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arXiv:1808.00766 · Europhysics Letters (EPL)

Schwinger's model of angular momentum with GUP

Harshit Verma, Toshali Mitra, Bhabani Prasad Mandal

Published Sep 5, 2018• 8 citations
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Abstract

We study the generalized uncertainty principle (GUP) modified simple harmonic oscillator (SHO) in the operator formalism by considering the appropriate form of the creation and annihilation operators . The angular-momentum algebra is then constructed using Schwinger's model of angular momentum with two independent GUP modified SHOs. With the GUP modified angular-momentum algebra, we discuss coupling of angular momentum for a two-particle composite system. Further, we calculate the Clebsch-Gordan (CG) coefficients for a two-particle system explicitly. Our results show that the CG coefficients do not receive any corrections up to quadratic GUP.

DOI
10.1209/0295-5075/123/30009

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Comment on “Schwinger's model of angular momentum with GUP” by Verma Harshit et al .

Pasquale Bosso, S. Das

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.

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Reply to the Comment by P. Bosso and S. Das

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Introduction.-Through this document, we wish to respond to the comments [1] made on our manuscript entitled "Schwinger's model of angular momentum with GUP".First of all, the authors have wrongly presumed that we have invalidated/questioned their results in [2].In the abstract section, they claim that their work is valid, on which, we made no claims suggesting the contrary in the paper [3].In fact we have used some of their results in our work.The only issue which we found was that the Clebsch-Gordan (CG) coefficients for the same state, obtained in their work by applying J -and J + operators were different.This is clearly unacceptable.This situation can be better described with the word "inconsistency" rather than with the word, "ambiguous".We feel that there was, in fact, no ambiguity in the mathematical calculations of CG coefficients using the operators J -and J + in their works.In our work we have removed this inconsistency.Now, we shall try to address the specific points raised by the authors.

Europhysics Letters (EPL)•Jan 29, 2019•0 citations

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We study the generalized uncertainty principle (GUP) modified simple harmonic oscillator (SHO) in the operator formalism by considering the appropriate form of the creation and annihilation operators . The angular-momentum algebra is then constructed using Schwinger's model of angular momentum with two independent GUP modified SHOs. With the GUP modified angular-momentum algebra, we discuss coupling of angular momentum for a two-particle composite system. Further, we calculate the Clebsch-Gordan (CG) coefficients for a two-particle system explicitly. Our results show that the CG coefficients do not receive any corrections up to quadratic GUP.
This work establishes a link between deformed angular momentum operators and modifications of the quantum uncertainty principle. Starting from an algebra in which the standard angular momentum operators are deformed by two parameters, [Formula: see text] and [Formula: see text], we construct the corresponding momentum operators via [Formula: see text] where the radial operator [Formula: see text] is determined by the requirement that the standard momentum operator is recovered in the limit [Formula: see text], [Formula: see text]. The resulting commutation relations take the form [Formula: see text] where [Formula: see text] is a position-dependent deformation matrix. This leads to an uncertainty relation [Formula: see text] which is state-dependent, anisotropic and position-dependent. We use the term Generalized Uncertainty Principle (GUP) in its broad sense referring to any modification of the Heisenberg relation motivated by quantum-gravity ideas. However, the model does not yield a fixed minimal length of the kind familiar from conventional GUP models (e.g. [Formula: see text]). In the isotropic, limit the trace of the deformation matrix simplifies the bound to [Formula: see text] which can be considered as a rescaling of the uncertainty constant. Our work thus provides a geometric origin for modified uncertainty relations from symmetry breaking in the rotation group, and offers a framework in which quantum-gravity effects manifest themselves through state- and direction-dependent measurement limits, without necessarily introducing a universal minimal length.
Abstract We review the irreducible representation of an angular momentum vector operator constructed in terms of spinor algebra. We generalize the idea of spinor approach to study the coupling of the eigenstates of two independent angular momentum vector operators. Utilizing the spinor algebra, we are able to develop a simple way for calculating the SU(2) Clebsch-Gordan (CG) coefficients. The explicit expression for the SU(2) CG coefficients is worked out, and some simple physical examples are presented to illustrate the spinor approach.
Introduction.-Through this document, we wish to respond to the comments [1] made on our manuscript entitled "Schwinger's model of angular momentum with GUP".First of all, the authors have wrongly presumed that we have invalidated/questioned their results in [2].In the abstract section, they claim that their work is valid, on which, we made no claims suggesting the contrary in the paper [3].In fact we have used some of their results in our work.The only issue which we found was that the Clebsch-Gordan (CG) coefficients for the same state, obtained in their work by applying J -and J + operators were different.This is clearly unacceptable.This situation can be better described with the word "inconsistency" rather than with the word, "ambiguous".We feel that there was, in fact, no ambiguity in the mathematical calculations of CG coefficients using the operators J -and J + in their works.In our work we have removed this inconsistency.Now, we shall try to address the specific points raised by the authors.
We analyze the issue of unitary equivalence within Generalized Uncertainty Principle (GUP) theories in the one-dimensional case. For a deformed Heisenberg algebra, its representation in terms of Hilbert space and conjugate operators is not uniquely determined, raising the question of whether different realizations of the same algebra are equivalent and describe the same physics. After proposing a definition of a quantum GUP theory, we establish conditions for unitary equivalence. Using this framework, we rigorously prove that two commonly used representations are unitarily equivalent, specifying the conditions under which this equivalence holds. We demonstrate this equivalence explicitly by providing a unitary map and showing how both GUP formulations yield the same physical results in two examples: the quantum harmonic oscillator and a free-falling particle. Finally, we discuss a case where equivalence fails, suggesting that a generalization of the Stone-von Neumann theorem may not be possible within the GUP framework under our definition of unitary equivalence.