Stochastic Levy Divergence and Maxwell's Equations
<p>One of the main reasons for interest in the Levy Laplacian and its analogues such as Levy d'Alembertian is a connection of these operators with gauge fields. The theorem proved by Accardi, Gibillisco and Volovich stated that a connection in a bundle over a Euclidean space or over a Min...
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| Format: | Article |
| Language: | Russian |
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MGTU im. N.È. Baumana
2015-01-01
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| Series: | Matematika i Matematičeskoe Modelirovanie |
| Subjects: | |
| Online Access: | http://mathm.elpub.ru/jour/article/view/29 |
| _version_ | 1818597869292093440 |
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| author | B. O. Volkov |
| author_facet | B. O. Volkov |
| author_sort | B. O. Volkov |
| collection | DOAJ |
| description | <p>One of the main reasons for interest in the Levy Laplacian and its analogues such as Levy d'Alembertian is a connection of these operators with gauge fields. The theorem proved by Accardi, Gibillisco and Volovich stated that a connection in a bundle over a Euclidean space or over a Minkowski space is a solution of the Yang-Mills equations if and only if the corresponding parallel transport to the connection is a solution of the Laplace equation for the Levy Laplacian or of the d'Alembert equation for the Levy d'Alembertian respectively (see [5, 6]). There are two approaches to define Levy type operators, both of which date back to the original works of Levy [7]. The first is that the Levy Laplacian (or Levy d'Alembertian) is defined as an integral functional generated by a special form of the second derivative. This approach is used in the works [5, 6], as well as in the paper [8] of Leandre and Volovich, where stochastic Levy-Laplacian is discussed. Another approach to the Levy Laplacian is defining it as the Cesaro mean of second order derivatives along the family of vectors, which is an orthonormal basis in the Hilbert space. This definition of the Levy Laplacian is used for the description of solutions of the Yang-Mills equations in the paper [10].</p><p>The present work shows that the definitions of the Levy Laplacian and the Levy d'Alembertian based on Cesaro averaging of the second order directional derivatives can be transferred to the stochastic case. In the article the values of these operators on a stochastic parallel transport associated with a connection (vector potential) are found. In this case, unlike the deterministic case and the stochastic case of Levy Laplacian from [8], these values are not equal to zero if the vector potential corresponding to the stochastic parallel transport is a solution of the Maxwell's equations. As a result, two approaches to definition of the Levy Laplacian in the stochastic case give different operators. This situation is different from the flat deterministic case, which is discussed in [11]. It can be expected that the work can be summarized in the non-commutative case of the Yang-Mills theory</p> |
| first_indexed | 2024-12-16T11:54:39Z |
| format | Article |
| id | doaj.art-348546ae8faa44609bd9ea65198e9374 |
| institution | Directory Open Access Journal |
| issn | 2412-5911 |
| language | Russian |
| last_indexed | 2024-12-16T11:54:39Z |
| publishDate | 2015-01-01 |
| publisher | MGTU im. N.È. Baumana |
| record_format | Article |
| series | Matematika i Matematičeskoe Modelirovanie |
| spelling | doaj.art-348546ae8faa44609bd9ea65198e93742022-12-21T22:32:35ZrusMGTU im. N.È. BaumanaMatematika i Matematičeskoe Modelirovanie2412-59112015-01-010511628Stochastic Levy Divergence and Maxwell's EquationsB. O. Volkov0Bauman Moscow State Technical University; Steklov Mathematical Institute of RAS<p>One of the main reasons for interest in the Levy Laplacian and its analogues such as Levy d'Alembertian is a connection of these operators with gauge fields. The theorem proved by Accardi, Gibillisco and Volovich stated that a connection in a bundle over a Euclidean space or over a Minkowski space is a solution of the Yang-Mills equations if and only if the corresponding parallel transport to the connection is a solution of the Laplace equation for the Levy Laplacian or of the d'Alembert equation for the Levy d'Alembertian respectively (see [5, 6]). There are two approaches to define Levy type operators, both of which date back to the original works of Levy [7]. The first is that the Levy Laplacian (or Levy d'Alembertian) is defined as an integral functional generated by a special form of the second derivative. This approach is used in the works [5, 6], as well as in the paper [8] of Leandre and Volovich, where stochastic Levy-Laplacian is discussed. Another approach to the Levy Laplacian is defining it as the Cesaro mean of second order derivatives along the family of vectors, which is an orthonormal basis in the Hilbert space. This definition of the Levy Laplacian is used for the description of solutions of the Yang-Mills equations in the paper [10].</p><p>The present work shows that the definitions of the Levy Laplacian and the Levy d'Alembertian based on Cesaro averaging of the second order directional derivatives can be transferred to the stochastic case. In the article the values of these operators on a stochastic parallel transport associated with a connection (vector potential) are found. In this case, unlike the deterministic case and the stochastic case of Levy Laplacian from [8], these values are not equal to zero if the vector potential corresponding to the stochastic parallel transport is a solution of the Maxwell's equations. As a result, two approaches to definition of the Levy Laplacian in the stochastic case give different operators. This situation is different from the flat deterministic case, which is discussed in [11]. It can be expected that the work can be summarized in the non-commutative case of the Yang-Mills theory</p>http://mathm.elpub.ru/jour/article/view/29Levy LaplacianLevy DivergenceMaxwell's equationsstochastic parallel transportchiral fields |
| spellingShingle | B. O. Volkov Stochastic Levy Divergence and Maxwell's Equations Matematika i Matematičeskoe Modelirovanie Levy Laplacian Levy Divergence Maxwell's equations stochastic parallel transport chiral fields |
| title | Stochastic Levy Divergence and Maxwell's Equations |
| title_full | Stochastic Levy Divergence and Maxwell's Equations |
| title_fullStr | Stochastic Levy Divergence and Maxwell's Equations |
| title_full_unstemmed | Stochastic Levy Divergence and Maxwell's Equations |
| title_short | Stochastic Levy Divergence and Maxwell's Equations |
| title_sort | stochastic levy divergence and maxwell s equations |
| topic | Levy Laplacian Levy Divergence Maxwell's equations stochastic parallel transport chiral fields |
| url | http://mathm.elpub.ru/jour/article/view/29 |
| work_keys_str_mv | AT bovolkov stochasticlevydivergenceandmaxwellsequations |