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Hendrik De Bie - One of the best experts on this subject based on the ideXlab platform.
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The total angular momentum algebra related to the S3 Dunkl Dirac equation
Annals of Physics, 2018Co-Authors: Hendrik De Bie, Roy Oste, Joris Van Der JeugtAbstract:Abstract We consider the symmetry algebra generated by the total angular momentum operators, appearing as constants of motion of the S 3 Dunkl Dirac equation. The latter is a deformation of the Dirac equation by means of Dunkl operators, in our case associated to the root system A 2 , with corresponding Weyl group S 3 , the symmetric group on three elements. The explicit form of the symmetry algebra in this case is a one-parameter deformation of the classical total angular momentum algebra s o ( 3 ) , incorporating elements of S 3 . This was obtained using recent results on the symmetry algebra for a class of Dirac operators, containing in particular the Dirac–Dunkl operator for arbitrary root system. For this symmetry algebra, we classify all finite-dimensional, irreducible Representations and determine the conditions for the Representations to be unitarizable. The class of unitary irreducible Representations admits a natural realization acting on a Representation Space of eigenfunctions of the Dirac Hamiltonian. Using a Cauchy–Kowalevski extension theorem we obtain explicit expressions for these eigenfunctions in terms of Jacobi polynomials.
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the total angular momentum algebra related to the mathrm s _3 dunkl dirac equation
arXiv: Mathematical Physics, 2017Co-Authors: Hendrik De Bie, Roy Oste, Joris Van Der JeugtAbstract:We consider the symmetry algebra generated by the total angular momentum operators, appearing as constants of motion of the $\mathrm{S}_3$ Dunkl Dirac equation. The latter is a deformation of the Dirac equation by means of Dunkl operators, in our case associated to the root system $A_2$, with corresponding Weyl group $\mathrm{S}_3$, the symmetric group on three elements. The explicit form of the symmetry algebra in this case is a one-parameter deformation of the classical total angular momentum algebra $\mathfrak{so}(3)$, incorporating elements of $\mathrm{S}_3$. This was obtained using recent results on the symmetry algebra for a class of Dirac operators, containing in particular the Dirac-Dunkl operator for arbitrary root system. For this symmetry algebra, we classify all finite-dimensional, irreducible Representations and determine the conditions for the Representations to be unitarizable. The class of unitary irreducible Representations admits a natural realization acting on a Representation Space of eigenfunctions of the Dirac Hamiltonian. Using a Cauchy-Kowalevsky extension theorem we obtain explicit expressions for these eigenfunctions in terms of Jacobi polynomials.
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The metaplectic Howe duality and polynomial solutions for the symplectic Dirac operator
Journal of Geometry and Physics, 2014Co-Authors: Hendrik De Bie, Petr Somberg, Vladimír SoučekAbstract:Abstract We study various aspects of the metaplectic Howe duality realized by the Fischer decomposition for the metaplectic Representation Space of polynomials on R 2 n valued in the Segal–Shale–Weil Representation. As a consequence, we determine symplectic monogenics, i.e. the Space of polynomial solutions of the symplectic Dirac operator D s .
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The Howe duality and polynomial solutions for the symplectic Dirac operator
arXiv: Representation Theory, 2010Co-Authors: Hendrik De Bie, Petr Somberg, Vladimír SoučekAbstract:We study various aspects of the metaplectic Howe duality realized by Fischer decomposition for the metaplectic Representation Space of polynomials on $\mathbb{R}^{2n}$ valued in the Segal-Shale-Weil Representation. As a consequence, we determine symplectic monogenics, i.e., the Space of polynomial solutions of the symplectic Dirac operator.
Joris Van Der Jeugt - One of the best experts on this subject based on the ideXlab platform.
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The total angular momentum algebra related to the S3 Dunkl Dirac equation
Annals of Physics, 2018Co-Authors: Hendrik De Bie, Roy Oste, Joris Van Der JeugtAbstract:Abstract We consider the symmetry algebra generated by the total angular momentum operators, appearing as constants of motion of the S 3 Dunkl Dirac equation. The latter is a deformation of the Dirac equation by means of Dunkl operators, in our case associated to the root system A 2 , with corresponding Weyl group S 3 , the symmetric group on three elements. The explicit form of the symmetry algebra in this case is a one-parameter deformation of the classical total angular momentum algebra s o ( 3 ) , incorporating elements of S 3 . This was obtained using recent results on the symmetry algebra for a class of Dirac operators, containing in particular the Dirac–Dunkl operator for arbitrary root system. For this symmetry algebra, we classify all finite-dimensional, irreducible Representations and determine the conditions for the Representations to be unitarizable. The class of unitary irreducible Representations admits a natural realization acting on a Representation Space of eigenfunctions of the Dirac Hamiltonian. Using a Cauchy–Kowalevski extension theorem we obtain explicit expressions for these eigenfunctions in terms of Jacobi polynomials.
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the total angular momentum algebra related to the mathrm s _3 dunkl dirac equation
arXiv: Mathematical Physics, 2017Co-Authors: Hendrik De Bie, Roy Oste, Joris Van Der JeugtAbstract:We consider the symmetry algebra generated by the total angular momentum operators, appearing as constants of motion of the $\mathrm{S}_3$ Dunkl Dirac equation. The latter is a deformation of the Dirac equation by means of Dunkl operators, in our case associated to the root system $A_2$, with corresponding Weyl group $\mathrm{S}_3$, the symmetric group on three elements. The explicit form of the symmetry algebra in this case is a one-parameter deformation of the classical total angular momentum algebra $\mathfrak{so}(3)$, incorporating elements of $\mathrm{S}_3$. This was obtained using recent results on the symmetry algebra for a class of Dirac operators, containing in particular the Dirac-Dunkl operator for arbitrary root system. For this symmetry algebra, we classify all finite-dimensional, irreducible Representations and determine the conditions for the Representations to be unitarizable. The class of unitary irreducible Representations admits a natural realization acting on a Representation Space of eigenfunctions of the Dirac Hamiltonian. Using a Cauchy-Kowalevsky extension theorem we obtain explicit expressions for these eigenfunctions in terms of Jacobi polynomials.
Fairon Maxime - One of the best experts on this subject based on the ideXlab platform.
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On the Hamiltonian formulation of the trigonometric spin Ruijsenaars-Schneider system
'Springer Science and Business Media LLC', 2020Co-Authors: Chalykh Oleg, Fairon MaximeAbstract:We suggest a Hamiltonian formulation for the spin Ruijsenaars–Schneider system in the trigonometric case. Within this interpretation, the phase Space is obtained by a quasi-Hamiltonian reduction performed on (the cotangent bundle to) a Representation Space of a framed Jordan quiver. For arbitrary quivers, analogous varieties were introduced by Crawley-Boevey and Shaw, and their interpretation as quasi-Hamiltonian quotients was given by Van den Bergh. Using Van den Bergh’s formalism, we construct commuting Hamiltonian functions on the phase Space and identify one of the flows with the spin Ruijsenaars–Schneider system. We then calculate all the Poisson brackets between local coordinates, thus answering an old question of Arutyunov and Frolov. We also construct a complete set of commuting Hamiltonians and integrate all the flows explicitly
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On the Hamiltonian formulation of the trigonometric spin Ruijsenaars-Schneider system
'Springer Science and Business Media LLC', 2020Co-Authors: Chalykh Oleg, Fairon MaximeAbstract:We suggest a Hamiltonian formulation for the spin Ruijsenaars-Schneider system in the trigonometric case. Within this interpretation, the phase Space is obtained by a quasi-Hamiltonian reduction performed on (the cotangent bundle to) a Representation Space of a framed Jordan quiver. For arbitrary quivers, analogous varieties were introduced by Crawley-Boevey and Shaw, and their interpretation as quasi-Hamiltonian quotients was given by Van den Bergh. Using Van den Bergh's formalism, we construct commuting Hamiltonian functions on the phase Space and identify one of the flows with the spin Ruijsenaars-Schneider system. We then calculate all the Poisson brackets between local coordinates, thus answering an old question of Arutyunov and Frolov. We also construct a complete set of commuting Hamiltonians and integrate all the flows explicitly.Comment: 30 pages. v2: improved exposition in Section 3.1, added Remarks 5.6 and 5.7. v3: References added, accepted versio
Naoki Sasakura - One of the best experts on this subject based on the ideXlab platform.
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Space time uncertainty relation and lorentz invariance
Journal of High Energy Physics, 2000Co-Authors: Naoki SasakuraAbstract:We discuss a Lorentz covariant Space-time uncertainty relation, which agrees with that of Karolyhazy-Ng-van Dam when an observational time period δt is larger than the Planck time lP. At δt lP, this uncertainty relation takes roughly the form δtδx lP2, which can be derived from the condition prohibiting the multi-production of probes to a geometry. We show that there exists a minimal area rather than a minimal length in the four-dimensional case. We study also a three-dimensional free field theory on a non-commutative Space-time realizing the uncertainty relation. We derive the algebra among the coordinate and momentum operators and define a positive-definite norm of the Representation Space. In four-dimensional Space-time, the Jacobi identity should be violated in the algebraic Representation of the uncertainty relation.
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Space time uncertainty relation and lorentz invariance
arXiv: High Energy Physics - Theory, 2000Co-Authors: Naoki SasakuraAbstract:We discuss a Lorentz covariant Space-time uncertainty relation, which agrees with that of Karolyhazy-Ng-van Dam when an observational time period delta t is larger than the Planck time lp. At delta t lp^2, which can be derived from the condition prohibiting the multi-production of probes to a geometry. We show that there exists a minimal area rather than a minimal length in the four-dimensional case. We study also a three-dimensional free field theory on a non-commutative Space-time realizing the uncertainty relation. We derive the algebra among the coordinate and momentum operators and define a positive-definite norm of the Representation Space. In four-dimensional Space-time, the Jacobi identity should be violated in the algebraic Representation of the uncertainty relation.
Roy Oste - One of the best experts on this subject based on the ideXlab platform.
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The total angular momentum algebra related to the S3 Dunkl Dirac equation
Annals of Physics, 2018Co-Authors: Hendrik De Bie, Roy Oste, Joris Van Der JeugtAbstract:Abstract We consider the symmetry algebra generated by the total angular momentum operators, appearing as constants of motion of the S 3 Dunkl Dirac equation. The latter is a deformation of the Dirac equation by means of Dunkl operators, in our case associated to the root system A 2 , with corresponding Weyl group S 3 , the symmetric group on three elements. The explicit form of the symmetry algebra in this case is a one-parameter deformation of the classical total angular momentum algebra s o ( 3 ) , incorporating elements of S 3 . This was obtained using recent results on the symmetry algebra for a class of Dirac operators, containing in particular the Dirac–Dunkl operator for arbitrary root system. For this symmetry algebra, we classify all finite-dimensional, irreducible Representations and determine the conditions for the Representations to be unitarizable. The class of unitary irreducible Representations admits a natural realization acting on a Representation Space of eigenfunctions of the Dirac Hamiltonian. Using a Cauchy–Kowalevski extension theorem we obtain explicit expressions for these eigenfunctions in terms of Jacobi polynomials.
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the total angular momentum algebra related to the mathrm s _3 dunkl dirac equation
arXiv: Mathematical Physics, 2017Co-Authors: Hendrik De Bie, Roy Oste, Joris Van Der JeugtAbstract:We consider the symmetry algebra generated by the total angular momentum operators, appearing as constants of motion of the $\mathrm{S}_3$ Dunkl Dirac equation. The latter is a deformation of the Dirac equation by means of Dunkl operators, in our case associated to the root system $A_2$, with corresponding Weyl group $\mathrm{S}_3$, the symmetric group on three elements. The explicit form of the symmetry algebra in this case is a one-parameter deformation of the classical total angular momentum algebra $\mathfrak{so}(3)$, incorporating elements of $\mathrm{S}_3$. This was obtained using recent results on the symmetry algebra for a class of Dirac operators, containing in particular the Dirac-Dunkl operator for arbitrary root system. For this symmetry algebra, we classify all finite-dimensional, irreducible Representations and determine the conditions for the Representations to be unitarizable. The class of unitary irreducible Representations admits a natural realization acting on a Representation Space of eigenfunctions of the Dirac Hamiltonian. Using a Cauchy-Kowalevsky extension theorem we obtain explicit expressions for these eigenfunctions in terms of Jacobi polynomials.