Comments on "Schwinger's Model of Angular Momentum with GUP" by H. Verma et al, arXiv:1808.00766

In this note, we show that the methodology and conclusions of "Schwinger's Model of Angular Momentum with GUP" [arxiv:1808.00766] are flawed and that the conclusions of "Generalized Uncertainty Principle and angular momentum" (P. Bosso and S. Das) [arxiv:1607.01083] remain valid.Comment: 3 page

Generalized Uncertainty Principle and Angular Momentum

Various models of quantum gravity suggest a modification of the Heisenberg's Uncertainty Principle, to the so-called Generalized Uncertainty Principle, between position and momentum. In this work we show how this modification influences the theory of angular momentum in Quantum Mechanics. In particular, we compute Planck scale corrections to angular momentum eigenvalues, the Hydrogen atom spectrum, the Stern-Gerlach experiment and the Clebsch-Gordan coefficients. We also examine effects of the Generalized Uncertainty Principle on multi-particle systems.Comment: 21 pages, 2 figures, minor revisions, to appear in Annals of Physic

Lorentz invariant mass and length scales

We show that the standard Lorentz transformations admit an invariant mass (length) scale, such as the Planck scale. In other words, the frame independence of such scale is built-in within those transformations, and one does not need to invoke the principle of relativity for their invariance. This automatically ensures the frame-independence of the spectrum of geometrical operators in quantum gravity. Furthermore, we show that the above predicts a small but measurable difference between the inertial and gravitational mass of any object, regardless of its size or whether it is elementary or composite.Comment: 10 page

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

Rigorous Hamiltonian and Lagrangian analysis of classical and quantum theories with minimal length

GUP is a phenomenological model aimed for a description of a minimal length in quantum and classical systems. However, the analysis of problems in classical physics is usually approached preferring a different formalism than the one used for quantum systems, and vice versa. Potentially, the two approaches can result in inconsistencies. Here, we eliminate such inconsistencies proposing particular meanings and relations between the variables used to describe physical systems, resulting in a precise form of the Legendre transformation. Furthermore, we introduce two different sets of canonical variables and the relative map between them. These two sets allow for a complete and unambiguous description of classical and quantum systems.Comment: 15 pages, 4 figure

Generalized ladder operators for the perturbed harmonic oscillator

In this paper, we construct corrections to the raising and lowering (i.e. ladder) operators for a quantum harmonic oscillator subjected to a polynomial type perturbation of any degree and to any order in perturbation theory. We apply our formalism to a couple of examples, namely q and p 4 perturbations, and obtain the explicit form of those operators. We also compute the expectation values of position and momentum for the above perturbations. This construction is essential for defining coherent and squeezed states for the perturbed oscillator. Furthermore, this is the first time that corrections to ladder operators for a harmonic oscillator with a generic perturbation and to an arbitrary order of perturbation theory have been constructed.Comment: 11 page

The minimal length: a cut-off in disguise?

The minimal-length paradigm, a possible implication of quantum gravity at low energies, is commonly understood as a phenomenological modification of Heisenberg's uncertainty relation. We show that this modification is equivalent to a cut-off in the space conjugate to the position representation, i.e. the space of wave numbers, which does not necessarily correspond to momentum space. This result is generalized to several dimensions and noncommutative geometries once a suitable definition of the wave number is provided. Furthermore, we find a direct relation between the ensuing bound in wave-number space and the minimal-length scale. For scenarios in which the existence of the minimal length cannot be explicitly verified, the proposed framework can be used to clarify the situation. Indeed, applying it to common models, we find that one of them does, against all expectations, allow for arbitrary precision in position measurements. In closing, we comment on general implications of our findings for the field. In particular, we point out that the minimal length is purely kinematical such that, effectively, there is only one model of minimal-length quantum mechanics

Potential tests of the Generalized Uncertainty Principle in the advanced LIGO experiment

The generalized uncertainty principle and a minimum measurable length arise in various theories of gravity and predict Planck-scale modifications of the canonical position-momentum commutation relation. Postulating a similar modified commutator between the canonical variables of the electromagnetic field in quantum optics, we compute Planck-scale corrections to the radiation pressure noise and shot noise of Michelson-Morley interferometers, with particular attention to gravity wave detectors such as LIGO. We show that advanced LIGO is potentially sensitive enough to observe Planck-scale effects and thereby indirectly a minimal length. We also propose estimates for the bounds on quantum gravity parameters from current and future advanced LIGO experiments.Comment: 11 pages, 8 figure