Pub Date : 2019-01-01DOI: 10.7546/giq-20-2019-65-78
C. Acatrinei
A discretization scheme provided by the noncommutativity of space is reviewed. In the representation chosen here the radial coordinate is rendered discrete, allowing fields to be put on a lattice in a natural way. Noncommutativity is traded for a controllable type of nonlocality of the field dynamics, which in turn allows fermions to be free of lattice artefacts. Exact, singularity-free solutions are found interpreted, and their continuum limit is well-defined. MSC : 33E20, 39A12, 33C80, 33C45, 05A10
{"title":"Discretization in Noncommutative Field Theory","authors":"C. Acatrinei","doi":"10.7546/giq-20-2019-65-78","DOIUrl":"https://doi.org/10.7546/giq-20-2019-65-78","url":null,"abstract":"A discretization scheme provided by the noncommutativity of space is reviewed. In the representation chosen here the radial coordinate is rendered discrete, allowing fields to be put on a lattice in a natural way. Noncommutativity is traded for a controllable type of nonlocality of the field dynamics, which in turn allows fermions to be free of lattice artefacts. Exact, singularity-free solutions are found interpreted, and their continuum limit is well-defined. MSC : 33E20, 39A12, 33C80, 33C45, 05A10","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79279654","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 : 2019-01-01DOI: 10.7546/giq-20-2019-88-98
M. Arminjon
In the electrodynamics of special relativity (SR) or general relativity (GR), the energy tensors of the charged medium and its EM field add to give the total energy tensor that obeys the dynamical equation without external force. In the investigated scalar theory of gravitation ("SET"), this assumption leads to charge non-conservation, hence an additional, "interaction" energy tensor T inter has to be postulated. The present work aims at constraining this tensor. First we study the independent equations of electrodynamics and their number, beginning with SR and GR. As in SR and GR, the system of electrodynamics of SET is closed in the absence of T inter. Hence, with T inter , at least one additional equation must be provided. This is done by assuming that T inter is Lorentz-invariant in the situation of SR. We derive equations allowing one in principle to compute T inter in a given gravitational plus EM field. T inter may contribute to the dark matter.
{"title":"Interaction Energy of a Charged Medium and its EM Field in a Curved Spacetime","authors":"M. Arminjon","doi":"10.7546/giq-20-2019-88-98","DOIUrl":"https://doi.org/10.7546/giq-20-2019-88-98","url":null,"abstract":"In the electrodynamics of special relativity (SR) or general relativity (GR), the energy tensors of the charged medium and its EM field add to give the total energy tensor that obeys the dynamical equation without external force. In the investigated scalar theory of gravitation (\"SET\"), this assumption leads to charge non-conservation, hence an additional, \"interaction\" energy tensor T inter has to be postulated. The present work aims at constraining this tensor. First we study the independent equations of electrodynamics and their number, beginning with SR and GR. As in SR and GR, the system of electrodynamics of SET is closed in the absence of T inter. Hence, with T inter , at least one additional equation must be provided. This is done by assuming that T inter is Lorentz-invariant in the situation of SR. We derive equations allowing one in principle to compute T inter in a given gravitational plus EM field. T inter may contribute to the dark matter.","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"2 1-2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72472356","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 : 2019-01-01DOI: 10.7546/GIQ-20-2019-266-284
A. Ungar
It is known that entangled particles involve Lorentz symmetry violation. Hence, we pay attention to Lorentz transformations of signature $(m,n)$ for all positive integers $m$ and $n$. We show that these form the symmetry groups by which systems of $m$ entangled $n$-dimensional particles can be understood, just as the common Lorentz group of signature $(1,3)$ forms the symmetry group by which Einstein's special theory of relativity is understood. A novel, unified parametric realization of the Lorentz transformations of any signature $(m,n)$ shakes down the underlying matrix algebra into elegant and transparent results.
{"title":"Relativistic-Geometric Entanglement: Symmetry Groups of Systems of Entangled Particles","authors":"A. Ungar","doi":"10.7546/GIQ-20-2019-266-284","DOIUrl":"https://doi.org/10.7546/GIQ-20-2019-266-284","url":null,"abstract":"It is known that entangled particles involve Lorentz symmetry violation. Hence, we pay attention to Lorentz transformations of signature $(m,n)$ for all positive integers $m$ and $n$. We show that these form the symmetry groups by which systems of $m$ entangled $n$-dimensional particles can be understood, just as the common Lorentz group of signature $(1,3)$ forms the symmetry group by which Einstein's special theory of relativity is understood. A novel, unified parametric realization of the Lorentz transformations of any signature $(m,n)$ shakes down the underlying matrix algebra into elegant and transparent results.","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"65 4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77319908","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 : 2019-01-01DOI: 10.7546/giq-20-2019-208-214
G. Nugmanova, Akbota Myrzakul
Among nonlinear evolutionary equations integrable ones are of particular interest since only in this we case can theoretically study the model in detail and in-depth. In the present, we establish the geometric connection of the well-known integrable two-component Manakov system with a new two-layer spin system. This indicates that the latter system is also integrable. In this formalism, geometric invariants define some important conserved quantities associated with two interacting curves, and with the corresponding nonlinear evolution equations. MSC : 53C05, 53C35
{"title":"Integrability of the Two-Layer Spin System","authors":"G. Nugmanova, Akbota Myrzakul","doi":"10.7546/giq-20-2019-208-214","DOIUrl":"https://doi.org/10.7546/giq-20-2019-208-214","url":null,"abstract":"Among nonlinear evolutionary equations integrable ones are of particular interest since only in this we case can theoretically study the model in detail and in-depth. In the present, we establish the geometric connection of the well-known integrable two-component Manakov system with a new two-layer spin system. This indicates that the latter system is also integrable. In this formalism, geometric invariants define some important conserved quantities associated with two interacting curves, and with the corresponding nonlinear evolution equations. MSC : 53C05, 53C35","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"342 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78039421","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 : 2018-12-31DOI: 10.7546/giq-20-2019-161-183
F. J. Herranz, Á. Ballesteros, I. Gutierrez-Sagredo, M. Santander
The nine two-dimensional Cayley-Klein geometries are firstly reviewed by following a graded contraction approach. Each geometry is considered as a set of three symmetrical homogeneous spaces (of points and two kinds of lines), in such a manner that the graded contraction parameters determine their curvature and signature. Secondly, new Poisson homogeneous spaces are constructed by making use of certain Poisson-Lie structures on the corresponding motion groups. Therefore, the quantization of these spaces provides noncommutative analogues of the Cayley-Klein geometries. The kinematical interpretation for the semi-Riemannian and pseudo-Riemannian Cayley-Klein geometries is emphasized, since they are just Newtonian and Lorentzian spacetimes of constant curvature.
{"title":"Cayley--Klein Poisson Homogeneous Spaces","authors":"F. J. Herranz, Á. Ballesteros, I. Gutierrez-Sagredo, M. Santander","doi":"10.7546/giq-20-2019-161-183","DOIUrl":"https://doi.org/10.7546/giq-20-2019-161-183","url":null,"abstract":"The nine two-dimensional Cayley-Klein geometries are firstly reviewed by following a graded contraction approach. Each geometry is considered as a set of three symmetrical homogeneous spaces (of points and two kinds of lines), in such a manner that the graded contraction parameters determine their curvature and signature. Secondly, new Poisson homogeneous spaces are constructed by making use of certain Poisson-Lie structures on the corresponding motion groups. Therefore, the quantization of these spaces provides noncommutative analogues of the Cayley-Klein geometries. The kinematical interpretation for the semi-Riemannian and pseudo-Riemannian Cayley-Klein geometries is emphasized, since they are just Newtonian and Lorentzian spacetimes of constant curvature.","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84617285","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 : 2015-02-07DOI: 10.7546/GIQ-20-2019-215-226
N. Ogawa
The appearance of a geometric flow in the conservation law of particle number in classical particle diffusion and in the conservation law of probability in quantum mechanics is discussed in the geometrical environment of a two-dimensional curved surface with thickness $epsilon$ embedded in $R_3$. In such a system with a small thickness $epsilon$, the usual two-dimensional conservation law does not hold and we find an anomaly. The anomalous term is obtained by the expansion of $epsilon$. We find that this term has a Gaussian and mean curvature dependence and can be written as the total divergence of some geometric flow. We then have a new conservation law by adding the geometric flow to the original one. This fact holds in both classical diffusion and quantum mechanics when we confine particles to a curved surface with a small thickness.
{"title":"Geometric Flow Appearing in Conservation Law in Classical and Quantum Mechanics","authors":"N. Ogawa","doi":"10.7546/GIQ-20-2019-215-226","DOIUrl":"https://doi.org/10.7546/GIQ-20-2019-215-226","url":null,"abstract":"The appearance of a geometric flow in the conservation law of particle number in classical particle diffusion and in the conservation law of probability in quantum mechanics is discussed in the geometrical environment of a two-dimensional curved surface with thickness $epsilon$ embedded in $R_3$. \u0000In such a system with a small thickness $epsilon$, the usual two-dimensional conservation law does not hold and we find an anomaly. The anomalous term is obtained by the expansion of $epsilon$. We find that this term has a Gaussian and mean curvature dependence and can be written as the total divergence of some geometric flow. We then have a new conservation law by adding the geometric flow to the original one. This fact holds in both classical diffusion and quantum mechanics when we confine particles to a curved surface with a small thickness.","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2015-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83242151","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 : 1900-01-01DOI: 10.7546/giq-23-2022-75-98
I. Mladenov
A plethora of new explicit formulas that parameterize all three types of the Cassinian ovals via elliptic and circular functions are derived from the first principles. These formulas allow a detailed study of the geometry of the Cassinian curves which is persuaded to some extent here. Conversion formulas relating various sets of the geometrical parameters are presented. On the way some interesting relationships satisfied by the Jacobian elliptic functions were found. Besides, a few general identities between the complete elliptic integrals of the first and second kind were also established. An explicit universal formula for the total area within the Cassinians which is valid for all types of them is derived. Detailed derivation of the formulas for the volumes of the bodies obtained as a result of rotations of the Cassinian ovals is presented.
{"title":"New and Old Parameterizations of the Cassinian Ovals and Some Applications","authors":"I. Mladenov","doi":"10.7546/giq-23-2022-75-98","DOIUrl":"https://doi.org/10.7546/giq-23-2022-75-98","url":null,"abstract":"A plethora of new explicit formulas that parameterize all three types of the Cassinian ovals via elliptic and circular functions are derived from the first principles. These formulas allow a detailed study of the geometry of the Cassinian curves which is persuaded to some extent here. Conversion formulas relating various sets of the geometrical parameters are presented. On the way some interesting relationships satisfied by the Jacobian elliptic functions were found. Besides, a few general identities between the complete elliptic integrals of the first and second kind were also established. An explicit universal formula for the total area within the Cassinians which is valid for all types of them is derived. Detailed derivation of the formulas for the volumes of the bodies obtained as a result of rotations of the Cassinian ovals is presented.","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84194525","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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