A Subtle Aspect of Minimal Lengths in the Generalized Uncertainty Principle

In this work, we point out an overlooked and subtle feature of the generalized uncertainty principle (GUP) approach to quantizing gravity: namely that different pairs of modified operators with the same modified commutator, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathM...

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Main Authors: Michael Bishop, Joey Contreras, Douglas Singleton
Format: Article
Language:English
Published: MDPI AG 2022-03-01
Series:Universe
Subjects:
Online Access:https://www.mdpi.com/2218-1997/8/3/192
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author Michael Bishop
Joey Contreras
Douglas Singleton
author_facet Michael Bishop
Joey Contreras
Douglas Singleton
author_sort Michael Bishop
collection DOAJ
description In this work, we point out an overlooked and subtle feature of the generalized uncertainty principle (GUP) approach to quantizing gravity: namely that different pairs of modified operators with the same modified commutator, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo stretchy="false">[</mo><mover accent="true"><mi>X</mi><mo stretchy="false">^</mo></mover><mo>,</mo><mover accent="true"><mi>P</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">]</mo></mrow><mo>=</mo><mi>i</mi><mi>ħ</mi><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>β</mi><msup><mi>p</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, may have different physical consequences such as having no minimal length at all. These differences depend on how the position and/or momentum operators are modified rather than only on the resulting modified commutator. This provides guidance when constructing GUP models since it distinguishes those GUPs that have a minimal length scale, as suggested by some broad arguments about quantum gravity, versus GUPs without a minimal length scale.
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spelling doaj.art-212900d590bc4d43a78e6044496b3c542023-11-30T22:41:04ZengMDPI AGUniverse2218-19972022-03-018319210.3390/universe8030192A Subtle Aspect of Minimal Lengths in the Generalized Uncertainty PrincipleMichael Bishop0Joey Contreras1Douglas Singleton2Mathematics Department, California State University Fresno, Fresno, CA 93740, USAPhysics Department, California State University Fresno, Fresno, CA 93740, USAPhysics Department, California State University Fresno, Fresno, CA 93740, USAIn this work, we point out an overlooked and subtle feature of the generalized uncertainty principle (GUP) approach to quantizing gravity: namely that different pairs of modified operators with the same modified commutator, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo stretchy="false">[</mo><mover accent="true"><mi>X</mi><mo stretchy="false">^</mo></mover><mo>,</mo><mover accent="true"><mi>P</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">]</mo></mrow><mo>=</mo><mi>i</mi><mi>ħ</mi><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>β</mi><msup><mi>p</mi><mn>2</mn></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, may have different physical consequences such as having no minimal length at all. These differences depend on how the position and/or momentum operators are modified rather than only on the resulting modified commutator. This provides guidance when constructing GUP models since it distinguishes those GUPs that have a minimal length scale, as suggested by some broad arguments about quantum gravity, versus GUPs without a minimal length scale.https://www.mdpi.com/2218-1997/8/3/192generalized uncertainty principlequantum gravityminimal length
spellingShingle Michael Bishop
Joey Contreras
Douglas Singleton
A Subtle Aspect of Minimal Lengths in the Generalized Uncertainty Principle
Universe
generalized uncertainty principle
quantum gravity
minimal length
title A Subtle Aspect of Minimal Lengths in the Generalized Uncertainty Principle
title_full A Subtle Aspect of Minimal Lengths in the Generalized Uncertainty Principle
title_fullStr A Subtle Aspect of Minimal Lengths in the Generalized Uncertainty Principle
title_full_unstemmed A Subtle Aspect of Minimal Lengths in the Generalized Uncertainty Principle
title_short A Subtle Aspect of Minimal Lengths in the Generalized Uncertainty Principle
title_sort subtle aspect of minimal lengths in the generalized uncertainty principle
topic generalized uncertainty principle
quantum gravity
minimal length
url https://www.mdpi.com/2218-1997/8/3/192
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