Geometric Methods in Physics XXXVI: Workshop and Summer School, Białowieża, Poland, 2017 PDF
Preview Geometric Methods in Physics XXXVI: Workshop and Summer School, Białowieża, Poland, 2017
Trends in Mathematics Piotr Kielanowski Anatol Odzijewicz Emma Previato Editors Geometric Methods in Physics XXXVI Workshop and Summer School, . Białowieza, Poland, 2017 Trends in Mathematics Trends in Mathematics is a series devoted to the publication of volumes arising from conferences and lecture series focusing on a particular topic from any area of mathematics. Its aim is to make current developments available to the community as rapidly as possible without compromise to quality and to archive these for reference. Proposals for volumes can be submitted using the Online Book Project Submission Form at our website www.birkhauser-science.com. Material submitted for publication must be screened and prepared as follows: All contributions should undergo a reviewing process similar to that carried out by journals and be checked for correct use of language which, as a rule, is English. Articles without proofs, or which do not contain any significantly new results, should be rejected. 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Moreinformationaboutthisseriesathttp://www.springer.com/series/4961 Piotr Kielanowski • Anatol Odzijewicz Emma Previato Editors Geometric Methods in Physics XXXVI Workshop and Summer School, Białowieża, Poland, 2017 Editors Piotr Kielanowski Anatol Odzijewicz Departamento de Física Institute of Mathematics CINVESTAV University of Białystok Ciudad de México, Mexico Białystok, Poland Emma Previato Department of Mathematics and Statistics Boston University Boston, MA, USA ISSN 2297-0215 ISSN 2297-024X (electronic) Trends in Mathematics ISBN 978-3-030-01155-0 ISBN 978-3-030-01156-7 (eBook) https://doi.org/10.1007/978-3-030-01156-7 Library of Congress Control Number: 2018968410 Mathematics Subject Classification (2010): 01-06, 01A70, 20N99, 58A50, 58Z05, 81P16, 33D80, 51P05 © Springer Nature Switzerla nd AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Contents Preface .................................................................. ix Part I: Quantum Mechanics and Mathematics – Twareque Ali in Memoriam G.A. Goldin In Memory of S. Twareque Ali ...................................... 3 F. Bagarello Two-dimensional Noncommutative Swanson Model and Its Bicoherent States ........................................... 9 R. Beneduci Universal Markov Kernels for Quantum Observables ................ 21 S. Berceanu Coherent States Associated to the Jacobi Group and Berezin Quantization of the Siegel–JacobiBall ...................... 31 J.P. Gazeau and R. Murenzi 1D & 2D Covariant Affine Integral Quantizations ................... 39 G.A. Goldin and D.H. Sharp Diffeomorphism Group Representations in Relativistic Quantum Field Theory ............................................. 47 Part II: Noncommutative Geometry M. Almulhem and T. Brzeziński Skew Derivations on Down-up Algebras ............................. 59 L. Dąbrowski On Noncommutative Geometry of the Standard Model: Fermion Multiplet as Internal Forms ................................ 69 vi Contents M.N. Hounkonnou, M.J. Landalidji and E. Balo¨ıtcha Recursion Operator in a Noncommutative Minkowski Phase Space ............................................. 83 A. Sitarz Decompactifying Spectral Triples ................................... 95 F.D.Garc´ıaand E. Wagner Dirac Operator on a Noncommutative Toeplitz Torus ............... 103 Part III: Quantization F. Kheirandish Field Quantization in the Presence of External Fields ............... 113 A.G. Sergeev Quantization of Mathematical Theory of Non-Smooth Strings ....... 119 D. Sternheimer The Reasonable Effectiveness of Mathematical Deformation Theory in Physics ..................................... 131 J. Tosiek States in Deformation Quantisation: Hopes and Difficulties .......... 139 N.A. Tyurin Exact LagrangianSubmanifolds and the Moduli Space of Special Bohr–Sommerfeld LagrangianCycles ..................... 147 A. Yoshioka Star Exponentials in Star Product Algebra .......................... 155 Part IV: Integrable Systems Y. Kosmann-Schwarzbach Beyond Recursion Operators ........................................ 167 G. Meng Kepler Problem and Jordan Algebras ............................... 181 A.E. Mironov and G.S. Mauleshova On Rank Two Algebro-Geometric Solutions of an Integrable Chain .............................................. 189 Contents vii Part V: Differential Geometry and Physics J. Attard The Dressing Field Method of Gauge Symmetry Reduction: Presentation and Examples ......................................... 199 E.M. Babalic, D. Doryn, C.I. Lazaroiu and M. Tavakol A Differential Model for B-type Landau–Ginzburg Theories ......... 207 B. Balcerzak On the Dirac Type Operators on Symmetric Tensors ................ 215 M. Fecko Surfaces Which Behave Like Vortex Lines ........................... 223 C.I. Lazaroiu and C.S. Shahbazi On the Spin Geometry of Supergravity and String Theory ........... 229 F. Pelletier Conic Sub-Hilbert–Finsler Structure on a Banach Manifold .......... 237 N. Sadeghzadeh On Spherically Symmetric Finsler Metrics ........................... 245 Part VI: Topics in Spectral Theory J. Derezin´ski Homogeneous Rank One Perturbations and Inverse Square Potentials ........................................... 253 A. Mostafazadeh Generalized Unitarity Relation for Linear Scattering Systems in One Dimension ................................................... 265 A. Shafarevich Differential Equations on Polytopes: Laplacians and LagrangianManifolds, Corresponding to Semiclassical Motion ....... 273 Part VII: Representation Theory D. Belti¸t˘a and A. Zergane Coadjoint Orbits in Representation Theory of pro-Lie Groups ....... 281 viii Contents T. Kobayashi Conformal Symmetry Breaking on Differential Forms and Some Applications .............................................. 289 E. Lytvynov Representations of the Anyon Commutation Relations ............... 309 Part VIII: Special Topics H. Baumg¨artel Remarks to the Resonance-Decay Problem in Quantum Mechanics from a Mathematical Point of View ................................. 331 M. Myronova and E. Bourret Dynamical Generation of Graphene ................................. 341 T. Czyżycki and J. Hrivna´k Eight Kinds of OrthogonalPolynomials of the Weyl Group C 2 and the Tau Method ................................................ 347 A.Yu. Orlov Links Between Quantum Chaos and Counting Problems ............. 355 PartIX:ExtendedAbstractsoftheLecturesat“SchoolonGeometryandPhysics” M. Fecko Integral Invariants (Poincar´e–Cartan)and Hydrodynamics .......... 377 B.K. Kwaśniewski Invitation to Hilbert C∗-modules and Morita–Rieffel Equivalence .... 383 Yu. Neretin After Plancherel Formula ........................................... 389 A. Sitarz A Glimpse of Noncommutative Geometry ........................... 403 A.B. Tumpach An Example of Banach and Hilbert Manifold: The Universal Teichmu¨ller Space .................................... 411 A. Shafarevich Extensions of Symmetric Operators and Evolution Equations on Singular Spaces .................................................. 419 GeometricMethodsinPhysics.XXXVIWorkshop2017 TrendsinMathematics,ix–x (cid:2)c SpringerNatureSwitzerlandAG2019 Preface ThisbookcontainsaselectionofpaperspresentedduringtheThirty-Sixth“Work- shop on Geometric Methods in Physics” (WGMPXXXVI) and abstracts of lec- tures given during the Sixth “School on Geometry and Physics”, both of which took place in Bial(cid:2)owiez˙a, Poland during the summer of 2017. These two coor- dinated activities constitute an annual event. Information on previous and up- coming schools and workshops, and related materials, can be found at the URL: http://wgmp.uwb.edu.pl. The volume opens with a chapter containing papers presented at the special session organized by A. Odzijewicz, G. Goldin, J.-P. Antoine, T. Bhattachryya, J.P.Gazeau,J.Harnad,andF.Schroeck,dedicatedtothememoryofS.Twareque Ali. Professor Ali, who died suddenly in 2016, was an active member of the Or- ganizing Committee of our workshop for many years. There follow chapters on “Noncommutative Geometry”, “Quantization”, “Integrable Systems”, “Differen- tial Geometry and Physics”, “Topics in Spectral Theory”, “Representation The- ory” and “Special Topics”, with papers based on the talks and posters presented attheworkshop.Thefinalchaptercontainsextendedabstractsofthelectureseries given during the “Sixth School on Geometry and Physics”. The WGMP is an international conference organized each year by the De- partment of Mathematical Physics in the Faculty of Mathematics and Computer ScienceoftheUniversityofBial(cid:2)ystok,Poland.Themainsubjectoftheworkshops, consistentwiththeirtitle,istheapplicationofgeometricmethodsinmathematical physics.Theyfrequentlyincludestudiesofnoncommutativegeometry,Poissonge- ometry, completely integrable systems, quantization, infinite-dimensional groups, supergroups and supersymmetry, quantum groups, Lie groupoids and algebroids, and related topics. Participation in the workshops is open; the participants typi- cally consistofphysicists and mathematicians fromcountriesacrossseveralconti- nents, who have a wide spectrum of interests. TheWorkshopandSchoolareheldinBia(cid:2)lowiez˙a,avillagelocatedintheeast ofPolandnearthe borderwithBelarus.Bia(cid:2)lowiez˙ais situatedinthe center ofthe renowned Bia(cid:2)lowiez˙a Forest. This forest, shared between Poland and Belarus, is one of the last remnants of the primeval forest that covered the European Plain beforehumansettlement. IthasbeendesignatedaUNESCOWorldHeritageSite. The peaceful atmosphere of a small village, together with natural beauty, affords auniqueenvironmentforlearning,cooperation,andcreativework.Asaresultthe
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