Alternative Derivation of the Non-Abelian Stokes Theorem in Two Dimensions
The relation between the holonomy along a loop with the curvature form is a well-known fact, where the small square loop approximation of aholonomy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi ma...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-10-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/11/2000 |
| _version_ | 1797457655260774400 |
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| author | Seramika Ariwahjoedi Freddy Permana Zen |
| author_facet | Seramika Ariwahjoedi Freddy Permana Zen |
| author_sort | Seramika Ariwahjoedi |
| collection | DOAJ |
| description | The relation between the holonomy along a loop with the curvature form is a well-known fact, where the small square loop approximation of aholonomy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold-italic">H</mi><mfenced separators="" open="(" close=")"><mi>γ</mi><mo>,</mo><mi mathvariant="script">O</mi></mfenced></mrow></semantics></math></inline-formula> is proportional to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold-italic">R</mi><mfenced open="(" close=")"><mi>σ</mi></mfenced></mrow></semantics></math></inline-formula>. In an attempt to generalize the relation for arbitrary loops, we encounter the following ambiguity. For a given loop <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula> embedded in a manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">M</mi></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold-italic">H</mi><mfenced separators="" open="(" close=")"><mi>γ</mi><mo>,</mo><mi mathvariant="script">O</mi></mfenced></mrow></semantics></math></inline-formula> is an element of a Lie group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula>; the curvature <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold-italic">R</mi><mfenced open="(" close=")"><mi>σ</mi></mfenced><mo>∈</mo><mi mathvariant="fraktur">g</mi></mrow></semantics></math></inline-formula> is an element of the Lie algebra of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula>. However, it turns out that the curvature form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold-italic">R</mi><mfenced open="(" close=")"><mi>σ</mi></mfenced></mrow></semantics></math></inline-formula> obtained from the small loop approximation is ambiguous, as the information of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold-italic">H</mi><mfenced separators="" open="(" close=")"><mi>γ</mi><mo>,</mo><mi mathvariant="script">O</mi></mfenced></mrow></semantics></math></inline-formula> are insufficient for determining a specific plane <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula> responsible for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold-italic">R</mi><mfenced open="(" close=")"><mi>σ</mi></mfenced></mrow></semantics></math></inline-formula>. To resolve this ambiguity, it is necessary to specify the surface <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">S</mi></semantics></math></inline-formula> enclosed by the loop <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula>; hence, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula> is defined as the limit of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">S</mi></semantics></math></inline-formula> when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula> shrinks to a point. In this article, we try to understand this problem more clearly. As a result, we obtain an exact relation between the holonomy along a loop with the integral of the curvature form over a surface that it encloses. The derivation of this result can be viewed as an alternative proof of the non-Abelian Stokes theorem in two dimensions with some generalizations. |
| first_indexed | 2024-03-09T16:25:10Z |
| format | Article |
| id | doaj.art-00d419770a244d07ada1d2e9d9e34c8d |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T16:25:10Z |
| publishDate | 2023-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-00d419770a244d07ada1d2e9d9e34c8d2023-11-24T15:08:42ZengMDPI AGSymmetry2073-89942023-10-011511200010.3390/sym15112000Alternative Derivation of the Non-Abelian Stokes Theorem in Two DimensionsSeramika Ariwahjoedi0Freddy Permana Zen1Asia Pacific Center for Theoretical Physics, Pohang University of Science and Technology, Pohang 37673, Republic of KoreaTheoretical Physics Laboratory, THEPi Division, Department of Physics, Institut Teknologi Bandung, Jl. Ganesha 10, Bandung 40132, IndonesiaThe relation between the holonomy along a loop with the curvature form is a well-known fact, where the small square loop approximation of aholonomy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold-italic">H</mi><mfenced separators="" open="(" close=")"><mi>γ</mi><mo>,</mo><mi mathvariant="script">O</mi></mfenced></mrow></semantics></math></inline-formula> is proportional to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold-italic">R</mi><mfenced open="(" close=")"><mi>σ</mi></mfenced></mrow></semantics></math></inline-formula>. In an attempt to generalize the relation for arbitrary loops, we encounter the following ambiguity. For a given loop <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula> embedded in a manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">M</mi></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold-italic">H</mi><mfenced separators="" open="(" close=")"><mi>γ</mi><mo>,</mo><mi mathvariant="script">O</mi></mfenced></mrow></semantics></math></inline-formula> is an element of a Lie group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula>; the curvature <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold-italic">R</mi><mfenced open="(" close=")"><mi>σ</mi></mfenced><mo>∈</mo><mi mathvariant="fraktur">g</mi></mrow></semantics></math></inline-formula> is an element of the Lie algebra of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">G</mi></semantics></math></inline-formula>. However, it turns out that the curvature form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold-italic">R</mi><mfenced open="(" close=")"><mi>σ</mi></mfenced></mrow></semantics></math></inline-formula> obtained from the small loop approximation is ambiguous, as the information of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold-italic">H</mi><mfenced separators="" open="(" close=")"><mi>γ</mi><mo>,</mo><mi mathvariant="script">O</mi></mfenced></mrow></semantics></math></inline-formula> are insufficient for determining a specific plane <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula> responsible for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold-italic">R</mi><mfenced open="(" close=")"><mi>σ</mi></mfenced></mrow></semantics></math></inline-formula>. To resolve this ambiguity, it is necessary to specify the surface <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">S</mi></semantics></math></inline-formula> enclosed by the loop <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula>; hence, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula> is defined as the limit of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">S</mi></semantics></math></inline-formula> when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula> shrinks to a point. In this article, we try to understand this problem more clearly. As a result, we obtain an exact relation between the holonomy along a loop with the integral of the curvature form over a surface that it encloses. The derivation of this result can be viewed as an alternative proof of the non-Abelian Stokes theorem in two dimensions with some generalizations.https://www.mdpi.com/2073-8994/15/11/2000holonomycurvature 2-formloop contractionhomotopynon-Abelian Stokes theorem |
| spellingShingle | Seramika Ariwahjoedi Freddy Permana Zen Alternative Derivation of the Non-Abelian Stokes Theorem in Two Dimensions Symmetry holonomy curvature 2-form loop contraction homotopy non-Abelian Stokes theorem |
| title | Alternative Derivation of the Non-Abelian Stokes Theorem in Two Dimensions |
| title_full | Alternative Derivation of the Non-Abelian Stokes Theorem in Two Dimensions |
| title_fullStr | Alternative Derivation of the Non-Abelian Stokes Theorem in Two Dimensions |
| title_full_unstemmed | Alternative Derivation of the Non-Abelian Stokes Theorem in Two Dimensions |
| title_short | Alternative Derivation of the Non-Abelian Stokes Theorem in Two Dimensions |
| title_sort | alternative derivation of the non abelian stokes theorem in two dimensions |
| topic | holonomy curvature 2-form loop contraction homotopy non-Abelian Stokes theorem |
| url | https://www.mdpi.com/2073-8994/15/11/2000 |
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