We study the Hamiltonian of two isotropic oscillators with weak coupling from an algebraic approach. We write the Hamiltonian of this problem in terms of the boson generators of the $SU(1,1)$ and $SU(2)$ groups. This allows us to apply two tilting transformations based on both group similarity transformations to obtain its energy spectrum and eigenfunctions. Then, we obtain the Mandel $Q-$parameter of the photon numbers $n_a$ and $n_b$. It is important to note that in our procedure we consider the case of weak coupling.
{"title":"$SU(1,1)times SU(2)$ approach and the Mandel parameter to the Hamiltonian of two oscillators with weak coupling","authors":"J. C. Vega, D. Ojeda-Guillén, R. D. Mota","doi":"arxiv-2409.08179","DOIUrl":"https://doi.org/arxiv-2409.08179","url":null,"abstract":"We study the Hamiltonian of two isotropic oscillators with weak coupling from\u0000an algebraic approach. We write the Hamiltonian of this problem in terms of the\u0000boson generators of the $SU(1,1)$ and $SU(2)$ groups. This allows us to apply\u0000two tilting transformations based on both group similarity transformations to\u0000obtain its energy spectrum and eigenfunctions. Then, we obtain the Mandel\u0000$Q-$parameter of the photon numbers $n_a$ and $n_b$. It is important to note\u0000that in our procedure we consider the case of weak coupling.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"142 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In order to define the complex exceptional Lie groups $ {F_4}^C, {E_6}^C, {E_7}^C, {E_8}^C $ and these compact real forms $ F_4,E_6,E_7,E_8 $, we usually use the Cayley algebra $ mathfrak{C} $. In the present article, we consider replacing the Cayley algebra $ mathfrak{C} $ with the field of real numbers $mathbb R$ in the definition of the groups above, and these groups are denoted as in title above. Our aim is to determine the structure of these groups. We call realization to determine the structure of the groups.
{"title":"On realizations of the complex Lie groups $ (F_{4,R})^C, (E_{6,R})^C, (E_{7,R})^C ,(E_{8,R})^C$ and those compact real forms $F_{4,R},E_{6,R},E_{7,R},E_{8,R}$","authors":"Toshikazu Miyashita","doi":"arxiv-2409.07760","DOIUrl":"https://doi.org/arxiv-2409.07760","url":null,"abstract":"In order to define the complex exceptional Lie groups $ {F_4}^C, {E_6}^C,\u0000{E_7}^C, {E_8}^C $ and these compact real forms $ F_4,E_6,E_7,E_8 $, we usually\u0000use the Cayley algebra $ mathfrak{C} $. In the present article, we consider\u0000replacing the Cayley algebra $ mathfrak{C} $ with the field of real numbers\u0000$mathbb R$ in the definition of the groups above, and these groups are denoted\u0000as in title above. Our aim is to determine the structure of these groups. We\u0000call realization to determine the structure of the groups.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"283 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216327","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study infinite dimensional Lie algebras, whose infinite dimensional mutually commuting subalgebras correspond with the symmetry algebra of $2d$ integrable models. These Lie algebras are defined by the set of infinitesimal, nonlinear, and higher derivative symmetry transformations present in theories with a left(right)-moving or (anti)-holomorphic current. We study a large class of such Lagrangian theories. We classify the commuting subalgebras of the $2d$ free massless scalar, and find the symmetries of the known integrable models such as sine-Gordon, Liouville, Bullough-Dodd, and Korteweg-de Vries. Along the way, we find several new sequences of commuting charges, which we conjecture are charges of integrable models which are new deformations of a single scalar. After quantizing, the Lie algebra is deformed, and so are their commuting subalgebras.
{"title":"On the space of $2d$ integrable models","authors":"Lukas W. Lindwasser","doi":"arxiv-2409.08266","DOIUrl":"https://doi.org/arxiv-2409.08266","url":null,"abstract":"We study infinite dimensional Lie algebras, whose infinite dimensional\u0000mutually commuting subalgebras correspond with the symmetry algebra of $2d$\u0000integrable models. These Lie algebras are defined by the set of infinitesimal,\u0000nonlinear, and higher derivative symmetry transformations present in theories\u0000with a left(right)-moving or (anti)-holomorphic current. We study a large class\u0000of such Lagrangian theories. We classify the commuting subalgebras of the $2d$\u0000free massless scalar, and find the symmetries of the known integrable models\u0000such as sine-Gordon, Liouville, Bullough-Dodd, and Korteweg-de Vries. Along the\u0000way, we find several new sequences of commuting charges, which we conjecture\u0000are charges of integrable models which are new deformations of a single scalar.\u0000After quantizing, the Lie algebra is deformed, and so are their commuting\u0000subalgebras.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216325","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Manuel de León, Jordi Gaset Rifà, Miguel C. Muñoz-Lecanda, Xavier Rivas, Narciso Román-Roy
Action-dependent field theories are systems where the Lagrangian or Hamiltonian depends on new variables that encode the action. After a friendly introduction, we make a quick presentation of a new mathematical framework for action-dependent field theory: multicontact geometry. The formalism is illustrated in a variety of action-dependent Lagrangians, some of which are regular and other singular, that come from some well-known theories whose Lagrangians have been modified to incorporate action-dependent terms. Detailed computations are provided, including the constraint algorithm, in both the Lagrangian and Hamiltonian formalisms. They include the Klein-Gordon equation, the Bosonic string theory, Metric-affine gravity, Maxwell's electromagnetism, (2+1)-dimensional gravity and Chern-Simons equation, and the heat equation and Burgers' equation.
{"title":"Practical Introduction to Action-Dependent Field Theories","authors":"Manuel de León, Jordi Gaset Rifà, Miguel C. Muñoz-Lecanda, Xavier Rivas, Narciso Román-Roy","doi":"arxiv-2409.08340","DOIUrl":"https://doi.org/arxiv-2409.08340","url":null,"abstract":"Action-dependent field theories are systems where the Lagrangian or\u0000Hamiltonian depends on new variables that encode the action. After a friendly\u0000introduction, we make a quick presentation of a new mathematical framework for\u0000action-dependent field theory: multicontact geometry. The formalism is\u0000illustrated in a variety of action-dependent Lagrangians, some of which are\u0000regular and other singular, that come from some well-known theories whose\u0000Lagrangians have been modified to incorporate action-dependent terms. Detailed\u0000computations are provided, including the constraint algorithm, in both the\u0000Lagrangian and Hamiltonian formalisms. They include the Klein-Gordon equation,\u0000the Bosonic string theory, Metric-affine gravity, Maxwell's electromagnetism,\u0000(2+1)-dimensional gravity and Chern-Simons equation, and the heat equation and\u0000Burgers' equation.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260526","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We explore the properties of fermionic convolution defined by fermionic Gaussian unitary. A key finding is the purity invariance of pure Gaussian states under this convolution. Leveraging this property, we propose an efficient protocol to test the fermionic Gaussianity of pure states by using 3 copies of the input states. Furthermore, we introduce a new family of measures called ``Non-Gaussian Entropy,'' designed to quantify the fermionic non-Gaussianity of states.
{"title":"Fermionic Gaussian Testing and Non-Gaussian Measures via Convolution","authors":"Nicholas Lyu, Kaifeng Bu","doi":"arxiv-2409.08180","DOIUrl":"https://doi.org/arxiv-2409.08180","url":null,"abstract":"We explore the properties of fermionic convolution defined by fermionic\u0000Gaussian unitary. A key finding is the purity invariance of pure Gaussian\u0000states under this convolution. Leveraging this property, we propose an\u0000efficient protocol to test the fermionic Gaussianity of pure states by using 3\u0000copies of the input states. Furthermore, we introduce a new family of measures\u0000called ``Non-Gaussian Entropy,'' designed to quantify the fermionic\u0000non-Gaussianity of states.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the exterior solution for a static, spherically symmetric source in Weyl conformal gravity in terms of the Newman--Penrose formalism. We first show that both the static, uncharged black hole solution of Mannheim and Kazanas and the static, charged Reissner--Nordstr"{o}m-like solution can be found more easily in this formalism than in the traditional coordinate-basis approach, where the metric tensor components are taken as the basic variables. Second, we show that the Newman-Penrose formalism offers a particularly convenient framework that is well suited for the discussion of conformal gravity solutions corresponding to Petrov ''type-D'' spacetimes. This is illustrated with a two-parameter class of wormhole solutions that includes the Ellis--Bronnikov wormhole solution of Einstein's gravity as a limiting case. Other salient issues, such as the gauge equivalence of solutions and the inclusion of the electromagnetic field are also discussed.
{"title":"Newman-Penrose formalism and exact vacuum solutions to conformal Weyl gravity","authors":"Petr Jizba, Kamil Mudruňka","doi":"arxiv-2409.08344","DOIUrl":"https://doi.org/arxiv-2409.08344","url":null,"abstract":"We study the exterior solution for a static, spherically symmetric source in\u0000Weyl conformal gravity in terms of the Newman--Penrose formalism. We first show\u0000that both the static, uncharged black hole solution of Mannheim and Kazanas and\u0000the static, charged Reissner--Nordstr\"{o}m-like solution can be found more\u0000easily in this formalism than in the traditional coordinate-basis approach,\u0000where the metric tensor components are taken as the basic variables. Second, we\u0000show that the Newman-Penrose formalism offers a particularly convenient\u0000framework that is well suited for the discussion of conformal gravity solutions\u0000corresponding to Petrov ''type-D'' spacetimes. This is illustrated with a\u0000two-parameter class of wormhole solutions that includes the Ellis--Bronnikov\u0000wormhole solution of Einstein's gravity as a limiting case. Other salient\u0000issues, such as the gauge equivalence of solutions and the inclusion of the\u0000electromagnetic field are also discussed.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Vladimir Dorodnitsyn, Roman Kozlov, Sergey Meleshko
The paper suggests a Hamiltonian formulation for delay ordinary differential equations (DODEs). Such equations are related to DODEs with a Lagrangian formulation via a delay analog of the Legendre transformation. The Hamiltonian delay operator identity is established. It states the relationship for the invariance of a delay Hamiltonian functional, appropriate delay variational equations, and their conserved quantities. The identity is used to formulate a Noether-type theorem, which provides first integrals for Hamiltonian DODEs with symmetries. The relationship between the invariance of the delay Hamiltonian functional and the invariance of the delay variational equations is also examined. Several examples illustrate the theoretical results.
{"title":"Delay ordinary differential equations: from Lagrangian approach to Hamiltonian approach","authors":"Vladimir Dorodnitsyn, Roman Kozlov, Sergey Meleshko","doi":"arxiv-2409.08165","DOIUrl":"https://doi.org/arxiv-2409.08165","url":null,"abstract":"The paper suggests a Hamiltonian formulation for delay ordinary differential\u0000equations (DODEs). Such equations are related to DODEs with a Lagrangian\u0000formulation via a delay analog of the Legendre transformation. The Hamiltonian\u0000delay operator identity is established. It states the relationship for the\u0000invariance of a delay Hamiltonian functional, appropriate delay variational\u0000equations, and their conserved quantities. The identity is used to formulate a\u0000Noether-type theorem, which provides first integrals for Hamiltonian DODEs with\u0000symmetries. The relationship between the invariance of the delay Hamiltonian\u0000functional and the invariance of the delay variational equations is also\u0000examined. Several examples illustrate the theoretical results.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216321","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper initiates a systematic study of operators arising as integrals of operator-valued functions with respect to positive operator-valued measures and utilizes these tools to provide relativization maps (Yen) for quantum reference frames (QRFs) defined on general homogeneous spaces. Properties of operator-valued integration are first studied and then employed to define general relativization maps and show their properties. The relativization maps presented here are defined for QRFs (systems of covariance) based on arbitrary homogeneous spaces of locally compact second countable topological groups and are shown to be contracting quantum channels, injective for localizable (norm-1 property) frames and multiplicative for the sharp ones (PVMs), extending the existing results.
{"title":"Quantum Reference Frames on Homogeneous Spaces","authors":"Jan Głowacki","doi":"arxiv-2409.07231","DOIUrl":"https://doi.org/arxiv-2409.07231","url":null,"abstract":"This paper initiates a systematic study of operators arising as integrals of\u0000operator-valued functions with respect to positive operator-valued measures and\u0000utilizes these tools to provide relativization maps (Yen) for quantum reference\u0000frames (QRFs) defined on general homogeneous spaces. Properties of\u0000operator-valued integration are first studied and then employed to define\u0000general relativization maps and show their properties. The relativization maps\u0000presented here are defined for QRFs (systems of covariance) based on arbitrary\u0000homogeneous spaces of locally compact second countable topological groups and\u0000are shown to be contracting quantum channels, injective for localizable (norm-1\u0000property) frames and multiplicative for the sharp ones (PVMs), extending the\u0000existing results.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"50 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216331","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Hari K. Kunduri, Juan Margalef-Bentabol, Sarah Muth
We establish the spacetime Penrose inequality in spherical symmetry in spacetime dimensions $n+1geq3$ with charge and cosmological constant from the initial data perspective. We also show that this result extends to the Gauss-Bonnet theory of gravity.
我们从初始数据的角度建立了球对称时空维$n+1geq3$中的彭罗斯不等式，它带有电荷和宇宙学常数。我们还证明了这一结果可以扩展到高斯-波奈引力理论（Gauss-Bonnet theory of gravity）。
{"title":"The Penrose inequality in spherical symmetry with charge and in Gauss-Bonnet gravity","authors":"Hari K. Kunduri, Juan Margalef-Bentabol, Sarah Muth","doi":"arxiv-2409.07639","DOIUrl":"https://doi.org/arxiv-2409.07639","url":null,"abstract":"We establish the spacetime Penrose inequality in spherical symmetry in\u0000spacetime dimensions $n+1geq3$ with charge and cosmological constant from the\u0000initial data perspective. We also show that this result extends to the\u0000Gauss-Bonnet theory of gravity.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Context: Thermal conductivity provides important contributions to the energy evolution of the upper solar atmosphere, behaving as a non-linear concentration-dependent diffusion equation. Recently, different methods have been offered as best-fit solutions to these problems in specific situations, but their effectiveness and limitations are rarely discussed. Aims. We rigorously test the different implementations of solving the conductivity flux, in the massively-parallel magnetohydrodynamics code, Bifrost, with the aim of specifying the best scenarios for the use of each method. Methods: We compare the differences and limitations of explicit versus implicit methods, and analyse the convergence of a hyperbolic approximation. Among the tests, we use a newly derived 1st-order self-similar approximation to compare the efficacy of each method analytically in a 1D pure-thermal test scenario. Results: We find that although the hyperbolic approximation proves the most accurate and the fastest to compute in long-running simulations, there is no optimal method to calculate the mid-term conductivity with both accuracy and efficiency. We also find that the solution of this approximation is sensitive to the initial conditions, and can lead to faster convergence if used correctly. Hyper-diffusivity is particularly useful in aiding the methods to perform optimally. Conclusions: We discuss recommendations for the use of each method within more complex simulations, whilst acknowledging the areas of opportunity for new methods to be developed.
{"title":"Implementation of thermal conduction energy transfer models in the Bifrost Solar atmosphere MHD code","authors":"George Cherry, Boris Gudiksen, Mikolaj Szydlarski","doi":"arxiv-2409.07074","DOIUrl":"https://doi.org/arxiv-2409.07074","url":null,"abstract":"Context: Thermal conductivity provides important contributions to the energy\u0000evolution of the upper solar atmosphere, behaving as a non-linear\u0000concentration-dependent diffusion equation. Recently, different methods have\u0000been offered as best-fit solutions to these problems in specific situations,\u0000but their effectiveness and limitations are rarely discussed. Aims. We\u0000rigorously test the different implementations of solving the conductivity flux,\u0000in the massively-parallel magnetohydrodynamics code, Bifrost, with the aim of\u0000specifying the best scenarios for the use of each method. Methods: We compare\u0000the differences and limitations of explicit versus implicit methods, and\u0000analyse the convergence of a hyperbolic approximation. Among the tests, we use\u0000a newly derived 1st-order self-similar approximation to compare the efficacy of\u0000each method analytically in a 1D pure-thermal test scenario. Results: We find\u0000that although the hyperbolic approximation proves the most accurate and the\u0000fastest to compute in long-running simulations, there is no optimal method to\u0000calculate the mid-term conductivity with both accuracy and efficiency. We also\u0000find that the solution of this approximation is sensitive to the initial\u0000conditions, and can lead to faster convergence if used correctly.\u0000Hyper-diffusivity is particularly useful in aiding the methods to perform\u0000optimally. Conclusions: We discuss recommendations for the use of each method\u0000within more complex simulations, whilst acknowledging the areas of opportunity\u0000for new methods to be developed.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"50 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142216361","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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