DOCSLIB.ORG

Localization and Diagonalization: a Review of Functional

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Localization and Diagonalization: a Review of Functional View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server IC/95/5 hep-th/9501075 1 Lo calization and Diagonalization A Review of Functional Integral Techniques for Low-Dimensional Gauge Theories and Top ological Field Theories 2 3 4 Matthias Blau ' and George Thompson ICTP P.O. Box 586 I-34014 Trieste Italy Abstract We review lo calization techniques for functional integrals which have recently b een used to p erform calculations in and gain insightinto the structure of certain top ological eld theories and low-dimensional processed by the SLAC/DESY Libraries on 18 Jan 1995. 〉 gauge theories. These are the functional integral counterparts of the Mathai-Quillen formalism, the Duistermaat-Heckman theorem, and the Weyl integral formula resp ectively. In each case, we rst intro duce PostScript the necessary mathematical background (Euler classes of vector bun- dles, equivariant cohomology, top ology of Lie groups), and describ e the nite dimensional integration formulae. We then discuss some applica- tions to path integrals and giveanoverview of the relevant literature. The applications we deal with include sup ersymmetric quantum me- chanics, cohomological eld theories, phase space path integrals, and two-dimensional Yang-Mills theory. 1 To app ear in the Journal of Mathematical Physics Sp ecial Issue on Functional Inte- gration (May 1995) 2 e-mail: [email protected] t, [email protected] s-lyo n.fr 3 Present Address: Lab. Physique Theorique, Ecole Normale Superieure de Lyon, France 4 e-mail: [email protected] t HEP-TH-9501075 Contents 1 Intro duction 1 2 Lo calization via the Mathai-Quillen Formalism 3 2.1 The Mathai-Quillen Formalism :: ::: :::: ::: :::: ::: :: 5 The Euler class of a nite dimensional vector bundle :::: ::: :: 5 The Mathai-Quillen representative of the Euler class :::: ::: :: 6 The Mathai-Quillen formalism for in nite dimensional vector bundles 9 2.2 The Regularized Euler Numb er of Lo op Space, or: Sup ersymmetric Quantum Mechanics ::: ::: :::: ::: :: 11 Geometry of lo op space :: :::: ::: :::: ::: :::: ::: :: 11 Sup ersymmetric quantum mechanics :: :::: ::: :::: ::: :: 12 2.3 Other Examples and Applications - an Overview :: :::: ::: :: 16 The basic strategy - illustrated by Donaldson theory :::: ::: :: 16 A top ological gauge theory of at connections : ::: :::: ::: :: 19 Top ological sigma mo dels : :::: ::: :::: ::: :::: ::: :: 21 BRST xed p oints and lo calization ::: :::: ::: :::: ::: :: 23 3 Equivariant Lo calization and the Stationary Phase Approxima- tion, or: The Duistermaat-Heckman Theorem 24 3.1 Equivariant Cohomology and Lo calization ::: ::: :::: ::: :: 27 Equivariant cohomology :: :::: ::: :::: ::: :::: ::: :: 27 The Berline-Vergne and Duistermaat-Heckman lo calization formulae 29 3.2 Lo calization Formulae for Phase Space Path Integrals ::: ::: :: 34 Equivariant cohomology for phase space path integrals ::: ::: :: 35 Lo calization formulae ::: :::: ::: :::: ::: :::: ::: :: 36 3.3 Other Examples and Applications - an Overview :: :::: ::: :: 40 Sup ersymmetric quantum mechanics and index theorems :: ::: :: 40 A lo calization formula for the square of the moment map : ::: :: 43 Character Formulae and other applications :: ::: :::: ::: :: 44 4 Gauge Invariance and Diagonalization - the Weyl Integral Formula 45 4.1 The Weyl Integral Formula :::: ::: :::: ::: :::: ::: :: 47 1 Background from the theory of Lie groups ::: ::: :::: ::: :: 47 Top ological obstructions to diagonalization :: ::: :::: ::: :: 50 The Weyl integral formula for path integrals :: ::: :::: ::: :: 55 4.2 Solving 2d Yang-Mills Theory via Ab elianization :: :::: ::: :: 57 Some Lie algebra theory :: :::: ::: :::: ::: :::: ::: :: 57 2d Yang-Mills theory : ::: :::: ::: :::: ::: :::: ::: :: 58 References 63 1 Intro duction In recentyears, the functional integral has b ecome a very p opular to ol in a branchofphysics lying on the interface b etween string theory, conformal eld theory and top ological eld theory on the one hand and top ology and algebraic geometry on the other. Not only has it b ecome p opular but it has also, b ecause of the consistent reliability of the results the functional integral can pro duce when handled with due care, acquired a certain degree of resp ectability among mathematicians. Here, however, our fo cus will not b e primarily on the results or predictions obtained by these metho ds, b ecause an appreciation of these results would require a rather detailed understanding of the mathematics and physics involved. Rather, wewant to explain some of the general features and prop erties the functional integrals app earing in this context have in com- mon. Foremost among these is the fact that, due to a large number of (sup er-)symmetries, these functional integrals essentially represent nite- dimensional integrals. The transition b etween the functional and nite dimensional integrals can then naturally b e regarded as a (rather drastic) lo calization of the original in nite dimensional integral. The purp ose of this article is to giveanintro- duction to and an overview of some functional integral tricks and techniques whichhave turned out to b e useful in understanding these prop erties and whichthus provide insightinto the structure of top ological eld theories and some of their close relatives in general. More sp eci cally,we fo cus on three techniques which are extensions to func- tional integrals of nite dimensional integration and lo calization formulae which are quite interesting in their own right, namely 2 1. the Mathai-Quillen formalism [1], dealing with integral representations of Euler classes of vector bundles; 2. the Duistermaat-Heckman theorem [2] on the exactness of the station- ary phase approximation for certain phase space path integrals, and its generalizations [3,4]; 3. the classical Weyl integral formula, relating integrals over a compact Lie group or Lie algebra to integrals over a maximal torus or a Cartan subalgebra. We will deal with these three techniques in sections 2-4 of this article resp ec- tively. In each case, we will rst try to provide the necessary mathematical background (Euler classes of vector bundles, equivariant cohomology, top ol- ogy of Lie groups). We then explain the integration formulae in their nite dimensional setting b efore wemove on b oldly to apply these techniques to some sp eci c in nite dimensional examples like sup ersymmetric quantum mechanics or two-dimensional Yang-Mills theory. Of course, the functional integrals we deal with in this article are very sp e- cial, corresp onding to theories with no eld theoretic degrees of freedom. Moreover, our treatment of functional integrals is completely formal as re- gards its functional analytic asp ects. So lest we lose the reader interested primarily in honest quantum mechanics or eld theory functional integrals at this p oint, we should p erhaps explain whywe b elieve that it is, nevertheless, worthwile to lo ok at these examples and techniques. First of all, precisely b ecause the integrals we deal with are essentially nite dimensional integrals, and the path integral manipulations are frequently known to pro duce the correct (meaning e.g. top ologically correct) results, any de nition of, or approach to, the functional integral worth its salt should b e able to repro duce these results and to incorp orate these techniques in some way. This applies in particular to the in nite dimensional analogue of the Mathai-Quillen formalism and to the Weyl integral formula. Secondly, the kinds of theories we deal with here allow one to study kinemat- ical (i.e. geometrical and top ological) asp ects of the path integral in isolation from their dynamical asp ects. In this sense, these theories are complemen- tary to, say, simple interacting theories. While the latter are typically kine- matically linear but dynamically non-linear, the former are usually dynami- cally linear (free eld theories) but kinematically highly non-linear (and the 3 entire non-triviality of the theories resides in this kinematic non-linearity). This is a feature shared by all the three techniques we discuss. Thirdly, in principle the techniques we review here are also applicable to theories with eld theoretic degrees of freedom, at least in the sense that they provide alternative approximation techniques to the usual p erturbative expansion. Here wehave in mind primarily the Weyl integral formula and the generalized WKB approximation techniques based on the Duistermaat- Heckman formula. Finally,wewant to draw attention to the fact that many top ological eld theories can b e interpreted as `twisted' versions of ordinary sup ersymmetric eld theories, and that this relation has already led to a dramatic improve- ment of our understanding of b oth typ es of theories, see e.g. [5]-[9] for some recent developments. Hoping to have p ersuaded the reader to stay with us, we will nowsketch brie y the organization of this pap er. As mentioned ab ove, we will deal with the three di erent techniques in three separate sections, each one of them b eginning with an intro duction to the mathematical background. The article is written in suchaway that these three sections can b e read indep endently of each other. Wehave also tried to set things up in suchaway that the reader primarily interested in the path integral applications should b e able to skip the mathematical intro duction up on rst reading and to move directly to the relevant section, going back to the more formal considerations with a solid example at hand. Furthermore, ample references are given so that the interested reader should b e able to trackdown most of the applications of the techniques that we describ e. Originally,we had intended to include a fth section explaining the interre- lations among the three seemingly rather di erent techniques we discuss in this article. These relations exist. A nice example to keep in mind is two- dimensional Yang-Mills theory which can b e solved either via a version of the Duistermaat-Heckman formula or using the Weyl integral formula, but which is also equivalent to a certain two-dimensional cohomological eld theory which provides a eld theoretic realization of the Mathai-Quillen formalism.
Load more

Recommended publications
  • Hardy-Type Inequalities for Integral Transforms Associated with Jacobi Operator
    HARDY-TYPE INEQUALITIES FOR INTEGRAL TRANSFORMS ASSOCIATED WITH JACOBI OPERATOR M. DZIRI AND L. T. RACHDI Received 8 April 2004 and in revised form 24 December 2004 We establish Hardy-type inequalities for the Riemann-Liouville and Weyl transforms as- sociated with the Jacobi operator by using Hardy-type inequalities for a class of integral operators. 1. Introduction It is well known that the Jacobi second-order differential operator is defined on ]0,+∞[ by 1 d du 2 ∆α,βu(x) = Aα,β(x) + ρ u(x), (1.1) Aα,β(x) dx dx where 2ρ 2α+1 2β+1 (i) Aα,β(x) = 2 sinh (x)cosh (x), (ii) α,β ∈ R; α ≥ β>−1/2, (iii) ρ = α + β +1. The Riemann-Liouville and Weyl transforms associated with Jacobi operator ∆α,β are, respectively, defined, for every nonnegative measurable function f ,by x Rα,β( f )(x) = kα,β(x, y) f (y)dy, 0 ∞ (1.2) Wα,β( f )(y) = kα,β(x, y) f (x)Aα,β(x)dx, y where kα,β is the nonnegative kernel defined, for x>y>0, by −1 2 2−α+3/2Γ(α +1) cosh(2x) − cosh(2y) α / kα,β(x, y) = √ πΓ(α +1/2)coshα+β(x)sinh2α(x) (1.3) 1 cosh(x) − cosh(y) × F α + β,α − β;α + ; 2 2cosh(x) Copyright © 2005 Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences 2005:3 (2005) 329–348 DOI: 10.1155/IJMMS.2005.329 330 Hardy-type inequalities and F is the Gaussian hypergeometric function.
    [Show full text]
  • K-Weyl Fractional Integral I Introduction and Preliminaries
    Int. Journal of Math. Analysis, Vol. 6, 2012, no. 34, 1685 - 1691 k-Weyl Fractional Integral Luis Guillermo Romero, Ruben Alejandro Cerutti and Gustavo Abel Dorrego Faculty of Exact Sciences National University of Nordeste. Avda. Libertad 5540 (3400) Corrientes, Argentina [email protected] [email protected] Abstract In this paper we present a generalization to the k-calculus of the Weyl Fractional Integral. Show the semigroup property and calculate their Fractional Fourier Transform. Mathematics Subject Classification: 26A33, 42A38 Keywords: k-Fractional Calculus. Weyl fractional integral I Introduction and Preliminaries In [1] Diaz and Pariguan has defined new functions called the k-Gamma function and the Pochhammer k-symbol that are generalization of the clas- sical Gamma function and the classical Pochhammer symbol. Later in 2012, Mubeen and Habibullah [11] has introduced the k-fractional integral of the Riemann-Liouville type. The purpose of this paper is introduce a fractional integral operator of Weyl type and study that may be posible to exprese this integral operator as certain convolution with the singular kernel of Riemann- Liouville. The classical Weyl integral operator of order α is defined (cf. [10]) in the form 1 ∞ W αf(u)= (t − u)α−1f(t)dt (I.1) Γ(α) u u ≥ 0, α>0 and f a function belonging to S(R) the Schwartzian space of functions, and Γ(z) is the Gamma Euler function given by the integral 1686 L. G. Romero, R. A. Cerutti and G. A. Dorrego ∞ Γ(z)= e−ttz−1dt, z ∈ C, Re(z) > 0, cf.
    [Show full text]
  • Multiplicity Formula and Stable Trace Formula 3
    MULTIPLICITY FORMULA AND STABLE TRACE FORMULA PENG ZHIFENG Abstract. Let G be a connected reductive group over Q. In this paper, we give the stabilization of the local trace formula. In particular, we construct the explicit form of the spectral side of the stable local trace formula in the Archimedean case, when one component of the test function is cuspidal. We also give the multiplicity formula for discrete series. At the same time, we obtain the stable version of L2-Lefschetz number formula. 1. Introduction Suppose that G is a connected reductive group over Q, and Γ is an arithmetic subgroup of G(R) defined by congruence conditions. Consider the regular representation R with G(R) acting on L2(Γ\G(R)) through the right translation. The fundamental problem is to decompose R into a direct sum of irreducible representations. In general, we decompose R into two parts R = Rdisc ⊕ Rcont, where Rdisc is the sum of discrete series, and Rcont is the continuous spectrum. The con- tinuous spectrum can be understood by Eisenstein series, which was studied by Langlands [23]. It suffices to study Rdisc. If πR ∈ Rdisc is an irreducible representation, we denote Rdisc(πR) for the πR-isotypical subspace of Rdisc. Then ⊕mdisc(πR) Rdisc(πR)= πR , where mdisc(πR) is the multiplicity. A classical problem is to find a finite summation formula for mdisc(πR). arXiv:1608.00055v3 [math.NT] 5 Jun 2018 If πR belongs to the square integrable discrete series, and Γ\G(R) is compact, then Langlands [21] gave a formula for mdisc(πR).
    [Show full text]
  • Efficient Numerical Methods for Fractional Differential Equations
    Von der Carl-Friedrich-Gauß-Fakultat¨ fur¨ Mathematik und Informatik der Technischen Universitat¨ Braunschweig genehmigte Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) von Marc Weilbeer Efficient Numerical Methods for Fractional Differential Equations and their Analytical Background 1. Referent: Prof. Dr. Kai Diethelm 2. Referent: Prof. Dr. Neville J. Ford Eingereicht: 23.01.2005 Prufung:¨ 09.06.2005 Supported by the US Army Medical Research and Material Command Grant No. DAMD-17-01-1-0673 to the Cleveland Clinic Contents Introduction 1 1 A brief history of fractional calculus 7 1.1 The early stages 1695-1822 . 7 1.2 Abel's impact on fractional calculus 1823-1916 . 13 1.3 From Riesz and Weyl to modern fractional calculus . 18 2 Integer calculus 21 2.1 Integration and differentiation . 21 2.2 Differential equations and multistep methods . 26 3 Integral transforms and special functions 33 3.1 Integral transforms . 33 3.2 Euler's Gamma function . 35 3.3 The Beta function . 40 3.4 Mittag-Leffler function . 42 4 Fractional calculus 45 4.1 Fractional integration and differentiation . 45 4.1.1 Riemann-Liouville operator . 45 4.1.2 Caputo operator . 55 4.1.3 Grunw¨ ald-Letnikov operator . 60 4.2 Fractional differential equations . 65 4.2.1 Properties of the solution . 76 4.3 Fractional linear multistep methods . 83 5 Numerical methods 105 5.1 Fractional backward difference methods . 107 5.1.1 Backward differences and the Grunw¨ ald-Letnikov definition . 107 5.1.2 Diethelm's fractional backward differences based on quadrature .
    [Show full text]
  • Weyl Character Formula in KK-Theory
    Weyl Character Formula in KK-Theory Jonathan Block1 and Nigel Higson2 1 Department of Mathematics, University of Pennsylvania, 209 South 33rd Street Philadelphia, PA 19104, USA. [email protected] 2 Department of Mathematics, Penn State University, University Park, PA 16803, USA. [email protected] 1 Introduction The purpose of this paper is to begin an exploration of connections between the Baum-Connes conjecture in operator K-theory and the geometric repre- sentation theory of reductive Lie groups. Our initial goal is very modest, and we shall not stray far from the realm of compact groups, where geometric representation theory amounts to elaborations of the Weyl character formula such as the Borel-Weil-Bott theorem. We shall recast the topological K-theory approach to the Weyl character formula, due basically to Atiyah and Bott, in the language of Kasparov's KK-theory [Kas80]. Then we shall show how, contingent on the Baum-Connes conjecture, our KK-theoretic Weyl character formula can be carried over to noncompact groups. The current form of the Baum-Connes conjecture is presented in [BCH94], and the case of Lie groups is discussed there in some detail. On the face of it, the conjecture is removed from traditional issues in representation theory, since it concerns the K-theory of group C∗-algebras, and therefore projective or quasi-projective modules over group C∗-algebras, rather than for example irreducible G-modules. But an important connection with the representation theory of reductive Lie groups was evident from the beginning. The conjec- ture uses the reduced group C∗-algebra, generated by convolution operators on L2(G), and as a result discrete series representations are projective in the appropriate C∗-algebraic sense.
    [Show full text]
  • A Brief History and Exposition of the Fundamental Theory of Fractional Calculus
    From "Fractional Calculus and its Applications", Springer Lecture Notes in Mathematics, volume 57, 1975, pp.1-36. A BRIEF HISTORY AND EXPOSITION OF THE FUNDAMENTAL THEORY OF FRACTIONAL CALCULUS BERTRAM ROSS Abstract: This opening lecture is intended to serve as a propaedeutic for the papers to be presented at this conference whose nonhomogeneous audience includes scientists, mathematicians, engineers and educators. This expository and developmental lecture, a case study of mathemati- cal growth, surveys the origin and development of a mathematical idea from its birth in intellectual curiosity to applications. The funda- mental structure of fractional calculus is outlined. The possibilities for the use of fractional calculus in applicab]e mathematics is indi- cated. The lecture closes with a statement of the purpose of the con- ference. Fractional calculus has its origin in the question of the ex- tension of meaning. A well known example is the extension of meaning of real numbers to complex numbers, and another is the extension of meaning of factorials of integers to factorials of complex numbers. In generalized integration and differentiation the question of the extension of meaning is: Can the meaning of derivatives of integral order dny/dx n be extended to have meaning where n is any number--- irrational, fractional or complex? Leibnitz invented the above notation. Perhaps, it was naive play with symbols that prompted L'Hospital to ask Leibnitz about the possibility that n be a fraction. "What if n be ½?", asked L'Hospital. Leibnitz [i] in 1695 replied, "It will lead to a paradox." But he added prophetically, "From this apparent paradox, one day use- ful consequences will be drawn." In 1697, Leibnitz, referring to Wallis's infinite product for ~/2, used the notation d2y and stated that differential calculus might have been used to achieve the same result.
    [Show full text]
  • Acoustic Mode, 284 Acoustic Cloaking, 228, 244 Inertial, 256 Acoustic
    Index acoustic mode, 284 Bloch wave mode, 281 acoustic cloaking, 228, 244 Bloch wave vector, 269, 272 inertial, 256 Bloch-Floquet theorem, 205, 269, 285, acoustic intensity level, 23 288 acoustic metafluid, 220, 255, 256 body force, 13, 196 acoustic metamaterial, 191 bound electron, 130 acoustic power, 23 Brewster’s angle, 66, 76 acoustic tensor, 52, 347 Brillouin zone, 284 effective, 274 bubbly liquid, 275 acoustics, 16, 63, 64, 107, 267 bulk modulus, 17, 267, 275 adiabatic, 17 complex, 181 adiabatic index, 17 bulk viscosity, 18 adjoint, 199 effective, 278 Airy equation, 312 Airy function, 312 capacitance, 141 Ament’s relation, 153 Cartesian coordinates Ampere’s law, 26 acoustics, 19 angular momentum, 4, 177 elastodynamics, 7 anisotropic mass, 167 electrodynamics, 31 anisotropic mass density, 40, 167, 326 Cauchy principal value, 171 anisotropy, 324, 325 Cauchy stress, 4, 5, 7, 8, 11, 14, 17, anomalous cloaking, 243 18, 56, 62, 69, 114, 182, 186, antiplane shear, 13, 37, 38, 54, 190 189, 196, 203, 211, 252, 271, artificial magnetism, 137 272, 277, 301, 325, 335, 336, attenuation length, 282 347 auxetic material, 214 unsymmetric, 213 axial vector, 175 Cauchy’s integral formula, 169 Cauchy-Goursat theorem, 171 Backus upscaling, 334 Cauchy-Riemann equations, 169 band gap, 190, 285 causality, 5, 17, 20, 34, 41, 140, 169, Bessel function, 97, 122, 136, 141, 312 184, 241 spherical, 107 chiral medium, 203 Bessel’s equation, 97, 117 Christoffel symbols, 9 bi-anisotropic medium, 202 cloaking, 227 Biot-Savart law, 26 acoustic, 244 Bloch condition,
    [Show full text]
  • Function Spaces Generated by Blocks and Spaces Of
    transactions of the american mathematical society Volume 309, Number 1, September 1988 FUNCTION SPACES GENERATED BY BLOCKS ASSOCIATED WITH SPHERES, LIE GROUPS AND SPACES OF HOMOGENEOUS TYPE ALE§ ZALOZNIK ABSTRACT. Functions generated by blocks were introduced by M. Taibleson and G. Weiss in the setting of the one-dimensional torus T [TW1]. They showed that these functions formed a space "close" to the class of integrable functions for which we have almost everywhere convergence of Fourier series. Together with S. Lu [LTW] they extended the theory to the n-dimensional torus where this convergence result (for Bochner-Riesz means at the critical index) is valid provided we also restrict ourselves to L log L. In this paper we show that this restriction is not needed if the underlying domain is a compact semisimple Lie group (or certain more general spaces of a homogeneous type). Other considerations (for example, these spaces form an interesting family of quasi-Banach spaces; they are connected with the notion of entropy) guide one in their study. We show how this point of view can be exploited in the setting of more general underlying domains. 1. Introduction. Function spaces generated by blocks were introduced by M. Taibleson and G. Weiss in the setting of the one-dimensional torus T (see [TW1]). The motivation for their study was the connection of these spaces with the almost everywhere convergence of Fourier series. Later S. Lu, M. Taibleson and G. Weiss extended the theory to the n-dimensional torus T" proving almost everywhere convergence results for the Bochner-Riesz means of multiple Fourier series (see [LTW]).
    [Show full text]
  • Inversion Formulas for Riemann-Liouville Transform and Its Dual Associated with Singular Partial Differential Operators
    INVERSION FORMULAS FOR RIEMANN-LIOUVILLE TRANSFORM AND ITS DUAL ASSOCIATED WITH SINGULAR PARTIAL DIFFERENTIAL OPERATORS C. BACCAR, N. B. HAMADI, AND L. T. RACHDI Received 21 May 2005; Revised 27 September 2005; Accepted 20 October 2005 t We define Riemann-Liouville transform ᏾α and its dual ᏾α associated with two singu- lar partial differential operators. We establish some results of harmonic analysis for the Fourier transform connected with ᏾α. Next, we prove inversion formulas for the opera- t t tors ᏾α, ᏾α and a Plancherel theorem for ᏾α. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved. 1. Introduction The mean operator is defined for a continuous function f on R2, even with respect to the first variable by 1 2π R0( f )(r,x) = f (r sinθ,x + r cosθ)dθ, (1.1) 2π 0 which means that R0( f )(r,x)isthemeanvalueof f on the circle centered at (0,x)and t radius r. The dual of the mean operator R0 is defined by t 1 2 2 R0( f )(r,x) = f r +(x − y) , y dy. (1.2) π R t ThemeanoperatorR0 and its dual R0 play an important role and have many applica- tions, for example, in image processing of the so-called synthetic aperture radar (SAR) data [11, 12] or in the linearized inverse scattering problem in acoustics [6]. Our purpose in this work is to define and study integral transforms which general- t ize the operators R0 and R0. More precisely, we consider the following singular partial differential operators: ∂ Δ = , 1 ∂x (1.3) ∂2 2α +1 ∂ ∂2 Δ = + − ,(r,x) ∈]0,+∞[×R, α 0.
    [Show full text]
  • The Plancherel Theorem for General Semisimple Groups Compositio Mathematica, Tome 57, No 3 (1986), P
    COMPOSITIO MATHEMATICA REBECCA A. HERB JOSEPH A. WOLF The Plancherel theorem for general semisimple groups Compositio Mathematica, tome 57, no 3 (1986), p. 271-355 <http://www.numdam.org/item?id=CM_1986__57_3_271_0> © Foundation Compositio Mathematica, 1986, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Compositio Mathematica 57 (1986) 271-355 © Martinus Nijhoff Publishers, Dordrecht - Printed in the Netherlands THE PLANCHEREL THEOREM FOR GENERAL SEMISIMPLE GROUPS Rebecca A. Herb * and Joseph A. Wolf ** §0. Introduction Harish-Chandra’s celebrated Plancherel formula (see [6b,c,d]) applies to real reductive Lie groups G whose component group G/G° is finite and such that [GO, GO] has finite center. That class includes all finite cover- ings of real reductive linear algebraic groups, but it does not include all real connected semisimple Lie groups. In this paper we obtain the Plancherel formula for a somewhat larger class of groups that does contain all real semisimple groups. The method is necessarily quite different from that of Harish-Chandra. We deal with the class of Lie groups introduced in [17a,b,c] by one of us in 1972. See §1 below for the precise definition. This is a class of reductive groups that contains all the connected real semisimple Lie groups and that contains all Levi components of cuspidal parabolic subgroups.
    [Show full text]
  • A Helmholtz' Theorem
    A Helmholtz’ Theorem Because 1 ∇2 = −4πδ(R) (A.1) R where R = r − r with magnitude R = |R| and where δ(R)=δ(r − r)= δ(x − x)δ(y − y)δ(z − z) is the three-dimensional Dirac delta function (see Appendix B), then any sufficiently well-behaved vector function F(r)= F(x, y, z) can be represented as F(r)= F(r)δ(r − r) d3r V 1 1 = − F(r)∇2 d3r 4π R V 1 F(r) = − ∇2 d3r, (A.2) 4π V R the integration extending over any region V that contains the point r. With the identity ∇×∇×= ∇∇ · −∇2, Eq. (A.2) may be written as 1 F(r) 1 F(r) F(r)= ∇×∇× d3r − ∇∇ · d3r. (A.3) 4π V R 4π V R Consider first the divergence term appearing in this expression. Because the vector differential operator ∇ does not operate on the primed coordinates, then 1 F(r) 1 1 ∇· d3r = F(r) ·∇ d3r. (A.4) 4π V R 4π V R Moreover, the integrand appearing in this expression may be expressed as 1 1 F(r) ·∇ = −F(r) ·∇ R R F(r) 1 = −∇ · + ∇ · F(r), (A.5) R R where the superscript prime on the vector differential operator ∇ denotes differentiation with respect to the primed coordinates alone. Substitution of 422 A Helmholtz’ Theorem Eq. (A.5) into Eq. (A.4) and application of the divergence theorem to the first term then yields 1 F(r) 1 F(r) 1 ∇ · F(r) ∇· d3r = − ∇ · d3r + d3r 4π R 4π R 4π R V V V 1 1 1 ∇ · F(r) = − F(r) · nˆd2r + d3r 4π S R 4π V R = φ(r), (A.6) whichisthedesiredformofthescalarpotentialφ(r) for the vector field F(r).
    [Show full text]
  • Variations À La Fourier-Weyl-Wigner on Quantizations of the Plane and the Half-Plane
    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 3 September 2018 doi:10.20944/preprints201809.0012.v1 Peer-reviewed version available at Entropy 2018, 20, 787; doi:10.3390/e20100787 Article Variations à la Fourier-Weyl-Wigner on quantizations of the plane and the half-plane Hervé Bergeron 1 and Jean-Pierre Gazeau2 ID 1 ISMO, UMR 8214 CNRS, Univ Paris-Sud, 91405 Orsay cedex France; [email protected] 2 APC, UMR 7164 CNRS, Univ Paris Diderot, Sorbonne Paris Cité, 75205 Paris, France; [email protected] * Correspondence: [email protected] 1 Abstract: Any quantization maps linearly functions on a phase space to symmetric operators in a 2 Hilbert space. Covariant integral quantization combines operator-valued measure with symmetry 3 group of the phase space. Covariant means that the quantization map intertwines classical (geometric 4 operation) and quantum (unitary transformations) symmetries. Integral means that we use all 5 ressources of integral calculus, in order to implement the method when we apply it to singular 6 functions, or distributions, for which the integral calculus is an essential ingredient. In this paper 7 we emphasize the deep connection between Fourier transform and covariant integral quantization 8 when the Weyl-Heisenberg and affine groups are involved. We show with our generalisations of the 9 Wigner-Weyl transform that many properties of the Weyl integral quantization, commonly viewed as 10 optimal, are actually shared by a large family of integral quantizations. 11 Keywords: Weyl-Heisenberg group; Affine group; Weyl quantization; Wigner function; Covariant 12 integral quantization 13 Contents 14 1 Introduction: a historical overview2 15 References 3 16 2 Covariant Integral quantization: a summary 3 17 2.1 General settings .........................................
    [Show full text]