G. Bergeron, Nicolas Cramp'e, S. Tsujimoto, L. Vinet, A. Zhedanov
This paper introduces and studies the Heun-Racah and Heun-Bannai-Ito algebras abstractly and establishes the relation between these new algebraic structures and generalized Heun-type operators derived from the notion of algebraic Heun operators in the case of the Racah and Bannai-Ito algebras.
{"title":"The Heun–Racah and Heun–Bannai–Ito algebras","authors":"G. Bergeron, Nicolas Cramp'e, S. Tsujimoto, L. Vinet, A. Zhedanov","doi":"10.1063/5.0008372","DOIUrl":"https://doi.org/10.1063/5.0008372","url":null,"abstract":"This paper introduces and studies the Heun-Racah and Heun-Bannai-Ito algebras abstractly and establishes the relation between these new algebraic structures and generalized Heun-type operators derived from the notion of algebraic Heun operators in the case of the Racah and Bannai-Ito algebras.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"66 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84039030","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}
There is a correspondence between integrable lattice models of statistical mechanics and discrete integrable equations which satisfy multidimensional consistency, where the latter may be found in a quasi-classical expansion of the former. This paper extends this correspondence to interaction-round-a-face (IRF) models, resulting in a new formulation of the consistency-around-a-cube (CAC) integrability condition that is applicable to five-point equations which are defined on a vertex and its four nearest-neighbours in the square lattice. Multidimensional consistency for these equations is formulated as consistency-around-a-face-centered-cube (CAFCC), which namely involves satisfying an overdetermined system of fourteen five-point lattice equations for eight unknown variables on the face-centered cubic unit cell. From the quasi-classical limit of IRF models, which are constructed from the continuous spin solutions of the star-triangle relations associated to the Adler-Bobenko-Suris (ABS) list, fifteen sets of equations are obtained which satisfy CAFCC.
{"title":"Interaction-round-a-face and consistency-around-a-face-centered-cube","authors":"A. P. Kels","doi":"10.1063/5.0024630","DOIUrl":"https://doi.org/10.1063/5.0024630","url":null,"abstract":"There is a correspondence between integrable lattice models of statistical mechanics and discrete integrable equations which satisfy multidimensional consistency, where the latter may be found in a quasi-classical expansion of the former. This paper extends this correspondence to interaction-round-a-face (IRF) models, resulting in a new formulation of the consistency-around-a-cube (CAC) integrability condition that is applicable to five-point equations which are defined on a vertex and its four nearest-neighbours in the square lattice. Multidimensional consistency for these equations is formulated as consistency-around-a-face-centered-cube (CAFCC), which namely involves satisfying an overdetermined system of fourteen five-point lattice equations for eight unknown variables on the face-centered cubic unit cell. From the quasi-classical limit of IRF models, which are constructed from the continuous spin solutions of the star-triangle relations associated to the Adler-Bobenko-Suris (ABS) list, fifteen sets of equations are obtained which satisfy CAFCC.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"76 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79967436","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}
Pub Date : 2020-03-19DOI: 10.1017/9781009320733.006
Michael S. Foskett, C. Tronci
In this paper we consider a new geometric approach to Madelung's quantum hydrodynamics (QHD) based on the theory of gauge connections. Unlike previous approaches, our treatment comprises a constant curvature thereby endowing QHD with intrinsic non-zero holonomy. In the hydrodynamic context, this leads to a fluid velocity which no longer is constrained to be irrotational and allows instead for vortex filaments solutions. After exploiting the Rasetti-Regge method to couple the Schrodinger equation to vortex filament dynamics, the latter is then considered as a source of geometric phase in the context of Born-Oppenheimer molecular dynamics. Similarly, we consider the Pauli equation for the motion of spin particles in electromagnetic fields and we exploit its underlying hydrodynamic picture to include vortex dynamics.
{"title":"Holonomy and vortex structures in quantum hydrodynamics","authors":"Michael S. Foskett, C. Tronci","doi":"10.1017/9781009320733.006","DOIUrl":"https://doi.org/10.1017/9781009320733.006","url":null,"abstract":"In this paper we consider a new geometric approach to Madelung's quantum hydrodynamics (QHD) based on the theory of gauge connections. Unlike previous approaches, our treatment comprises a constant curvature thereby endowing QHD with intrinsic non-zero holonomy. In the hydrodynamic context, this leads to a fluid velocity which no longer is constrained to be irrotational and allows instead for vortex filaments solutions. After exploiting the Rasetti-Regge method to couple the Schrodinger equation to vortex filament dynamics, the latter is then considered as a source of geometric phase in the context of Born-Oppenheimer molecular dynamics. Similarly, we consider the Pauli equation for the motion of spin particles in electromagnetic fields and we exploit its underlying hydrodynamic picture to include vortex dynamics.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":" 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141221575","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}
Pub Date : 2020-03-18DOI: 10.32523/2306-6172-2018-6-3-4-33
M. Belishev, A. Kaplun
An eikonal algebra ${mathfrak E}(Omega)$ is a C*-algebra related to a metric graph $Omega$. It is determined by trajectories and reachable sets of a dynamical system associated with the graph. The system describes the waves, which are initiated by boundary sources (controls) and propagate into the graph with finite velocity. Motivation and interest to eikonal algebras comes from the inverse problem of reconstruction of the graph via its dynamical and/or spectral boundary data. Algebra ${mathfrak E}(Omega)$ is determined by these data. In the mean time, its structure and algebraic invariants (irreducible representations) are connected with topology of $Omega$. We demonstrate such connections and study ${mathfrak E}(Omega)$ by the example of $Omega$ of a simple structure. Hopefully, in future, these connections will provide an approach to reconstruction.
{"title":"Eikonal algebra on a graph of simple structure","authors":"M. Belishev, A. Kaplun","doi":"10.32523/2306-6172-2018-6-3-4-33","DOIUrl":"https://doi.org/10.32523/2306-6172-2018-6-3-4-33","url":null,"abstract":"An eikonal algebra ${mathfrak E}(Omega)$ is a C*-algebra related to a metric graph $Omega$. It is determined by trajectories and reachable sets of a dynamical system associated with the graph. The system describes the waves, which are initiated by boundary sources (controls) and propagate into the graph with finite velocity. Motivation and interest to eikonal algebras comes from the inverse problem of reconstruction of the graph via its dynamical and/or spectral boundary data. Algebra ${mathfrak E}(Omega)$ is determined by these data. In the mean time, its structure and algebraic invariants (irreducible representations) are connected with topology of $Omega$. We demonstrate such connections and study ${mathfrak E}(Omega)$ by the example of $Omega$ of a simple structure. Hopefully, in future, these connections will provide an approach to reconstruction.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"37 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89613833","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 provide a proof of the first correction to the leading asymptotics of the minimal energy of pseudo-relativistic molecules in the presence of magnetic fields, the so-called "relativistic Scott correction", when $max{Z_kalpha} leq 2/pi$, where $Z_k$ is the charge of the $k$-th nucleus and $alpha$ is the fine structure constant. Our theorem extends a previous result by Erdős, Fournais, and Solovej to the critical constant $2/pi$ in the relativistic Hardy inequality $|p| - frac{2}{pi |x|} geq 0$.
{"title":"The magnetic Scott correction for relativistic matter at criticality","authors":"Gonzalo A. Bley, S. Fournais","doi":"10.1063/5.0007903","DOIUrl":"https://doi.org/10.1063/5.0007903","url":null,"abstract":"We provide a proof of the first correction to the leading asymptotics of the minimal energy of pseudo-relativistic molecules in the presence of magnetic fields, the so-called \"relativistic Scott correction\", when $max{Z_kalpha} leq 2/pi$, where $Z_k$ is the charge of the $k$-th nucleus and $alpha$ is the fine structure constant. Our theorem extends a previous result by Erdős, Fournais, and Solovej to the critical constant $2/pi$ in the relativistic Hardy inequality $|p| - frac{2}{pi |x|} geq 0$.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83563726","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}
Pub Date : 2020-03-11DOI: 10.1142/S0219887820502060
M. Trindade, S. Floquet, J. Vianna
In this work we explore the structure of Clifford algebras and the representations of the algebraic spinors in quantum information theory. Initially we present an general formulation through elements of left minimal ideals in tensor products of the Clifford algebra $Cl^{+}_{1,3}$. Posteriorly we perform some applications in quantum computation: qubits, entangled states, quantum gates, representations of the braid group, quantum teleportation, Majorana operators and supersymmetry. Finally, we discuss advantages related to standard Hilbert space formulation.
{"title":"A general formulation based on algebraic spinors for the quantum computation","authors":"M. Trindade, S. Floquet, J. Vianna","doi":"10.1142/S0219887820502060","DOIUrl":"https://doi.org/10.1142/S0219887820502060","url":null,"abstract":"In this work we explore the structure of Clifford algebras and the representations of the algebraic spinors in quantum information theory. Initially we present an general formulation through elements of left minimal ideals in tensor products of the Clifford algebra $Cl^{+}_{1,3}$. Posteriorly we perform some applications in quantum computation: qubits, entangled states, quantum gates, representations of the braid group, quantum teleportation, Majorana operators and supersymmetry. Finally, we discuss advantages related to standard Hilbert space formulation.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"197 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75539295","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}
Pub Date : 2020-03-09DOI: 10.21468/scipostphys.9.6.086
J. Maillet, G. Niccoli, L. Vignoli
Using the framework of the quantum separation of variables (SoV) for higher rank quantum integrable lattice models [1], we introduce some foundations to go beyond the obtained complete transfer matrix spectrum description, and open the way to the computation of matrix elements of local operators. This first amounts to obtain simple expressions for scalar products of the so-called separate states (transfer matrix eigenstates or some simple generalization of them). In the higher rank case, left and right SoV bases are expected to be pseudo-orthogonal, that is for a given SoV co-vector, there could be more than one non-vanishing overlap with the vectors of the chosen right SoV basis. For simplicity, we describe our method to get these pseudo-orthogonality overlaps in the fundamental representations of the $mathcal{Y}(gl_3)$ lattice model with $N$ sites, a case of rank 2. The non-zero couplings between the co-vector and vector SoV bases are exactly characterized. While the corresponding SoV-measure stays reasonably simple and of possible practical use, we address the problem of constructing left and right SoV bases which do satisfy standard orthogonality. In our approach, the SoV bases are constructed by using families of conserved charges. This gives us a large freedom in the SoV bases construction, and allows us to look for the choice of a family of conserved charges which leads to orthogonal co-vector/vector SoV bases. We first define such a choice in the case of twist matrices having simple spectrum and zero determinant. Then, we generalize the associated family of conserved charges and orthogonal SoV bases to generic simple spectrum and invertible twist matrices. Under this choice of conserved charges, and of the associated orthogonal SoV bases, the scalar products of separate states simplify considerably and take a form similar to the $mathcal{Y}(gl_2)$ rank one case.
{"title":"On scalar products in higher rank quantum separation of variables","authors":"J. Maillet, G. Niccoli, L. Vignoli","doi":"10.21468/scipostphys.9.6.086","DOIUrl":"https://doi.org/10.21468/scipostphys.9.6.086","url":null,"abstract":"Using the framework of the quantum separation of variables (SoV) for higher rank quantum integrable lattice models [1], we introduce some foundations to go beyond the obtained complete transfer matrix spectrum description, and open the way to the computation of matrix elements of local operators. This first amounts to obtain simple expressions for scalar products of the so-called separate states (transfer matrix eigenstates or some simple generalization of them). In the higher rank case, left and right SoV bases are expected to be pseudo-orthogonal, that is for a given SoV co-vector, there could be more than one non-vanishing overlap with the vectors of the chosen right SoV basis. For simplicity, we describe our method to get these pseudo-orthogonality overlaps in the fundamental representations of the $mathcal{Y}(gl_3)$ lattice model with $N$ sites, a case of rank 2. The non-zero couplings between the co-vector and vector SoV bases are exactly characterized. While the corresponding SoV-measure stays reasonably simple and of possible practical use, we address the problem of constructing left and right SoV bases which do satisfy standard orthogonality. In our approach, the SoV bases are constructed by using families of conserved charges. This gives us a large freedom in the SoV bases construction, and allows us to look for the choice of a family of conserved charges which leads to orthogonal co-vector/vector SoV bases. We first define such a choice in the case of twist matrices having simple spectrum and zero determinant. Then, we generalize the associated family of conserved charges and orthogonal SoV bases to generic simple spectrum and invertible twist matrices. Under this choice of conserved charges, and of the associated orthogonal SoV bases, the scalar products of separate states simplify considerably and take a form similar to the $mathcal{Y}(gl_2)$ rank one case.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82288987","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}
A four-wave mixing Hamiltonian system on the classical as well as on the quantum level is investigated. In the classical case, if one assumes the frequency resonance condition of the form $omega_0 -omega_1 +omega_2 -omega_3=0$, this Hamiltonian system is integrated in quadratures and the explicit formulas of solutions are presented. Under the same condition the spectral decomposition of quantum Hamiltonian is found and thus, the Heisenberg equation for this system is solved. Some applications of the obtained results in non-linear optics are disscused.
{"title":"An integrable (classical and quantum) four-wave mixing Hamiltonian system","authors":"A. Odzijewicz, E. Wawreniuk","doi":"10.1063/5.0006887","DOIUrl":"https://doi.org/10.1063/5.0006887","url":null,"abstract":"A four-wave mixing Hamiltonian system on the classical as well as on the quantum level is investigated. In the classical case, if one assumes the frequency resonance condition of the form $omega_0 -omega_1 +omega_2 -omega_3=0$, this Hamiltonian system is integrated in quadratures and the explicit formulas of solutions are presented. Under the same condition the spectral decomposition of quantum Hamiltonian is found and thus, the Heisenberg equation for this system is solved. Some applications of the obtained results in non-linear optics are disscused.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85257407","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}
Pub Date : 2020-03-06DOI: 10.1215/00192082-8746137
David Hobby, E. Shemyakova
We identify conditions giving large natural classes of partial differential operators for which it is possible to construct a complete set of Laplace invariants. In order to do that we investigate general properties of differential invariants of partial differential operators under gauge transformations and introduce a sufficient condition for a set of invariants to be complete. We also give a some mild conditions that guarantee the existence of such a set. The proof is constructive. The method gives many examples of invariants previously known in the literature as well as many new examples including multidimensional.
{"title":"Laplace invariants of differential operators","authors":"David Hobby, E. Shemyakova","doi":"10.1215/00192082-8746137","DOIUrl":"https://doi.org/10.1215/00192082-8746137","url":null,"abstract":"We identify conditions giving large natural classes of partial differential operators for which it is possible to construct a complete set of Laplace invariants. In order to do that we investigate general properties of differential invariants of partial differential operators under gauge transformations and introduce a sufficient condition for a set of invariants to be complete. We also give a some mild conditions that guarantee the existence of such a set. The proof is constructive. The method gives many examples of invariants previously known in the literature as well as many new examples including multidimensional.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74833615","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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