eq (5), suffers from the drawback that the p2 term translates to p2 = p20 − 2ap30 + O(a2) = −h 2∇2 + i2ah 3∇3/2 + O(a2), of which the second term is non-local. The Dirac equation can address both issues at once
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