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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MDPI AG
2022-03-01
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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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issn | 2218-1997 |
language | English |
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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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