Introduction to symplectic dirac operators 2006 (LNM 1887 habermann k habermann L)(ISBN 3540334203)(131s)

Lecture Notes in Mathematics Editors: J.-M Morel, Cachan F Takens, Groningen B Teissier, Paris 1887 K Habermann · L Habermann Introduction to Symplectic Dirac Operators ABC Authors Katharina Habermann State and University Library Göttingen Platz der Göttinger Sieben 37073 Göttingen Germany e-mail: habermann@sub.uni-goettingen.de Lutz Habermann Department of Mathematics University of Hannover Welfengarten 30167 Hannover Germany e-mail: habermann@math.uni-hannover.de Library of Congress Control Number: 2006924i23 Mathematics Subject Classification (2000): 53-02, 53Dxx, 58-02, 58Jxx ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN-10 3-540-33420-3 Springer Berlin Heidelberg New York ISBN-13 978-3-540-33420-0 Springer Berlin Heidelberg New York DOI 10.1007/b138212 This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable for prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, 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 A EX package Typesetting: by the authors and SPI Publisher Services using a Springer LT Cover design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN: 11736684 V A 41/3100/ SPI 543210 To Karen Preface This book aims to give a systematic and self-contained introduction to the theory of symplectic Dirac operators and to reflect the current state of the subject At the same time, it is intended to establish the idea that symplectic spin geometry and symplectic Dirac operators may give valuable tools in symplectic geometry and symplectic topology, which have become important fields and very active areas of mathematical research The basic idea of symplectic spin geometry goes back to the early 1970s, when Bertram Kostant introduced symplectic spinors in order to give the construction of the half-form bundle and the half-form pairings in the context of geometric quantization [37] During the next two decades, however, almost no attention has been given to a closer study of symplectic spin geometry itself In 1995, the first author introduced symplectic Dirac operators [24] and started a systematical investigation [25, 26, 27] These symplectic Dirac operators are called Dirac operators, since they are defined in an analogous way as the classical Riemannian Dirac operator known from Riemannian spin geometry (cf e.g [21]) They are called symplectic because they are constructed by use of the symplectic setting of the underlying symplectic manifold All tools which were necessary for that construction have already been known and accepted, mainly in mathematical physics These are the symplectic Clifford algebra (also known as Weyl algebra), the metaplectic group, the metaplectic representation (Segal–Shale–Weil representation) acting on L2 (Rn ), metaplectic structures, and symplectic connections One of the basic ideas in differential geometry is that the study of analytic properties of certain differential operators acting on sections of vector bundles yields geometric and topological properties of the underlying base manifold There are several classical results in that direction An example is Hodge– de Rham theory Here, one considers the Hodge–Laplace–Beltrami operator ∆ VIII Preface acting on differential forms This operator is one of the most studied operators in global Riemannian geometry and his spectrum gives important topological invariants In particular, the dimension of the kernel of ∆ on p-forms over a closed Riemannian manifold is the p-th Betti number Other well known and well studied operators are the Kodaira–Hodge–Laplace operator on differential forms with values in a holomorphic vector bundle or the classical Dirac operator on Riemannian manifolds Now, symplectic spinor fields are sections in an L2 (Rn )-Hilbert space bundle over a symplectic manifold and symplectic Dirac operators, acting on symplectic spinor fields, are associated to the symplectic manifold in a very natural way Hence they may be expected to give interesting applications in symplectic geometry and symplectic topology It is our opinion that, besides the already stated, there are further close relations to mathematical physics Some steps towards this direction have been made by the first author and Andreas Klein in [28, 29, 30] Another perspective could be the extension of Clifford analysis and spin geometry to super differential geometry According to the most used version of super geometry developed by Bertram Kostant, geometrical structures over a supermanifold consist of Z2 -graded objects and thus have an even as well as an odd part Then one can imagine that a metric has to satisfy some kind of graded symmetry which, roughly speaking, corresponds to a symmetric object on the even part as well as to a skew symmetric object on the odd part of the supermanifold For the first one, we have the classical Riemannian spin geometry, whereas the second one is basically given by symplectic spin geometry The main aspects of that idea are treated in a paper by Frank Sommen [45] Although the construction of symplectic Dirac operators follows the same procedure as for the classical Riemannian Dirac operator, using the symplectic structure of the underlying manifold instead of the Riemannian metric, there are essential differences to the Riemannian case These are caused by the fact that the algebraic structure of the symplectic Clifford algebra is completely different from that of Riemannian spin geometry For the classical Clifford algebra, we have the relation v2 = − v , whereas the algebraic structure of the symplectic Clifford algebra is given by v · w − w · v = −ω0 (v, w) This implies essentially different properties for the Clifford multiplications, which enter into the definition of the Dirac operators Moreover, the non-compactness of the symplectic group leads to analytic difficulties Namely, since the typical fiber of the symplectic spinor bundle is the Hilbert space L2 (Rn ), we deal with operators acting on sections of a vector bundle of infinite rank For elliptic formally self-adjoint pseudo-differential operators with positive definite leading symbol acting on sections in a vector bundle of finite rank, one has a completely developed theory So, in order to be able to apply these techniques, we are interested in equivariance properties Preface IX of our operators with respect to a certain decomposition of the symplectic spinor bundle into a series of subbundles of finite rank It turns out that an associated second order operator respects this decomposition provided that a technical assumption, which always can be realized, holds true Let us now briefly describe the content of each chapter of this book The first chapter is introductory It contains preliminaries and basic material needed for our considerations Chapter is devoted to symplectic connections In particular, we introduce a further Ricci tensor, which we will call symplectic Ricci tensor To our knowledge, no attention has been given to this tensor in previous studies In fact, this symplectic Ricci tensor is a new object in the case of non-vanishing torsion To date, mostly only torsion-free symplectic connections have been considered It turns out that, in our context, it is convenient to work also with symplectic connections with torsion and that the symplectic Ricci tensor is more suitable for our purposes The next chapter introduces the symplectic spinor bundle and the spinor derivative and analyzes a splitting property of the spinor bundle In Chapter 4, we give the definition of symplectic Dirac operators and describe in detail how these operators depend on the objects from which they are built Chapter is concerned with an associated second order operator of Laplace type and addresses properties of this operator The objective of Chapter is the situation for a special class of symplectic manifolds, namely Kă ahler manifolds Here, we also investigate the example of CP The aim of Chapter is to construct a Fourier transform for symplectic spinor fields and to derive consequences for the symplectic Dirac operators The last chapter focuses on relations to mathematical physics, in particular to quantization This closes the circle to the beginnings by Bertram Kostant The present text is originated in research ideas of the first author and provides an extended version of her “Habilitationsschrift”, which was never published separately Starting with helpful discussions from the beginning of the investigations and proposing many improvements in the selection and presentation of the material, the second author became more and more involved into the subject We decided to write this book tree years ago and, for this book, he made a thorough revision of the material, in particular, to improve the strictness of the presentation Then it took time to compose it in a natural and organic way Furthermore, our working places are no longer as close to each other as before and it became difficult to keep our discussions at its intensive level Now we consider the text ready for publication and hope to present the reader a mature work During the work on the book, we received financial support from DFG, the German Research Foundation, contract HA 3056/1-1,2 Many thanks are due to our students Paul Rosenthal and Steffen Rudnick for proof-reading the LATEX-type-written manuscript X Preface We dedicate this book to our daughter Karen She accepts our mathematical family circle and enriches it with her interest We are grateful to being able to give her an understanding of the fascination of mathematics Karen, you are a wonderful teenager which makes us enjoying the challenge of teaching mathematics Gă ottingen and Greifswald, November 2004 Katharina & Lutz Habermann Contents Background on Symplectic Spinors 1.1 Symplectic Group and Clifford Algebra 1.2 The Stone–von Neumann Theorem 1.3 Metaplectic Representation 1.4 Symplectic Clifford Multiplication 11 1.5 Hermite Functions 16 Symplectic Connections 21 2.1 Symplectic Manifolds 21 2.2 Constructions and Torsion 25 2.3 Symplectic Curvature and Ricci Tensors 29 Symplectic Spinor Fields 35 3.1 Metaplectic Structures 35 3.2 Symplectic Spinor Bundle 37 3.3 Splitting of the Spinor Bundle 43 Symplectic Dirac Operators 49 4.1 Definition of the Operators 49 4.2 Dependence on the Symplectic Connection 52 4.3 Dependence on the Metaplectic Structure 57 4.4 Dependence on the Almost Complex Structure 62 4.5 Formal Self-Adjointness 64 XII Contents An Associated Second Order Operator 67 5.1 Definition and Ellipticity 67 5.2 A Weitzenbă ock Formula 68 5.3 Splitting of the Operator 74 The Kă ahler Case 81 6.1 The Operator P on Kă ahler Manifolds 81 6.2 Lower Bound Estimates 86 6.3 The Spectrum of P on CP 87 Fourier Transform for Symplectic Spinors 97 7.1 Definition of the Transform 97 7.2 Basic Properties 98 7.3 Symmetry of the Spectra of D and D 99 Lie Derivative and Quantization 101 8.1 Lie Derivative of Symplectic Spinor Fields 101 8.2 Schră odinger Equation for Quadratic Hamiltonians 109 8.3 Lie Derivative as Quantization 111 References 115 Index 119 8.2 Schră odinger Equation for Quadratic Hamiltonians 109 According to Proposition 3.2.9 and Lemma 8.1.11, i 2n ∇X ∇Jej Y · ej − ∇Y ∇Jej X · ej − ∇Jej [X, Y ] · ej · ϕ j=1 = i 2n R(X, Y )(Jej ) · ej + ∇[X,Jej ] Y · ej − ∇[Y,Jej ] X · ej · ϕ j=1 = RQ (X, Y )ϕ + i 2n ∇[X,Jej ] Y · ej − ∇[Y,Jej ] X · ej · ϕ j=1 By Lemmas 8.1.9 and 8.1.10 and the torsion-freeness of ∇, it follows that [LX , LY ]ϕ − L[X,Y ] ϕ = − i 2n ∇Jej Y · ∇X ej − ∇Jej X · ∇Y ej j=1 +∇[X,Jej ] Y · ej − ∇[Y,Jej ] X · ej · ϕ − (∇X · ∇Y · ϕ − ∇Y · ∇X · ϕ) = i 2n ∇∇X (Jej ) Y · ej − ∇∇Y (Jej ) X · ej j=1 −∇[X,Jej ] Y · ej + ∇[Y,Jej ] X · ej −∇∇Jej X Y · ej + ∇∇Jej Y X · ej · ϕ =0, and the proposition is proved 8.2 Schră odinger Equation for Quadratic Hamiltonians We consider R2n with its standard symplectic structure ω0 and a quadratic Hamiltonian u on R2n , i.e a function u : R2n → Rn defined by u(v) = Υ v, v , where Υ is any symmetric real 2n × 2n-matrix First we note that the Hamiltonian vector field Xu of u is given by an element of the Lie algebra sp(n, R) Lemma 8.2.1 (1) One has Xu (v) = −2J0 Υ v for all v ∈ R2n 110 Lie Derivative and Quantization (2) J0 Υ and Υ J0 lie in sp(n, R) Proof Assertion (1) follows from (du)v (w) = Υ v, w = −2ω0 (J0 Υ v, w) for any v, w ∈ R2n To verify (2), use the fact that a real 2n × 2n-matrix A is an element of sp(n, R) if and only if AT J0 + J0 A = Let Hu denote the Hamilton operator associated to u via normal ordering quantization That means that n ( Υ , aj Qi ◦ Qj + Υ , bj (Qi ◦ Pj + Pj ◦ Qi ) Hu = i,j=1 + Υ b i , b j Pi ◦ Pj ) , where Qi and Pi are again the position and momentum operators Lemma 8.2.2 One has the relation i m∗ ◦ ρ−1 ∗ (Υ J0 ) = − Hu Proof By Lemma 1.3.1, ρ−1 ∗ (Υ J0 ) = = n (ai · Υ + bi · Υ bi ) i=1 n ( Υ , a j a i · a j + Υ a i , b j a i · b j i,j=1 + Υ bi , aj bi · aj + Υ bi , bj bi · bj ) n = ( Υ , aj · aj + Υ , bj (ai · bj + bj · ) i,j=1 + Υ bi , bj bi · bj ) Now apply Equation (1.4.2) Let (AΥt ) be the one parameter subgroup of Sp(n, R) generated by 2Υ J0 , i.e AΥt = exp(2tΥ J0 ) for t ∈ R, and let (qtΥ ) denote the lift of (AΥt ) to Mp(n, R) 8.3 Lie Derivative as Quantization 111 Proposition 8.2.3 For any f ∈ S(Rn ), the curve (m(qtΥ )f ) in S(Rn ) satises the Schră odinger equation d m(qtΥ )f = −iHu m(qtΥ )f dt Proof We have ρ∗ d Υ q ds s Hence = s=0 d ρ(q Υ ) ds s d Υ q ds s = s=0 d Υ A ds s = 2Υ J0 s=0 = ρ−1 ∗ (Υ J0 ) s=0 By means of Lemma 8.2.2, we get d d m(qtΥ )f = m(qsΥ )m(qtΥ )f dt ds s=0 d Υ q = m∗ m(qtΥ )f ds s s=0 Υ = m∗ ρ−1 ∗ (Υ J0 ) m(qt )f = −iHu m(qtΥ )f for any f ∈ S(Rn ) 8.3 Lie Derivative as Quantization We use the symplectic standard basis (a1 , , an , b1 , , bn ) understood as a global section of the symplectic frame bundle R of (R2n , ω0 ) to identify R2n × Sp(n, R) with R That means, we assign to (v, A) ∈ R2n × Sp(n, R) the symplectic basis (Aa1 , , Aan , Ab1 , , Abn ) of Tv R2n = R2n The metaplectic structure of (R2n , ω0 ) is then simply the product bundle P = R2n × Mp(n, R) , where the map FP : P → R is given by FP (v, q) = (v, ρ(q)) We identify R2n × L2 (Rn ) with the symplectic spinor bundle Q via (v, f ) ∈ R2n × L2 (Rn ) → v, e+ , f ∈ Q , where e+ denotes the unit element of Mp(n, R) Then sections of Q are maps ϕ : R2n → L2 (Rn ) In the next lemma, we use that elements of Sp(n, R) can also be considered as symplectomorphisms of R2n 112 Lie Derivative and Quantization Lemma 8.3.1 Let B ∈ Sp(n, R) and let qB ∈ Mp(n, R) such that ρ(qB ) = B Then one has ˜ : R → R induced by B is given by (1) The isomorphism B ˜ B(v, A) = (Bv, BA) for (v, A) ∈ R2n × Sp(n, R) ¯ : P → P defined by (2) The isomorphism B ¯ B(v, q) = (Bv, qB q) ˜ for (v, q) ∈ R2n × Mp(n, R) is a lift of B (3) It is ¯ −1 )∗ ϕ (v) = m q −1 ϕ(Bv) (B B for ϕ ∈ Γ(Q) and v ∈ R2n Proof Assertion (1) follows from (dB)v (w) = Bw for v, w ∈ R2n Assertion (2) is easily checked To see (3), first observe that the Mp(n, R)-equivariant mapping ϕˆ : P → L2 (Rn ) corresponding to ϕ ∈ Γ(Q) is given by ϕ(v, ˆ q) = m q −1 ϕ(v) for v ∈ R2n and q ∈ Mp(n, R) Hence, by (2), −1 ¯ ϕ(Bv) ϕˆ ◦ B(v, q) = m q −1 m qB This implies the stated relation We now give an interpretation of the Lie derivative in the direction of an Hamiltonian vector field in the language of normal ordering quantization For this, let f for f ∈ S(Rn ) denote the symplectic spinor field over R2n that is constantly equal to f Proposition 8.3.2 Let u be a quadratic Hamiltonian on R2n Then LXu f = −i F −1 ◦ Hu ◦ F(f ) for any f ∈ S(Rn ) Proof Let Υ , (AΥt ) and (qtΥ ) be as in Section 8.2 By Lemma 8.2.1, the flow (θt ) of the Hamiltonian vector field Xu is θt = exp(−2tJ0 Υ ) = J0 AΥ−t J0−1 8.3 Lie Derivative as Quantization 113 By Proposition 1.3.7(1) and Lemma 8.3.1, it follows that (θ¯t−1 )∗ f = F −1 ◦ m(qtΥ ) ◦ F(f ) for f ∈ S(Rn ) Now, using Proposition 8.2.3, one gets the assertion Although the above relation cannot be carried over to arbitrary manifolds, we can prove a Heisenberg relation in the general situation Proposition 8.3.3 Let (M, ω) be a symplectic manifold that admits a metaplectic structure Setting q(u)ϕ = iLXu ϕ , one has [q(u), q(v)]ϕ = −iq({u, v})ϕ for all u, v ∈ C ∞ (M ) and all symplectic spinor fields ϕ over M with respect to any metaplectic structure Proof By Propositions 2.1.9 and 8.1.8, one concludes that [q(u), q(v)]ϕ = − [LXu , LXv ] ϕ = −L[Xu ,Xv ] ϕ = LX{u,v} ϕ = −iq({u, v})ϕ References A.O Barut, R Raczka: Theory of group representations and applications 2nd rev ed Singapore: World Scientic (1986) H Baum: Spin-Strukturen und Dirac-Operatoren u ăber pseudoriemannschen Mannigfaltigkeiten Teubner-Texte zur Mathematik, Bd 41 Leipzig: BSB B G Teubner Verlagsgesellschaft (1981) N Berline, E Getzler, M Vergne: Heat Kernels and Dirac Operators Grundlehren der Mathematischen Wissenschaften 298 Berlin etc.: SpringerVerlag (1992) R.J Blattner: Some remarks on quantization Symplectic geometry and mathematical physics, Proc Colloq., Aix-en-Provence/Fr 1990, Prog Math 99, 37-47 (1991) A Borel, N Wallach: Continuous cohomology, discrete subgroups, and representations of reductive groups 2nd ed Mathematical Surveys and Monographs 67 Providence, RI: American Mathematical Society (AMS) (2000) F Bourgeois, M Cahen: Can one define a preferred symplectic connection? Rep Math Phys 43, No.1-2, 35-42 (1999) F Bourgeois, M Cahen: A variational principle for symplectic connections J Geom Phys 30, No.3, 233-265 (1999) M Cahen, S Gutt, J Rawnsley: Symplectic connections with parallel Ricci tensor Grabowski, Janusz (ed.) et al., Poisson geometry Stanislaw Zakrzewski in memoriam Warszawa: Polish Academy of Sciences, Institute of Mathematics, Banach Cent Publ 51, 31-41 (2000) M Cahen, S Gutt, J Rawnsley: Symmetric symplectic spaces with Ricci-type curvature Dito, Giuseppe (ed.) et al., Conf´erence Mosh´e Flato 1999: Quantization, deformations, and symmetries, Dijon, France, September 5-8, 1999 Volume II Dordrecht: Kluwer Academic Publishers Math Phys Stud 22, 81-91 (2000) 10 M Cahen, S Gutt, J Horowitz, J Rawnsley: Homogeneous symplectic manifolds with Ricci-type curvature J Geom Phys 38, No.2, 140-151 (2001) 11 A Cannas da Silva: Lectures on symplectic geometry Lecture Notes in Mathematics 1764 Berlin: Springer (2001) 12 M de Gosson: Maslov classes, metaplectic representation and Lagrangian quantization Mathematical Research 95 Berlin: Akademie Verlag (1997) 13 C Emmrich, A Weinstein: The differential geometry of Fedosov’s quantization Brylinski, Jean-Luc (ed.) et al., Lie theory and geometry: in honor of Bertram 116 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 References Kostant on the occasion of his 65th birthday Invited papers, some originated at a symposium held at MIT, Cambridge, MA, USA in May 1993 Boston, MA: Birkhă auser Prog Math 123, 217-239 (1994) B.V Fedosov: A simple geometrical construction of deformation quantization J Differ Geom 40, No.2, 213-238 (1994) B.V Fedosov: Deformation quantization and index theory Mathematical Topics Berlin: Akademie Verlag (1996) G.B Folland: Harmonic analysis in phase space Annals of Mathematics Studies, 122 Princeton, NJ: Princeton University Press (1989) M Forger, H Hess: Universal metaplectic structures and geometric quantization Commun Math Phys 64, 269-278 (1979) Th Friedrich: Einige differentialgeometrische Untersuchungen des DiracOperators einer Riemannschen Mannigfaltigkeit Habilitation thesis, Humboldt University Berlin (1979) Th Friedrich: Der erste Eigenwert des Dirac-Operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkră ummung Math Nachr 97, 117-146 (1980) Th Friedrich: Zur Abhă angigkeit des Dirac Operators von der Spin-Struktur Colloq Math 48, 57-62 (1984) Th Friedrich: Dirac operators in Riemannian geometry Graduate Studies in Mathematics 25 Providence, RI: American Mathematical Society (2000) I Gelfand, V Retakh, M Shubin: Fedosov Manifolds Adv Math 136, No.1, 104-140 (1998) S Gutt: Variations on deformation quantization Dito, Giuseppe (ed.) et al., Conf´erence Mosh´e Flato 1999: Quantization, deformation, and symmetries, Dijon, France, September 5-8, 1999 Volume I Dordrecht: Kluwer Academic Publishers Math Phys Stud 21, 217-254 (2000) K Habermann: The Dirac Operator on Symplectic Spinors Ann Global Anal Geom 13, 155-168 (1995) K Habermann: Basic Properties of Symplectic Dirac Operators Commun Math Phys 184, 629-652 (1997) K Habermann: Harmonic Symplectic Spinors on Riemann Surfaces Manuscr Math 94, 465-484 (1997) K Habermann: Symplectic Dirac Operators on Kă ahler Manifolds Math Nachr 211, 37-62 (2000) K Habermann: An M pC -spinorial approach to a geometric-type quantization for symplectic manifolds Gesztesy, Fritz (ed.) et al., Stochastic processes, physics and geometry: New interplays II A volume in honor of Sergio Albeverio Proceedings of the conference on infinite dimensional (stochastic) analysis and quantum physics, Leipzig, Germany, January 18-22, 1999 Providence, RI: American Mathematical Society (AMS) CMS Conf Proc 29, 259-268 (2000) K Habermann, A Klein: A Fourier transform for symplectic spinors and applications Preprint, Greifswald University (2002) K Habermann, A Klein: Lie derivative of symplectic spinor fields, metaplectic representation, and quantization Rostocker Math Kolloq 57, 71-91 (2003) H Hofer, E Zehnder: Symplectic invariants and Hamiltonian dynamics Birkhă auser Advanced Texts Basel: Birkhă auser (1994) M Kashiwara, M Vergne: On the Segal-Shale-Weil representations and harmonic polynomials Invent Math 44, 1-47 (1978) References 117 33 A.A Kirillov: Geometric quantization Dynamical systems IV Symplectic geometry and its applications, Encycl Math Sci 4, 137-172 (1990); translation from Itogi Nauki Tekh., Ser Sovrem Probl Mat., Fundam Napravleniya 4, 141-178 (1985) 34 A Klein: Eine Fouriertransformation fă ur symplektische Spinoren und Anwendungen in der Quantisierung Diploma Thesis, Technische Universită at Berlin (2000) 35 S Kobayashi, K Nomizu: Foundations of differential geometry I New YorkLondon: Interscience Publishers, a division of John Wiley & Sons XI (1963) 36 S Kobayashi, K Nomizu: Foundations of Differential Geometry Vol II New York-London-Sydney: Interscience Publishers a division of John Wiley and Sons (1969) 37 B Kostant: Symplectic Spinors Symp math 14, Geom simplett., Fis mat., Teor geom Integr Var minim., Convegni 1973, 139-152 (1974) 38 A Lichnerowicz: Spineurs harmoniques C R Acad Sci., Paris 257, 7-9 (1963) 39 D McDuff, D Salamon: Introduction to symplectic topology Oxford Mathematical Monographs Oxford: Clarendon Press (1995) 40 I Niven, H.S Zuckerman: An introduction to the theory of numbers 3rd ed New York etc.: John Wiley & Sons, Inc XII (1972) 41 M.J Pflaum: The normal symbol on Riemannian manifolds New York J Math 4, 97-125, electronic only (1998) 42 P.L Robinson, J.H Rawnsley: The metaplectic representation, M pc structures and geometric quantization Mem Am Math Soc 410 (1989) 43 E Schră odinger: Diracsches Elektron im Schwerefeld I Sitzungsber Preuß Akad Wiss., Phys.-Math Kl 1932, No 11/12, 105-128 (1932) 44 M Shubin: A sequence of connections and a characterization of Kă ahler manifolds Farber, Michael (ed.) et al., Tel Aviv topology conference: Rothenberg Festschrift Proceedings of the international conference on topology, Tel Aviv, Israel, June 1-5, 1998 dedicated to Mel Rothenberg on the occasion of his 65th birthday Providence, RI: American Mathematical Society Contemp Math 231, 265-270 (1999) 45 F Sommen: An extension of Clifford analysis towards super-symmetry Ryan, John (ed.) et al., Clifford algebras and their applications in mathematical physics Papers of the 5th international conference, Ixtapa-Zihuatanejo, Mexico, June 27-July 4, 1999 Volume 2: Clifford analysis Boston, MA: Birkhă auser Prog Phys 19, 199-224 (2000) 46 Ph Tondeur: Ane Zusammenhă ange auf Mannigfaltigkeiten mit fast-symplektischer Struktur Comment Math Helv 36, 234-244 (1961) 47 I Vaisman: Symplectic curvature tensors Monatsh Math 100, 299-327 (1985) 48 N.R Wallach: Harmonic Analysis on Homogeneous Spaces Pure and Applied Mathematics 19 New York: Marcel Dekker, Inc (1973) 49 N.R Wallach: Symplectic Geometry and Fourier Analysis With an appendix on quantum mechanics by Robert Hermann Lie Groups: History, Frontiers and Applications Vol V Brookline, Mass.: Math Sci Press (1977) 50 A Weil: Sur certains groupes d’op´ erateurs unitaires Acta Math 111, 143-211 (1964) 51 N.M.J Woodhouse: Geometric quantization 2nd ed Oxford Mathematical Monographs Oxford: Clarendon Press (1997) 52 Ch Wyss: Symplektische Diracoperatoren auf dem komplexen projektiven Raum Diploma Thesis, Universită at Bremen (2003) 118 References ˇ 53 D.P Zelobenko: Compact Lie Groups and their representations Translated from the Russian by Israel Program for Scientific Translations Translations of Mathematical Monographs Vol 40 Providence, R.I.: American Mathematical Society (AMS) VIII (1973) Index almost complex structure, 23 ω-compatible, 23 anti-canonical line bundle, 46 Baum, 49 Bianchi identity, 29 Bourgeois, 21 Cahen, 21 Casimir operator, 90 Cauchy–Riemann equation generalized, 79 Chern class first, 36 complex projective space, 36, 87, 92 complex torus, 85 connection canonical, 90 Hermitian, 27 Levi–Civita, 34, 81 symplectic, 25 reducible, 33 torsion-free, 27 cotangent bundle, 22, 36 curvature, 29 of the spinor derivative, 42 Darboux theorem, 22 deformation of metaplectic structures, 58 divergence, 27 Einstein manifold, 84 Fedosov, 21 flow, 102, 112 formally self-adjoint, 64, 66, 81 Fourier transform, for symplectic spinors, 97 Friedrich, 49 Frobenius reciprocity, 91, 92 Fubini–Study metric, 36, 88 G˚ arding, Gelfand, 21 Gutt, 21 Hamiltonian quadratic, 109 vector field, 24 harmonic oscillator, 17 Heisenberg group, relation, 113 Hermite function, 17, 89 Hermitian connection, 27 structure, 39 holomorphic sectional curvature constant, 33, 84, 94 invariant subspace, Kă ahler form, 22, 81 manifold, 22, 81 Lie derivative, 102 120 Index of symplectic spinor fields, 103 metaplectic group, representation, 10 differential of, 14 structure, 35 momentum operator, 11 normal ordering quantization, 110, 112 orthogonal group, Pell’s equation, 95 Poisson bracket, 24 position operator, 11 quadratic Hamiltonian, 109 quantization, 97, 101 normal ordering, 110, 112 Rawnsley, 21 representation, adjoint, 88 differential of, equivalent, unitary, irreducible, isotropy, 88 metaplectic, 10 differential of, 14 projective unitary, unitary, Retakh, 21 Ricci tensor, 31 symplectic, 31 scalar curvature, 34 symplectic, 34 Schră odinger equation, 111 representation, Schur’s lemma, 5, 93 Schwartz space, sectional curvature, 84 holomorphic, 33, 84 Segal–Shale–Weil representation, 10 Shubin, 21 smooth section, 37 vector, spectrum, 93, 100 spinor derivative, 39 curvature of, 42 Stiefel–Whitney class first, 36 second, 36 Stone–von Neumann theorem, symmetric space, 88 symplectic basis, 1, 23 Clifford algebra, multiplication, 13, 38 connection, 25 curvature tensor, 29 Dirac operator, 50 form, frame, 23 frame bundle, 23 gradient, 24 group, Lie algebra, manifold, 22 Ricci tensor, 31 scalar curvature, 34 spinor, 10 bundle, 37 field, 37 Laplacian, 70 structure, 22 vector field, 102 symplectomorphic, 22 symplectomorphism, 22 Tondeur, 21 torsion, 27 vector field, 27 unitary frame, 24 group, Vaisman, 21 vector field Hamiltonian, 24 symplectic, 102 volume element, 28 weight, 90, 92 Weil, Weitzenbă ock formula, 72, 82 Weyl algebra, Lecture Notes in Mathematics For information about earlier volumes please contact your bookseller or Springer LNM Online archive: springerlink.com Vol 1681: G J Wirsching, The Dynamical System Generated by the 3n+1 Function (1998) Vol 1682: H.-D Alber, Materials with Memory (1998) Vol 1683: A Pomp, The Boundary-Domain Integral Method for Elliptic Systems (1998) Vol 1684: C A Berenstein, P F Ebenfelt, S G Gindikin, S Helgason, A E Tumanov, Integral Geometry, Radon Transforms and Complex Analysis Firenze, 1996 Editors: E Casadio Tarabusi, M A Picardello, G Zampieri (1998) Vol 1685: S König, A Zimmermann, Derived Equivalences for Group Rings (1998) Vol 1686: J Azéma, M Émery, M Ledoux, M Yor (Eds.), Séminaire de Probabilités XXXII (1998) Vol 1687: F Bornemann, Homogenization in Time of Singularly Perturbed Mechanical Systems (1998) Vol 1688: S Assing, W Schmidt, Continuous Strong Markov Processes in Dimension One (1998) Vol 1689: W Fulton, P Pragacz, Schubert Varieties and Degeneracy Loci (1998) Vol 1690: M T Barlow, D Nualart, Lectures on Probability Theory and Statistics Editor: P Bernard (1998) Vol 1691: R Bezrukavnikov, M Finkelberg, V Schechtman, Factorizable Sheaves and Quantum Groups (1998) Vol 1692: T M W Eyre, Quantum Stochastic Calculus and Representations of Lie Superalgebras (1998) Vol 1694: A Braides, Approximation of Free-Discontinuity Problems (1998) Vol 1695: D J Hart el, Markov Set-Chains ( 1998) Vol 1696: E Bouscaren (Ed.): Model Theory and Algebraic Geometry (1998) Vol 1697: B Cockburn, C Johnson, C.-W Shu, E Tadmor, Advanced Numerical Approximation of Nonlinear Hyperbolic Equations Cetraro, Italy, 1997 Editor: A Quarteroni (1998) Vol 1698: M Bhattacharjee, D Macpherson, R G Möller, P Neumann, Notes on In nite Permutation Groups (1998) Vol 1699: A Inoue,Tomita-Takesaki Theory in Algebras of Unbounded Operators (1998) Vol 1700: W A Woyczy´nski, Burgers-KPZ Turbulence (1998) Vol 1701: Ti-Jun Xiao, J Liang, The Cauchy Problem of Higher Order Abstract Differential Equations (1998) Vol 1702: J Ma, J Yong, Forward-Backward Stochastic Differential Equations and Their Applications (1999) Vol 1703: R M Dudley, R Norvaiša, Differentiability of Six Operators on Nonsmooth Functions and pVariation (1999) Vol 1704: H Tamanoi, Elliptic Genera and Vertex Operator Super-Algebras (1999) Vol 1705: I Nikolaev, E Zhuzhoma, Flows in 2-dimensional Manifolds (1999) Vol 1706: S Yu Pilyugin, Shadowing in Dynamical Systems (1999) Vol 1707: R Pytlak, Numerical Methods for Optimal Control Problems with State Constraints (1999) Vol 1708: K Zuo, Representations of Fundamental Groups of Algebraic Varieties (1999) Vol 1709: J Azéma, M Émery, M Ledoux, M Yor (Eds.), Séminaire de Probabilités XXXIII (1999) Vol 1710: M Koecher, The Minnesota Notes on Jordan Algebras and Their Applications (1999) Vol 1711: W Ricker, Operator Algebras Generated by Commuting Proje´ctions: A Vector Measure Approach (1999) Vol 1712: N Schwartz, J J Madden, Semi-algebraic Function Rings and Re ectors of Partially Ordered Rings (1999) Vol 1713: F Bethuel, G Huisken, S Müller, K Steffen, Calculus of Variations and Geometric Evolution Problems Cetraro, 1996 Editors: S Hildebrandt, M Struwe (1999) Vol 1714: O Diekmann, R Durrett, K P Hadeler, P K Maini, H L Smith, Mathematics Inspired by Biology Martina Franca, 1997 Editors: V Capasso, O Diekmann (1999) Vol 1715: N V Krylov, M Röckner, J Zabczyk, Stochastic PDE’s and Kolmogorov Equations in In nite Dimensions Cetraro, 1998 Editor: G Da Prato (1999) Vol 1716: J Coates, R Greenberg, K A Ribet, K Rubin, Arithmetic Theory of Elliptic Curves Cetraro, 1997 Editor: C Viola (1999) Vol 1717: J Bertoin, F Martinelli, Y Peres, Lectures on Probability Theory and Statistics Saint-Flour, 1997 Editor: P Bernard (1999) Vol 1718: A Eberle, Uniqueness and Non-Uniqueness of Semigroups Generated by Singular Diffusion Operators (1999) Vol 1719: K R Meyer, Periodic Solutions of the N-Body Problem (1999) Vol 1720: D Elworthy, Y Le Jan, X-M Li, On the Geometry of Diffusion Operators and Stochastic Flows (1999) Vol 1721: A Iarrobino, V Kanev, Power Sums, Gorenstein Algebras, and Determinantal Loci (1999) Vol 1722: R McCutcheon, Elemental Methods in Ergodic Ramsey Theory (1999) Vol 1723: J P Croisille, C Lebeau, Diffraction by an Immersed Elastic Wedge (1999) Vol 1724: V N Kolokoltsov, Semiclassical Analysis for Diffusions and Stochastic Processes (2000) Vol 1725: D A Wolf-Gladrow, Lattice-Gas Cellular Automata and Lattice Boltzmann Models (2000) Vol 1726: V Mari´c, Regular Variation and Differential Equations (2000) Vol 1727: P Kravanja M Van Barel, Computing the Zeros of Analytic Functions (2000) Vol 1728: K Gatermann Computer Algebra Methods for Equivariant Dynamical Systems (2000) Vol 1729: J Azéma, M Émery, M Ledoux, M Yor (Eds.) Séminaire de Probabilités XXXIV (2000) Vol 1730: S Graf, H Luschgy, Foundations of Quantization for Probability Distributions (2000) Vol 1731: T Hsu, Quilts: Central Extensions, Braid Actions, and Finite Groups (2000) Vol 1732: K Keller, Invariant Factors, Julia Equivalences and the (Abstract) Mandelbrot Set (2000) Vol 1733: K Ritter, Average-Case Analysis of Numerical Problems (2000) Vol 1734: M Espedal, A Fasano, A Mikeli´c, Filtration in Porous Media and Industrial Applications Cetraro 1998 Editor: A Fasano 2000 Vol 1735: D Yafaev, Scattering Theory: Some Old and New Problems (2000) Vol 1736: B O Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces (2000) Vol 1737: S Wakabayashi, Classical Microlocal Analysis in the Space of Hyperfunctions (2000) Vol 1738: M Émery, A Nemirovski, D Voiculescu, Lectures on Probability Theory and Statistics (2000) Vol 1739: R Burkard, P Deu hard, A Jameson, J.-L Lions, G Strang, Computational Mathematics Driven by Industrial Problems Martina Franca, 1999 Editors: V Capasso, H Engl, J Periaux (2000) Vol 1740: B Kawohl, O Pironneau, L Tartar, J.-P Zolesio, Optimal Shape Design Tróia, Portugal 1999 Editors: A Cellina, A Ornelas (2000) Vol 1741: E Lombardi, Oscillatory Integrals and Phenomena Beyond all Algebraic Orders (2000) Vol 1742: A Unterberger, Quantization and Nonholomorphic Modular Forms (2000) Vol 1743: L Habermann, Riemannian Metrics of Constant Mass and Moduli Spaces of Conformal Structures (2000) Vol 1744: M Kunze, Non-Smooth Dynamical Systems (2000) Vol 1745: V D Milman, G Schechtman (Eds.), Geometric Aspects of Functional Analysis Israel Seminar 19992000 (2000) Vol 1746: A Degtyarev, I Itenberg, V Kharlamov, Real Enriques Surfaces (2000) Vol 1747: L W Christensen, Gorenstein Dimensions (2000) Vol 1748: M Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory (2001) Vol 1749: M Fuchs, G Seregin, Variational Methods for Problems from Plasticity Theory and for Generalized Newtonian Fluids (2001) Vol 1750: B Conrad, Grothendieck Duality and Base Change (2001) Vol 1751: N J Cutland, Loeb Measures in Practice: Recent Advances (2001) Vol 1752: Y V Nesterenko, P Philippon, Introduction to Algebraic Independence Theory (2001) Vol 1753: A I Bobenko, U Eitner, Painlevé Equations in the Differential Geometry of Surfaces (2001) Vol 1754: W Bertram, The Geometry of Jordan and Lie Structures (2001) Vol 1755: J Azéma, M Émery, M Ledoux, M Yor (Eds.), Séminaire de Probabilités XXXV (2001) Vol 1756: P E Zhidkov, Korteweg de Vries and Nonlinear Schrödinger Equations: Qualitative Theory (2001) Vol 1757: R R Phelps, Lectures on Choquet’s Theorem (2001) Vol 1758: N Monod, Continuous Bounded Cohomology of Locally Compact Groups (2001) Vol 1759: Y Abe, K Kopfermann, Toroidal Groups (2001) Vol 1760: D Filipovi´c, Consistency Problems for HeathJarrow-Morton Interest Rate Models (2001) Vol 1761: C Adelmann, The Decomposition of Primes in Torsion Point Fields (2001) Vol 1762: S Cerrai, Second Order PDE’s in Finite and In nite Dimension ( 2001) Vol 1763: J.-L Loday, A Frabetti, F Chapoton, F Goichot, Dialgebras and Related Operads (2001) Vol 1764: A Cannas da Silva, Lectures on Symplectic Geometry (2001) Vol 1765: T Kerler, V V Lyubashenko, Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with Corners (2001) Vol 1766: H Hennion, L Hervé, Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness (2001) Vol 1767: J Xiao, Holomorphic Q Classes (2001) Vol 1768: M.J P aum, Analytic and Geometric Study of Strati ed Spaces (2001) Vol 1769: M Alberich-Carramiñana, Geometry of the Plane Cremona Maps (2002) Vol 1770: H Gluesing-Luerssen, Linear DelayDifferential Systems with Commensurate Delays: An Algebraic Approach (2002) Vol 1771: M Émery, M Yor (Eds.), Séminaire de Probabilités 1967-1980 A Selection in Martingale Theory (2002) Vol 1772: F Burstall, D Ferus, K Leschke, F Pedit, U Pinkall, Conformal Geometry of Surfaces in S4 (2002) Vol 1773: Z Arad, M Muzychuk, Standard Integral Table Algebras Generated by a Non-real Element of Small Degree (2002) Vol 1774: V Runde, Lectures on Amenability (2002) Vol 1775: W H Meeks, A Ros, H Rosenberg, The Global Theory of Minimal Surfaces in Flat Spaces Martina Franca 1999 Editor: G P Pirola (2002) Vol 1776: K Behrend, C Gomez, V Tarasov, G Tian, Quantum Comohology Cetraro 1997 Editors: P de Bartolomeis, B Dubrovin, C Reina (2002) Vol 1777: E García-Río, D N Kupeli, R VázquezLorenzo, Osserman Manifolds in Semi-Riemannian Geometry (2002) Vol 1778: H Kiechle, Theory of K-Loops (2002) Vol 1779: I Chueshov, Monotone Random Systems (2002) Vol 1780: J H Bruinier, Borcherds Products on O(2,1) and Chern Classes of Heegner Divisors (2002) Vol 1781: E Bolthausen, E Perkins, A van der Vaart, Lectures on Probability Theory and Statistics Ecole d’ Eté de Probabilités de Saint-Flour XXIX-1999 Editor: P Bernard (2002) Vol 1782: C.-H Chu, A T.-M Lau, Harmonic Functions on Groups and Fourier Algebras (2002) Vol 1783: L Grüne, Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization (2002) Vol 1784: L.H Eliasson, S B Kuksin, S Marmi, J.-C Yoccoz, Dynamical Systems and Small Divisors Cetraro, Italy 1998 Editors: S Marmi, J.-C Yoccoz (2002) Vol 1785: J Arias de Reyna, Pointwise Convergence of Fourier Series (2002) Vol 1786: S D Cutkosky, Monomialization of Morphisms from 3-Folds to Surfaces (2002) Vol 1787: S Caenepeel, G Militaru, S Zhu, Frobenius and Separable Functors for Generalized Module Categories and Nonlinear Equations (2002) Vol 1788: A Vasil’ev, Moduli of Families of Curves for Conformal and Quasiconformal Mappings (2002) Vol 1789: Y Sommerhäuser, Yetter-Drinfel’d Hopf algebras over groups of prime order (2002) Vol 1790: X Zhan, Matrix Inequalities (2002) Vol 1791: M Knebusch, D Zhang, Manis Valuations and Prüfer Extensions I: A new Chapter in Commutative Algebra (2002) Vol 1792: D D Ang, R Goren o, V K Le, D D Trong, Moment Theory and Some Inverse Problems in Potential Theory and Heat Conduction (2002) Vol 1793: J Cortés Monforte, Geometric, Control and Numerical Aspects of Nonholonomic Systems (2002) Vol 1794: N Pytheas Fogg, Substitution in Dynamics, Arithmetics and Combinatorics Editors: V Berthé, S Ferenczi, C Mauduit, A Siegel (2002) Vol 1795: H Li, Filtered-Graded Transfer in Using Noncommutative Gröbner Bases (2002) Vol 1796: J.M Melenk, hp-Finite Element Methods for Singular Perturbations (2002) Vol 1797: B Schmidt, Characters and Cyclotomic Fields in Finite Geometry (2002) Vol 1798: W.M Oliva, Geometric Mechanics (2002) Vol 1799: H Pajot, Analytic Capacity, Recti ability, Menger Curvature and the Cauchy Integral (2002) Vol 1800: O Gabber, L Ramero, Almost Ring Theory (2003) Vol 1801: J Azéma, M Émery, M Ledoux, M Yor (Eds.), Séminaire de Probabilités XXXVI (2003) Vol 1802: V Capasso, E Merzbach, B.G Ivanoff, M Dozzi, R Dalang, T Mountford, Topics in Spatial Stochastic Processes Martina Franca, Italy 2001 Editor: E Merzbach (2003) Vol 1803: G Dolzmann, Variational Methods for Crystalline Microstructure – Analysis and Computation (2003) Vol 1804: I Cherednik, Ya Markov, R Howe, G Lusztig, Iwahori-Hecke Algebras and their Representation Theory Martina Franca, Italy 1999 Editors: V Baldoni, D Barbasch (2003) Vol 1805: F Cao, Geometric Curve Evolution and Image Processing (2003) Vol 1806: H Broer, I Hoveijn G Lunther, G Vegter, Bifurcations in Hamiltonian Systems Computing Singularities by Gröbner Bases (2003) Vol 1807: V D Milman, G Schechtman (Eds.), Geometric Aspects of Functional Analysis Israel Seminar 20002002 (2003) Vol 1808: W Schindler, Measures with Symmetry Properties (2003) Vol 1809: O Steinbach, Stability Estimates for Hybrid Coupled Domain Decomposition Methods (2003) Vol 1810: J Wengenroth, Derived Functors in Functional Analysis (2003) Vol 1811: J Stevens, Deformations of Singularities (2003) Vol 1812: L Ambrosio, K Deckelnick, G Dziuk, M Mimura, V A Solonnikov, H M Soner, Mathematical Aspects of Evolving Interfaces Madeira, Funchal, Portugal 2000 Editors: P Colli, J F Rodrigues (2003) Vol 1813: L Ambrosio, L A Caffarelli, Y Brenier, G Buttazzo, C Villani, Optimal Transportation and its Applications Martina Franca, Italy 2001 Editors: L A Caffarelli, S Salsa (2003) Vol 1814: P Bank, F Baudoin, H Föllmer, L.C.G Rogers, M Soner, N Touzi, Paris-Princeton Lectures on Mathematical Finance 2002 (2003) Vol 1815: A M Vershik (Ed.), Asymptotic Combinatorics with Applications to Mathematical Physics St Petersburg, Russia 2001 (2003) Vol 1816: S Albeverio, W Schachermayer, M Talagrand, Lectures on Probability Theory and Statistics Ecole d’Eté de Probabilités de Saint-Flour XXX-2000 Editor: P Bernard (2003) Vol 1817: E Koelink, W Van Assche(Eds.), Orthogonal Polynomials and Special Functions Leuven 2002 (2003) Vol 1818: M Bildhauer, Convex Variational Problems with Linear, nearly Linear and/or Anisotropic Growth Conditions (2003) Vol 1819: D Masser, Yu V Nesterenko, H P Schlickewei, W M Schmidt, M Waldschmidt, Diophantine Approximation Cetraro, Italy 2000 Editors: F Amoroso, U Zannier (2003) Vol 1820: F Hiai, H Kosaki, Means of Hilbert Space Operators (2003) Vol 1821: S Teufel, Adiabatic Perturbation Theory in Quantum Dynamics (2003) Vol 1822: S.-N Chow, R Conti, R Johnson, J MalletParet, R Nussbaum, Dynamical Systems Cetraro, Italy 2000 Editors: J W Macki, P Zecca (2003) Vol 1823: A M Anile, W Allegretto, C Ringhofer, Mathematical Problems in Semiconductor Physics Cetraro, Italy 1998 Editor: A M Anile (2003) Vol 1824: J A Navarro González, J B Sancho de Salas, C ∞ – Differentiable Spaces (2003) Vol 1825: J H Bramble, A Cohen, W Dahmen, Multiscale Problems and Methods in Numerical Simulations, Martina Franca, Italy 2001 Editor: C Canuto (2003) Vol 1826: K Dohmen, Improved Bonferroni Inequalities via Abstract Tubes Inequalities and Identities of Inclusion-Exclusion Type VIII, 113 p, 2003 Vol 1827: K M Pilgrim, Combinations of Complex Dynamical Systems IX, 118 p, 2003 Vol 1828: D J Green, Gröbner Bases and the Computation of Group Cohomology XII, 138 p, 2003 Vol 1829: E Altman, B Gaujal, A Hordijk, DiscreteEvent Control of Stochastic Networks: Multimodularity and Regularity XIV, 313 p, 2003 Vol 1830: M I Gil’, Operator Functions and Localization of Spectra XIV, 256 p, 2003 Vol 1831: A Connes, J Cuntz, E Guentner, N Higson, J E Kaminker, Noncommutative Geometry, Martina Franca, Italy 2002 Editors: S Doplicher, L Longo (2004) Vol 1832: J Azéma, M Émery, M Ledoux, M Yor (Eds.), Séminaire de Probabilités XXXVII (2003) Vol 1833: D.-Q Jiang, M Qian, M.-P Qian, Mathematical Theory of Nonequilibrium Steady States On the Frontier of Probability and Dynamical Systems IX, 280 p, 2004 Vol 1834: Yo Yomdin, G Comte, Tame Geometry with Application in Smooth Analysis VIII, 186 p, 2004 Vol 1835: O.T Izhboldin, B Kahn, N.A Karpenko, A Vishik, Geometric Methods in the Algebraic Theory of Quadratic Forms Summer School, Lens, 2000 Editor: J.P Tignol (2004) Vol 1836: C Nastasescu, F Van Oystaeyen, Methods of Graded Rings XIII, 304 p, 2004 Vol 1837: S Tavaré, O Zeitouni, Lectures on Probability Theory and Statistics Ecole d’Eté de Probabilités de Saint-Flour XXXI-2001 Editor: J Picard (2004) Vol 1838: A.J Ganesh, N.W O’Connell, D.J Wischik, Big Queues XII, 254 p, 2004 Vol 1839: R Gohm, Noncommutative Stationary Processes VIII, 170 p, 2004 Vol 1840: B Tsirelson, W Werner, Lectures on Probability Theory and Statistics Ecole d’Eté de Probabilités de Saint-Flour XXXII-2002 Editor: J Picard (2004) Vol 1841: W Reichel, Uniqueness Theorems for Variational Problems by the Method of Transformation Groups (2004) Vol 1842: T Johnsen, A.L Knutsen, K3 Projective Models in Scrolls (2004) Vol 1843: B Jefferies, Spectral Properties of Noncommuting Operators (2004) Vol 1844: K.F Siburg, The Principle of Least Action in Geometry and Dynamics (2004) Vol 1845: Min Ho Lee, Mixed Automorphic Forms, Torus Bundles, and Jacobi Forms (2004) Vol 1846: H Ammari, H Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements (2004) Vol 1847: T.R Bielecki, T Björk, M Jeanblanc, M Rutkowski, J.A Scheinkman, W Xiong, Paris-Princeton Lectures on Mathematical Finance 2003 (2004) Vol 1848: M Abate, J E Fornaess, X Huang, J P Rosay, A Tumanov, Real Methods in Complex and CR Geometry, Martina Franca, Italy 2002 Editors: D Zaitsev, G Zampieri (2004) Vol 1849: Martin L Brown, Heegner Modules and Elliptic Curves (2004) Vol 1850: V D Milman, G Schechtman (Eds.), Geometric Aspects of Functional Analysis Israel Seminar 20022003 (2004) Vol 1851: O Catoni, Statistical Learning Theory and Stochastic Optimization (2004) Vol 1852: A.S Kechris, B.D Miller, Topics in Orbit Equivalence (2004) Vol 1853: Ch Favre, M Jonsson, The Valuative Tree (2004) Vol 1854: O Saeki, Topology of Singular Fibers of Differential Maps (2004) Vol 1855: G Da Prato, P.C Kunstmann, I Lasiecka, A Lunardi, R Schnaubelt, L Weis, Functional Analytic Methods for Evolution Equations Editors: M Iannelli, R Nagel, S Piazzera (2004) Vol 1856: K Back, T.R Bielecki, C Hipp, S Peng, W Schachermayer, Stochastic Methods in Finance, Bressanone/Brixen, Italy, 2003 Editors: M Fritelli, W Runggaldier (2004) Vol 1857: M Émery, M Ledoux, M Yor (Eds.), Séminaire de Probabilités XXXVIII (2005) Vol 1858: A.S Cherny, H.-J Engelbert, Singular Stochastic Differential Equations (2005) Vol 1859: E Letellier, Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras (2005) Vol 1860: A Borisyuk, G.B Ermentrout, A Friedman, D Terman, Tutorials in Mathematical Biosciences I Mathematical Neurosciences (2005) Vol 1861: G Benettin, J Henrard, S Kuksin, Hamiltonian Dynamics – Theory and Applications, Cetraro, Italy, 1999 Editor: A Giorgilli (2005) Vol 1862: B Helffer, F Nier, Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians (2005) Vol 1863: H Fürh, Abstract Harmonic Analysis of Continuous Wavelet Transforms (2005) Vol 1864: K Efstathiou, Metamorphoses of Hamiltonian Systems with Symmetries (2005) Vol 1865: D Applebaum, B.V R Bhat, J Kustermans, J M Lindsay, Quantum Independent Increment Processes I From Classical Probability to Quantum Stochastic Calcu- lus Editors: M Schürmann, U Franz (2005) Vol 1866: O.E Barndorff-Nielsen, U Franz, R Gohm, B Kümmerer, S Thorbjønsen, Quantum Independent Increment Processes II Structure of Quantum Lévy Processes, Classical Probability, and Physics Editors: M Schürmann, U Franz, (2005) Vol 1867: J Sneyd (Ed.), Tutorials in Mathematical Biosciences II Mathematical Modeling of Calcium Dynamics and Signal Transduction (2005) Vol 1868: J Jorgenson, S Lang, Posn (R) and Eisenstein Sereies (2005) Vol 1869: A Dembo, T Funaki, Lectures on Probability Theory and Statistics Ecole d’Eté de Probabilités de Saint-Flour XXXIII-2003 Editor: J Picard (2005) Vol 1870: V.I Gurariy, W Lusky, Geometry of Müntz Spaces and Related Questions (2005) Vol 1871: P Constantin, G Gallavotti, A.V Kazhikhov, Y Meyer, S Ukai, Mathematical Foundation of Turbulent Viscous Flows, Martina Franca, Italy, 2003 Editors: M Cannone, T Miyakawa (2006) Vol 1872: A Friedman (Ed.), Tutorials in Mathematical Biosciences III Cell Cycle, Proliferation, and Cancer (2006) Vol 1873: R Mansuy, M Yor, Random Times and Enlargements of Filtrations in a Brownian Setting (2006) Vol 1874: M Yor, M Émery (Eds.), In Memoriam PaulAndré Meyer - Séminaire de Probabilités XXXIX (2006) Vol 1875: J Pitman, Combinatorial Stochastic Processes Ecole d’Eté de Probabilités de Saint-Flour XXXII-2002 Editor: J Picard (2006) Vol 1876: H Herrlich, Axiom of Choice (2006) Vol 1877: J Steuding, Value Distributions of L-Functions (2006) Vol 1878: R Cerf, The Wulff Crystal in Ising and Percolation Models, Ecole d'Eté de Probabilit és de Saint-Flour XXXIV-2004 Editor: Jean Picard (2006) Vol 1879: G Slade, The Lace Expansion and its Applications, Ecole d'Eté de Probabilités de Saint-Flour XXXIV-2004 Editor: Jean Picard (2006) Vol 1880: S Attal, A Joye, C.-A Pillet, Open Quantum Systems I, The Hamiltonian Approach (2006) Vol 1881: S Attal, A Joye, C.-A Pillet, Open Quantum Systems II, The Markovian Approach (2006) Vol 1882: S Attal, A Joye, C.-A Pillet, Open Quantum Systems III, Recent Developments (2006) Vol 1883: W Van Assche, F Marcellàn (Eds.), Orthogonal Polynomials and Special Functions, Computation and Application (2006) Vol 1884: N Hayashi, E.I Kaikina, P.I Naumkin, I.A Shishmarev, Asymptotics for Dissipative Nonlinear Equations (2006) Vol 1885: A Telcs, The Art of Random Walks (2006) Vol 1886: S Takamura, Splitting Deformations of Degenerations of Complex Curves (2006) Vol 1887: K Habermann, L Habermann, Introduction to Symplectic Dirac Operators (2006) Recent Reprints and New Editions Vol 1618: G Pisier, Similarity Problems and Completely Bounded Maps 1995 – Second, Expanded Edition (2001) Vol 1629: J.D Moore, Lectures on Seiberg-Witten Invariants 1997 – Second Edition (2001) Vol 1638: P Vanhaecke, Integrable Systems in the realm of Algebraic Geometry 1996 – Second Edition (2001) Vol 1702: J Ma, J Yong, Forward-Backward Stochastic Differential Equations and their Applications 1999 – Corrected 3rd printing (2005) .. .K Habermann · L Habermann Introduction to Symplectic Dirac Operators ABC Authors Katharina Habermann State and University Library Göttingen Platz... 11736684 V A 41/3100/ SPI 543210 To Karen Preface This book aims to give a systematic and self-contained introduction to the theory of symplectic Dirac operators and to reflect the current state of... (1.5.8) Symplectic Connections In order to define symplectic Dirac operators, we have to fix a symplectic connection on the underlying symplectic manifold Every symplectic manifold admits symplectic