Lecture 15 1 Nov, 2016
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On the Von Neumann Rule in Quantization
On the von Neumann rule in quantization Olaf M¨uller∗ June 25, 2019 Abstract We show that any linear quantization map into the space of self-adjoint operators in a Hilbert space violates the von Neumann rule on post- composition with real functions. 1 Introduction and main results Physics today has the ambition to be entirely mathematically derivable from two fundamental theories: gravity and the standard model of parti- cle physics. Whereas the former is a classical field theory with only mildly paradoxical features such as black holes, the latter is not so much a closed theory as rather a toolbox full of complex algorithms and ill-defined objects. Moreover, as it uses a form of canonical quantization, the non-mathematical, interpretational subtleties of quantum mechanics, such as the measurement problem, can also be found in the standard model. Despite of these prob- lems, the standard model is very successful if used by experts inasfar as its predictions are in accordance with a large class of experiments to an un- precedented precision. Its canonical quantization uses a quantization map 0 arXiv:1903.10494v3 [math-ph] 22 Jun 2019 Q : G LSA(H) from some nonempty subset G C (C) of classical → ⊂ observables, where C is the classical phase space, usually diffeomorphic to the space of solutions, and LSA(H) is the space of linear self-adjoint (s.-a.) maps of a Hilbert space to itself.1 There is a widely accepted list of desirable properties for a quantization map going back to Weyl, von Neumann and Dirac ([24], [18], [6]): ∗Humboldt-Universit¨at, Institut f¨ur Mathematik, Unter den Linden 6, 10099 Berlin 1More exactly, the image of Q consists of essentially s.-a. -
An Introduction to Pseudo-Differential Operators
An introduction to pseudo-differential operators Jean-Marc Bouclet1 Universit´ede Toulouse 3 Institut de Math´ematiquesde Toulouse [email protected] 2 Contents 1 Background on analysis on manifolds 7 2 The Weyl law: statement of the problem 13 3 Pseudodifferential calculus 19 3.1 The Fourier transform . 19 3.2 Definition of pseudo-differential operators . 21 3.3 Symbolic calculus . 24 3.4 Proofs . 27 4 Some tools of spectral theory 41 4.1 Hilbert-Schmidt operators . 41 4.2 Trace class operators . 44 4.3 Functional calculus via the Helffer-Sj¨ostrandformula . 50 5 L2 bounds for pseudo-differential operators 55 5.1 L2 estimates . 55 5.2 Hilbert-Schmidt estimates . 60 5.3 Trace class estimates . 61 6 Elliptic parametrix and applications 65 n 6.1 Parametrix on R ................................ 65 6.2 Localization of the parametrix . 71 7 Proof of the Weyl law 75 7.1 The resolvent of the Laplacian on a compact manifold . 75 7.2 Diagonalization of ∆g .............................. 78 7.3 Proof of the Weyl law . 81 A Proof of the Peetre Theorem 85 3 4 CONTENTS Introduction The spirit of these notes is to use the famous Weyl law (on the asymptotic distribution of eigenvalues of the Laplace operator on a compact manifold) as a case study to introduce and illustrate one of the many applications of the pseudo-differential calculus. The material presented here corresponds to a 24 hours course taught in Toulouse in 2012 and 2013. We introduce all tools required to give a complete proof of the Weyl law, mainly the semiclassical pseudo-differential calculus, and then of course prove it! The price to pay is that we avoid presenting many classical concepts or results which are not necessary for our purpose (such as Borel summations, principal symbols, invariance by diffeomorphism or the G˚ardinginequality). -
Properties of Field Functionals and Characterization of Local Functionals
This is a repository copy of Properties of field functionals and characterization of local functionals. White Rose Research Online URL for this paper: https://eprints.whiterose.ac.uk/126454/ Version: Accepted Version Article: Brouder, Christian, Dang, Nguyen Viet, Laurent-Gengoux, Camille et al. (1 more author) (2018) Properties of field functionals and characterization of local functionals. Journal of Mathematical Physics. ISSN 0022-2488 Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ Properties of field functionals and characterization of local functionals Christian Brouder,1, a) Nguyen Viet Dang,2 Camille Laurent-Gengoux,3 and Kasia Rejzner4 1)Sorbonne Universit´es, UPMC Univ. Paris 06, UMR CNRS 7590, Institut de Min´eralogie, de Physique des Mat´eriaux et de Cosmochimie, Mus´eum National d’Histoire Naturelle, IRD UMR 206, 4 place Jussieu, F-75005 Paris, France. 2)Institut Camille Jordan (UMR CNRS 5208) Universit´eClaude Bernard Lyon 1, Bˆat. -
Covariant Schrödinger Semigroups on Riemannian Manifolds1
Covariant Schr¨odinger semigroups on Riemannian manifolds1 Batu G¨uneysu Humboldt-Universit¨at zu Berlin arXiv:1801.01335v2 [math.DG] 23 Apr 2018 1This is a shortened version (the Chapters VII - XIII have been removed). The full version has appeared in December 2017 as a monograph in the Birkh¨auser series Operator Theory: Advances and Applications, and only the latter version should be cited. The full version can be obtained on: http://www.springer.com/de/book/9783319689029 For my daughter Elena ♥ Contents Introduction 5 Chapter I. Sobolev spaces on vector bundles 16 1. Differential operators with smooth coefficients and weak derivatives on vector bundles 16 2. Some remarks on covariant derivatives 25 3. Generalized Sobolev spaces and Meyers-Serrin theorems 28 Chapter II. Smooth heat kernels on vector bundles 35 Chapter III. Basic differential operators in Riemannian manifolds 40 1. Preleminaries from Riemannian geometry 40 2. Riemannian Sobolev spaces and Meyers-Serrin theorems 49 3. The Friedrichs realization of † /2 59 ∇ ∇ Chapter IV. Some specific results for the minimal heat kernel 62 Chapter V. Wiener measure and Brownian motion on Riemannian manifolds 84 1. Introduction 84 2. Path spaces as measurable spaces 85 3. The Wiener measure on Riemannian manifolds 89 Chapter VI. Contractive Dynkin and Kato potentials 98 1. Generally valid results 98 2. Specific results under some control on the geometry 112 Chapter VII. Foundations of covariant Schr¨odinger semigroups 115 1. Notation and preleminaries 115 2. Scalar Schr¨odinger semigroups and the Feynman-Kac (FK) formula 118 3. Kato-Simon inequality 118 1 Chapter VIII. Compactness of V (H∇ + 1)− 119 Chapter IX. -
Some Aspects of Differential Theories Respectfully Dedicated to the Memory of Serge Lang
Handbook of Global Analysis 1071 Demeter Krupka and David Saunders c 2007 Elsevier All rights reserved Some aspects of differential theories Respectfully dedicated to the memory of Serge Lang Jozsef´ Szilasi and Rezso˝ L. Lovas Contents Introduction 1 Background 2 Calculus in topological vector spaces and beyond 3 The Chern-Rund derivative Introduction Global analysis is a particular amalgamation of topology, geometry and analysis, with strong physical motivations in its roots, its development and its perspectives. To formulate a problem in global analysis exactly, we need (1) a base manifold, (2) suitable fibre bundles over the base manifold, (3) a differential operator between the topological vector spaces consisting of the sec- tions of the chosen bundles. In quantum physical applications, the base manifold plays the role of space-time; its points represent the location of the particles. The particles obey the laws of quantum physics, which are encoded in the vector space structure attached to the particles in the mathematical model. A particle carries this vector space structure with itself as it moves. Thus we arrive at the intuitive notion of vector bundle, which had arisen as the ‘repere` mobile’ in Elie´ Cartan’s works a few years before quantum theory was discovered. To describe a system (e.g. in quantum physics) in the geometric framework of vector bundles effectively, we need a suitably flexible differential calculus. We have to differen- tiate vectors which change smoothly together with the vector space carrying them. The primitive idea of differentiating the coordinates of the vector in a fixed basis is obviously not satisfactory, since there is no intrinsic coordinate system which could guarantee the invariance of the results. -
Tempered Distributions and the Fourier Transform
CHAPTER 1 Tempered distributions and the Fourier transform Microlocal analysis is a geometric theory of distributions, or a theory of geomet- ric distributions. Rather than study general distributions { which are like general continuous functions but worse { we consider more specific types of distributions which actually arise in the study of differential and integral equations. Distri- butions are usually defined by duality, starting from very \good" test functions; correspondingly a general distribution is everywhere \bad". The conormal dis- tributions we shall study implicitly for a long time, and eventually explicitly, are usually good, but like (other) people have a few interesting faults, i.e. singulari- ties. These singularities are our principal target of study. Nevertheless we need the general framework of distribution theory to work in, so I will start with a brief in- troduction. This is designed either to remind you of what you already know or else to send you off to work it out. (As noted above, I suggest Friedlander's little book [4] - there is also a newer edition with Joshi as coauthor) as a good introduction to distributions.) Volume 1 of H¨ormander'streatise [8] has all that you would need; it is a good general reference. Proofs of some of the main theorems are outlined in the problems at the end of the chapter. 1.1. Schwartz test functions To fix matters at the beginning we shall work in the space of tempered distribu- tions. These are defined by duality from the space of Schwartz functions, also called the space of test functions of rapid decrease. -
The Schur Lemma
Appendix A The Schur Lemma Schur’s lemma provides sufficient conditions for linear operators to be bounded on Lp. Moreover, for positive operators it provides necessary and sufficient such condi- tions. We discuss these situations. A.1 The Classical Schur Lemma We begin with an easy situation. Suppose that K(x,y) is a locally integrable function on a product of two σ-finite measure spaces (X, μ) × (Y,ν), and let T be a linear operator given by T( f )(x)= K(x,y) f (y)dν(y) Y when f is bounded and compactly supported. It is a simple consequence of Fubini’s theorem that for almost all x ∈ X the integral defining T converges absolutely. The following lemma provides a sufficient criterion for the Lp boundedness of T. Lemma. Suppose that a locally integrable function K(x,y) satisfies sup |K(x,y)|dν(y)=A < ∞, x∈X Y sup |K(x,y)|dμ(x)=B < ∞. y∈Y X Then the operator T extends to a bounded operator from Lp(Y) to Lp(X) with norm − 1 1 A1 p B p for 1 ≤ p ≤ ∞. Proof. The second condition gives that T maps L1 to L1 with bound B, while the first condition gives that T maps L∞ to L∞ with bound A. It follows by the Riesz– − 1 1 Thorin interpolation theorem that T maps Lp to Lp with bound A1 p B p . This lemma can be improved significantly when the operators are assumed to be positive. A.2 Schur’s Lemma for Positive Operators We have the following necessary and sufficient condition for the Lp boundedness of positive operators. -
Lecture Notes: Introduction to Microlocal Analysis with Applications
LECTURE NOTES: INTRODUCTION TO MICROLOCAL ANALYSIS WITH APPLICATIONS MIHAJLO CEKIC´ Contents Course Summary2 Notation 4 1. Recap: LCS topologies, distributions, Fourier transforms5 1.1. Topological vector spaces5 1.2. Space of distributions6 1.3. Schwartz kernel theorem8 1.4. Fourier transform9 2. Symbol classes, oscillatory integrals and Fourier Integral Operators 10 2.1. Symbol classes 10 2.2. Oscillatory integrals 13 3. Method of stationary phase 16 4. Algebra of pseudodifferential operators 18 4.1. Changes of variables (Kuranishi trick) 22 5. Elliptic operators and L2-continuity 23 5.1. L2-continuity 25 5.2. Sobolev spaces and PDOs 26 6. The wavefront set of a distribution 29 6.1. Operations with distributions: pullbacks and products 33 7. Manifolds 36 8. Quantum ergodicity 38 8.1. Ergodic theory 39 8.2. Statement and ingredients 41 8.3. Main proof of Quantum Ergodicity 44 9. Semiclassical calculus and Weyl's law 46 9.1. Proof of Weyl's law 48 References 50 1 2 M. CEKIC´ Course Summary Microlocal analysis studies singularities of distributions in phase space, by describing the behaviour of the singularity in both position and direction. It is a part of the field of partial differential equations, created by H¨ormander, Kohn, Nirenberg and others in 1960s and 1970s, and is used to study questions such as solvability, regularity and propagation of singularities of solutions of PDEs. To name a few other classical applications, it can be used to study asymptotics of eigenfunctions for elliptic operators, trace formulas and inverse problems. There have been recent exciting advances in the field and many applications to geometry and dynamical systems. -
Peetre's Theorem
Humboldt-Universität zu Berlin Institut für Mathematik Wintersemester 2011/2012 Peetre's Theorem Bodo Graumann April 3, 2014 Contents 1 Basics 2 2 Peetre's Theorem 5 References 7 cba This document was published on http://bodograumann.de/studium. It is pro- vided under the Creative Commons Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0) license. X TE EX The code was written in gvim with the help of vim-latex and compiled with xelatex, all on Gentoo Linux. Thanks to the free software community and to the fellow TEX users on TEX.SX for their great help and advice. If you want to make any comments or suggest corrections to this script, please con- tact me! You can either email me directly or use the contact form on my website. 1 Peetre’s Theorem 1 Basics Page 2 In 1959 Jaak Peetre first published his astonishing result, that any local (i.e. support-non- increasing) operator is a differential operator, [Pee59]. Unfortunately his proof contained an error and so in 1960 he had to publish a correction, [Pee60]. There, instead of just patching the error of the previous paper, he proposed a more general version of his theorem using distribution theory. We will take this second approach as it seems more natural. So before we can proof Peetre’s Theorem we need to introduce the necessary results of distribution theory. Most of those are taken from [Hö90]. For a more abstract context [RR73] is a good source. 1 Basics Choose any open set 훺 ⊂ ℝ푛 and a field 핂 ∈ {ℝ, ℂ}. -
Characterizations of Continuous Operators on with the Strict Topology
Tusi Mathematical Ann. Funct. Anal. (2021) 12:28 Research https://doi.org/10.1007/s43034-021-00112-1 Group ORIGINAL PAPER Characterizations of continuous operators on Cb(X) with the strict topology Marian Nowak1 · Juliusz Stochmal2 Received: 2 September 2020 / Accepted: 7 January 2021 / Published online: 16 February 2021 © The Author(s) 2021 Abstract C X Let X be a completely regular Hausdorf space and b( ) be the space of all bounded continuous functions on X, equipped with the strict topology . We study some ⋅ C X important classes of (, ‖ ‖E)-continuous linear operators from b( ) to a Banach E ⋅ space ( , ‖ ‖E) : -absolutely summing operators, compact operators and -nuclear operators. We characterize compact operators and -nuclear operators in terms of their representing measures. It is shown that dominated operators and -absolutely T C X → E summing operators ∶ b( ) coincide and if, in particular, E has the Radon– Nikodym property, then -absolutely summing operators and -nuclear operators coincide. We generalize the classical theorems of Pietsch, Tong and Uhl concern- ing the relationships between absolutely summing, dominated, nuclear and compact operators on the Banach space C(X), where X is a compact Hausdorf space. Keywords Spaces of bounded continuous functions · k-spaces · Radon vector measures · Strict topologies · Absolutely summing operators · Dominated operators · Nuclear operators · Compact operators · Generalized DF-spaces · Projective tensor product Mathematics Subject Classifcation 46G10 · 28A32 · 47B10 Communicated by Raymond Mortini. * Juliusz Stochmal [email protected] Marian Nowak [email protected] 1 Institute of Mathematics, University of Zielona Góra, ul. Szafrana 4A, 65-516 Zielona Gora, Poland 2 Institute of Mathematics, Kazimierz Wielki University, ul. -
Arxiv:1503.04086V1 [Math-Ph] 13 Mar 2015 Schwartz Operators
Schwartz operators M. Keyl TU München, Fakultät Mathematik, Boltzmannstr. 3, 85748 Garching, Germany [email protected] J. Kiukas School of Math. Sci., Univ. Nottingham, University Park, Nottingham, NG7 2RD, UK [email protected] R. F. Werner Inst. Theor. Physik, Leibniz Univ. Hannover, Appelstr. 2, 30167 Hannover, Germany [email protected] March 16, 2015 In this paper we introduce Schwartz operators as a non-commutative analog of Schwartz functions and provide a detailed discussion of their properties. We equip them in particular with a number of different (but equivalent) families of seminorms which turns the space of Schwartz operators into a Frechet space. The study of the topological dual leads to non-commutative tempered distributions which are discussed in detail as well. We show in particular that the latter can be identified with a certain class of quadratic forms, therefore making operations like products with bounded (and also some unbounded) operators and quantum harmonic analysis available to objects which are otherwise too singular for being a Hilbert space operator. Finally we show how the new methods can be applied by studying operator moment problems and convergence properties of fluctuation operators. 1 Introduction arXiv:1503.04086v1 [math-ph] 13 Mar 2015 The theory of tempered distributions is used extensively in various areas of mathematical physics, in order to regularise singular objects, most notably "delta-functions" that often appear as a result of some convenient idealisation (e.g. plane wave, or point interaction). The well-known intuitive idea is to make sense of the functional R2N φ(f) = S f(x)φ(x)dx, f ∈ S( ) (1) 1 to cases where φ is no longer a function, by making use of the highly regular behaviour of the 2N Schwartz functions f ∈ S(R ). -
Elliptic Problems with Boundary Operators of Higher Orders in H
ELLIPTIC PROBLEMS WITH BOUNDARY OPERATORS OF HIGHER ORDERS IN HORMANDER–ROITBERG¨ SPACES TETIANA KASIRENKO AND ALEKSANDR MURACH Abstract. We investigate elliptic boundary-value problems for which the maxi- mum of the orders of the boundary operators is equal to or greater than the order of the elliptic differential equation. We prove that the operator corresponding to an arbitrary problem of this kind is bounded and Fredholm between appropriate Hilbert spaces which form certain two-sided scales and are built on the base of isotropic H¨ormander spaces. The differentiation order for these spaces is given by an arbitrary real number and positive function which varies slowly at infinity in the sense of Karamata. We establish a local a priori estimate for the generalized solutions to the problem and investigate their local regularity (up to the bound- ary) on these scales. As an application, we find sufficient conditions under which the solutions have continuous classical derivatives of a given order. 1. Introduction Among elliptic boundary-value problems occur problems with boundary condi- tions whose orders are equal to or greater than the order of the elliptic differential equation for which the problem is posed. Such problems appear specifically in acoustic, hydrodynamics, and the theory of stochastic processes [26, 73, 74]. The known Ventcel’ elliptic boundary-value problem [73] apparently was the first such a problem arisen in applications. It consists of an elliptic differential equation of the second order and a boundary condition of the same order and was applied ini- tially to investigation of diffusion processes. A number of papers are devoted to this problem; see, e.g.