DOCSLIB.ORG

Mathematical Methods of Theoretical Physics V

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

KARLSVOZIL MATHEMATICALMETH- ODSOFTHEORETICAL PHYSICS arXiv:1203.4558v8 [math-ph] 1 Feb 2019 EDITIONFUNZL Copyright © 2019 Karl Svozil Published by Edition Funzl For academic use only. You may not reproduce or distribute without permission of the author. First Edition, October 2011 Second Edition, October 2013 Third Edition, October 2014 Fourth Edition, October 2016 Fifth Edition, October 2018 Sixth Edition, February 2019 Contents Unreasonable effectiveness of mathematics in the natural sciences xi Part I: Linear vector spaces 1 1 Finite-dimensional vector spaces and linear algebra 3 1.1 Conventions and basic definitions3 1.1.1 Fields of real and complex numbers,5.—1.1.2 Vectors and vector space, 5. 1.2 Linear independence6 1.3 Subspace7 1.3.1 Scalar or inner product,7.—1.3.2 Hilbert space,9. 1.4 Basis9 1.5 Dimension 10 1.6 Vector coordinates or components 11 1.7 Finding orthogonal bases from nonorthogonal ones 13 1.8 Dual space 15 1.8.1 Dual basis,16.—1.8.2 Dual coordinates,19.—1.8.3 Representation of a functional by inner product,20.—1.8.4 Double dual space, 23. 1.9 Direct sum 23 1.10 Tensor product 24 1.10.1 Sloppy definition, 24.—1.10.2 Definition,24.—1.10.3 Representation, 25. 1.11 Linear transformation 26 1.11.1 Definition, 26.—1.11.2 Operations, 26.—1.11.3 Linear transforma- tions as matrices, 28. 1.12 Change of basis 29 1.12.1 Settlement of change of basis vectors by definition, 29.—1.12.2 Scale change of vector components by contra-variation,31. 1.13 Mutually unbiased bases 33 iv Karl Svozil 1.14 Completeness or resolution of the identity operator in terms of base vectors 34 1.15 Rank 35 1.16 Determinant 36 1.16.1 Definition, 36.—1.16.2 Properties, 37. 1.17 Trace 39 1.17.1 Definition, 39.—1.17.2 Properties, 40.—1.17.3 Partial trace,40. 1.18 Adjoint or dual transformation 42 1.18.1 Definition, 42.—1.18.2 Adjoint matrix notation, 42.— 1.18.3 Properties, 43. 1.19 Self-adjoint transformation 43 1.20 Positive transformation 44 1.21 Unitary transformation and isometries 44 1.21.1 Definition, 44.—1.21.2 Characterization in terms of orthonormal basis,47. 1.22 Orthonormal (orthogonal) transformation 48 1.23 Permutation 48 1.24 Projection or projection operator 49 1.24.1 Definition, 50.—1.24.2 Orthogonal (perpendicular) projections, 51.— 1.24.3 Construction of orthogonal projections from single unit vectors, 53.— 1.24.4 Examples of oblique projections which are not orthogonal projections, 55. 1.25 Proper value or eigenvalue 56 1.25.1 Definition, 56.—1.25.2 Determination,56. 1.26 Normal transformation 60 1.27 Spectrum 60 1.27.1 Spectral theorem, 60.—1.27.2 Composition of the spectral form, 62. 1.28 Functions of normal transformations 64 1.29 Decomposition of operators 65 1.29.1 Standard decomposition, 65.—1.29.2 Polar decomposition, 66.—1.29.3 Decomposition of isometries, 66.—1.29.4 Singular value decomposition, 67.—1.29.5 Schmidt decomposition of the tensor product of two vectors, 67. 1.30 Purification 68 1.31 Commutativity 69 1.32 Measures on closed subspaces 73 1.32.1 Gleason’s theorem,74.—1.32.2 Kochen-Specker theorem, 75. 2 Multilinear Algebra and Tensors 79 2.1 Notation 80 2.2 Change of Basis 81 2.2.1 Transformation of the covariant basis, 81.—2.2.2 Transformation of the contravariant coordinates, 82.—2.2.3 Transformation of the contravariant (dual) basis, 83.—2.2.4 Transformation of the covariant coordinates, 85.— 2.2.5 Orthonormal bases,85. Mathematical Methods of Theoretical Physics v 2.3 Tensor as multilinear form 85 2.4 Covariant tensors 86 2.4.1 Transformation of covariant tensor components, 86. 2.5 Contravariant tensors 87 2.5.1 Definition of contravariant tensors, 87.—2.5.2 Transformation of con- travariant tensor components, 87. 2.6 General tensor 87 2.7 Metric 88 2.7.1 Definition, 88.—2.7.2 Construction from a scalar product, 88.— 2.7.3 What can the metric tensor do for you?, 89.—2.7.4 Transformation of the metric tensor, 90.—2.7.5 Examples,90. 2.8 Decomposition of tensors 94 2.9 Form invariance of tensors 94 2.10 The Kronecker symbol ± 100 2.11 The Levi-Civita symbol " 100 2.12 Nabla, Laplace, and D’Alembert operators 101 2.13 Tensor analysis in orthogonal curvilinear coordinates 102 2.13.1 Curvilinear coordinates, 102.—2.13.2 Curvilinear bases, 104.— 2.13.3 Infinitesimal increment, line element, and volume, 104.—2.13.4 Vector differential operator and gradient, 106.—2.13.5 Divergence in three dimen- sional orthogonal curvilinear coordinates, 107.—2.13.6 Curl in three dimen- sional orthogonal curvilinear coordinates, 108.—2.13.7 Laplacian in three di- mensional orthogonal curvilinear coordinates, 108. 2.14 Index trickery and examples 109 2.15 Some common misconceptions 117 2.15.1 Confusion between component representation and “the real thing”, 117.—2.15.2 Matrix as a representation of a tensor of type (order, degree, rank) two, 118. 3 Groups as permutations 119 3.1 Basic definition and properties 119 3.1.1 Group axioms, 119.—3.1.2 Discrete and continuous groups, 120.— 3.1.3 Generators and relations in finite groups, 120.—3.1.4 Uniqueness of identity and inverses, 121.—3.1.5 Cayley or group composition table, 121.— 3.1.6 Rearrangement theorem, 122. 3.2 Zoology of finite groups up to order 6 123 3.2.1 Group of order 2, 123.—3.2.2 Group of order 3, 4 and 5, 123.— 3.2.3 Group of order 6, 124.—3.2.4 Cayley’s theorem, 124. 3.3 Representations by homomorphisms 125 3.4 Lie theory 126 3.4.1 Generators, 126.—3.4.2 Exponential map, 126.—3.4.3 Lie algebra, 126. 3.5 Zoology of some important continuous groups 126 3.5.1 General linear group GL(n,C), 126.—3.5.2 Orthogonal group over the reals O(n,R) O(n), 126.—3.5.3 Rotation group SO(n), 127.—3.5.4 Unitary Æ group U(n,C) U(n), 127.—3.5.5 Special unitary group SU(n), 128.— Æ 3.5.6 Symmetric group S(n), 128.—3.5.7 Poincaré group, 128. vi Karl Svozil 4 Projective and incidence geometry 129 4.1 Notation 129 4.2 Affine transformations map lines into lines as well as parallel lines to parallel lines 129 4.2.1 One-dimensional case, 132. 4.3 Similarity transformations 132 4.4 Fundamental theorem of affine geometry revised 132 4.5 Alexandrov’s theorem 132 Part II: Functional analysis 135 5 Brief review of complex analysis 137 5.1 Geometric representations of complex numbers and functions thereof 139 5.1.1 The complex plane, 139.—5.1.2 Multi-valued relationships, branch points, and branch cuts, 139. 5.2 Riemann surface 140 5.3 Differentiable, holomorphic (analytic) function 141 5.4 Cauchy-Riemann equations 141 5.5 Definition analytical function 142 5.6 Cauchy’s integral theorem 143 5.7 Cauchy’s integral formula 143 5.8 Series representation of complex differentiable functions 144 5.9 Laurent series 145 5.10 Residue theorem 147 5.11 Some special functional classes 150 5.11.1 Criterion for coincidence, 150.—5.11.2 Entire function, 150.— 5.11.3 Liouville’s theorem for bounded entire function, 150.—5.11.4 Picard’s theorem, 151.—5.11.5 Meromorphic function, 151. 5.12 Fundamental theorem of algebra 152 6 Brief review of Fourier transforms 153 6.0.1 Functional spaces, 153.—6.0.2 Fourier series, 154.—6.0.3 Exponential Fourier series, 156.—6.0.4 Fourier transformation, 157. 7 Distributions as generalized functions 161 7.1 Heuristically coping with discontinuities and singularities 161 7.2 General distribution 162 7.2.1 Duality, 163.—7.2.2 Linearity, 163.—7.2.3 Continuity, 164. 7.3 Test functions 164 7.3.1 Desiderata on test functions, 164.—7.3.2 Test function class I, 165.— Mathematical Methods of Theoretical Physics vii 7.3.3 Test function class II, 166.—7.3.4 Test function class III: Tempered dis- tributions and Fourier transforms, 166.—7.3.5 Test function class C 1, 168. 7.4 Derivative of distributions 168 7.5 Fourier transform of distributions 169 7.6 Dirac delta function 169 7.6.1 Delta sequence, 170.—7.6.2 ±£'¤ distribution, 172.—7.6.3 Useful formulæ involving ±, 172.—7.6.4 Fourier transform of ±, 178.— 7.6.5 Eigenfunction expansion of ±, 178.—7.6.6 Delta function expansion, 179. 7.7 Cauchy principal value 179 1 7.7.1 Definition, 179.—7.7.2 Principle value and pole function x distribution, 180. 7.8 Absolute value distribution 180 7.9 Logarithm distribution 181 7.9.1 Definition, 181.—7.9.2 Connection with pole function, 181. 1 7.10 Pole function xn distribution 182 1 7.11 Pole function x i® distribution 182 § 7.12 Heaviside or unit step function 183 7.12.1 Ambiguities in definition, 183.—7.12.2 Unit step function sequence, 184.—7.12.3 Useful formulæ involving H, 185.—7.12.4 H £'¤ distribution, 186.—7.12.5 Regularized unit step function, 186.—7.12.6 Fourier transform of the unit step function, 186. 7.13 The sign function 187 7.13.1 Definition, 187.—7.13.2 Connection to the Heaviside function, 187.— 7.13.3 Sign sequence, 188.—7.13.4 Fourier transform of sgn, 188.
Recommended publications
  • The Future of Fundamental Physics

    The Future of Fundamental Physics

    The Future of Fundamental Physics Nima Arkani-Hamed Abstract: Fundamental physics began the twentieth century with the twin revolutions of relativity and quantum mechanics, and much of the second half of the century was devoted to the con- struction of a theoretical structure unifying these radical ideas. But this foundation has also led us to a number of paradoxes in our understanding of nature. Attempts to make sense of quantum mechanics and gravity at the smallest distance scales lead inexorably to the conclusion that space- Downloaded from http://direct.mit.edu/daed/article-pdf/141/3/53/1830482/daed_a_00161.pdf by guest on 23 September 2021 time is an approximate notion that must emerge from more primitive building blocks. Further- more, violent short-distance quantum fluctuations in the vacuum seem to make the existence of a macroscopic world wildly implausible, and yet we live comfortably in a huge universe. What, if anything, tames these fluctuations? Why is there a macroscopic universe? These are two of the central theoretical challenges of fundamental physics in the twenty-½rst century. In this essay, I describe the circle of ideas surrounding these questions, as well as some of the theoretical and experimental fronts on which they are being attacked. Ever since Newton realized that the same force of gravity pulling down on an apple is also responsible for keeping the moon orbiting the Earth, funda- mental physics has been driven by the program of uni½cation: the realization that seemingly disparate phenomena are in fact different aspects of the same underlying cause. By the mid-1800s, electricity and magnetism were seen as different aspects of elec- tromagnetism, and a seemingly unrelated phenom- enon–light–was understood to be the undulation of electric and magnetic ½elds.
  • Research Statement

    Research Statement

    D. A. Williams II June 2021 Research Statement I am an algebraist with expertise in the areas of representation theory of Lie superalgebras, associative superalgebras, and algebraic objects related to mathematical physics. As a post- doctoral fellow, I have extended my research to include topics in enumerative and algebraic combinatorics. My research is collaborative, and I welcome collaboration in familiar areas and in newer undertakings. The referenced open problems here, their subproblems, and other ideas in mind are suitable for undergraduate projects and theses, doctoral proposals, or scholarly contributions to academic journals. In what follows, I provide a technical description of the motivation and history of my work in Lie superalgebras and super representation theory. This is then followed by a description of the work I am currently supervising and mentoring, in combinatorics, as I serve as the Postdoctoral Research Advisor for the Mathematical Sciences Research Institute's Undergraduate Program 2021. 1 Super Representation Theory 1.1 History 1.1.1 Superalgebras In 1941, Whitehead dened a product on the graded homotopy groups of a pointed topological space, the rst non-trivial example of a Lie superalgebra. Whitehead's work in algebraic topol- ogy [Whi41] was known to mathematicians who dened new graded geo-algebraic structures, such as -graded algebras, or superalgebras to physicists, and their modules. A Lie superalge- Z2 bra would come to be a -graded vector space, even (respectively, odd) elements g = g0 ⊕ g1 Z2 found in (respectively, ), with a parity-respecting bilinear multiplication termed the Lie g0 g1 superbracket inducing1 a symmetric intertwining map of -modules.
  • Mathematical Methods of Classical Mechanics-Arnold V.I..Djvu

    Mathematical Methods of Classical Mechanics-Arnold V.I..Djvu

    V.I. Arnold Mathematical Methods of Classical Mechanics Second Edition Translated by K. Vogtmann and A. Weinstein With 269 Illustrations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest V. I. Arnold K. Vogtmann A. Weinstein Department of Department of Department of Mathematics Mathematics Mathematics Steklov Mathematical Cornell University University of California Institute Ithaca, NY 14853 at Berkeley Russian Academy of U.S.A. Berkeley, CA 94720 Sciences U.S.A. Moscow 117966 GSP-1 Russia Editorial Board J. H. Ewing F. W. Gehring P.R. Halmos Department of Department of Department of Mathematics Mathematics Mathematics Indiana University University of Michigan Santa Clara University Bloomington, IN 47405 Ann Arbor, MI 48109 Santa Clara, CA 95053 U.S.A. U.S.A. U.S.A. Mathematics Subject Classifications (1991): 70HXX, 70005, 58-XX Library of Congress Cataloging-in-Publication Data Amol 'd, V.I. (Vladimir Igorevich), 1937- [Matematicheskie melody klassicheskoi mekhaniki. English] Mathematical methods of classical mechanics I V.I. Amol 'd; translated by K. Vogtmann and A. Weinstein.-2nd ed. p. cm.-(Graduate texts in mathematics ; 60) Translation of: Mathematicheskie metody klassicheskoi mekhaniki. Bibliography: p. Includes index. ISBN 0-387-96890-3 I. Mechanics, Analytic. I. Title. II. Series. QA805.A6813 1989 531 '.01 '515-dcl9 88-39823 Title of the Russian Original Edition: M atematicheskie metody klassicheskoi mekhaniki. Nauka, Moscow, 1974. Printed on acid-free paper © 1978, 1989 by Springer-Verlag New York Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, NY 10010, U.S.A.), except for brief excerpts in connection with reviews or scholarly analysis.
  • Quantum Field Theory*

    Quantum Field Theory*

    Quantum Field Theory y Frank Wilczek Institute for Advanced Study, School of Natural Science, Olden Lane, Princeton, NJ 08540 I discuss the general principles underlying quantum eld theory, and attempt to identify its most profound consequences. The deep est of these consequences result from the in nite number of degrees of freedom invoked to implement lo cality.Imention a few of its most striking successes, b oth achieved and prosp ective. Possible limitation s of quantum eld theory are viewed in the light of its history. I. SURVEY Quantum eld theory is the framework in which the regnant theories of the electroweak and strong interactions, which together form the Standard Mo del, are formulated. Quantum electro dynamics (QED), b esides providing a com- plete foundation for atomic physics and chemistry, has supp orted calculations of physical quantities with unparalleled precision. The exp erimentally measured value of the magnetic dip ole moment of the muon, 11 (g 2) = 233 184 600 (1680) 10 ; (1) exp: for example, should b e compared with the theoretical prediction 11 (g 2) = 233 183 478 (308) 10 : (2) theor: In quantum chromo dynamics (QCD) we cannot, for the forseeable future, aspire to to comparable accuracy.Yet QCD provides di erent, and at least equally impressive, evidence for the validity of the basic principles of quantum eld theory. Indeed, b ecause in QCD the interactions are stronger, QCD manifests a wider variety of phenomena characteristic of quantum eld theory. These include esp ecially running of the e ective coupling with distance or energy scale and the phenomenon of con nement.
  • Automatic Linearity Detection

    Automatic Linearity Detection

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Mathematical Institute Eprints Archive Report no. [13/04] Automatic linearity detection Asgeir Birkisson a Tobin A. Driscoll b aMathematical Institute, University of Oxford, 24-29 St Giles, Oxford, OX1 3LB, UK. [email protected]. Supported for this work by UK EPSRC Grant EP/E045847 and a Sloane Robinson Foundation Graduate Award associated with Lincoln College, Oxford. bDepartment of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA. [email protected]. Supported for this work by UK EPSRC Grant EP/E045847. Given a function, or more generally an operator, the question \Is it lin- ear?" seems simple to answer. In many applications of scientific computing it might be worth determining the answer to this question in an automated way; some functionality, such as operator exponentiation, is only defined for linear operators, and in other problems, time saving is available if it is known that the problem being solved is linear. Linearity detection is closely connected to sparsity detection of Hessians, so for large-scale applications, memory savings can be made if linearity information is known. However, implementing such an automated detection is not as straightforward as one might expect. This paper describes how automatic linearity detection can be implemented in combination with automatic differentiation, both for stan- dard scientific computing software, and within the Chebfun software system. The key ingredients for the method are the observation that linear operators have constant derivatives, and the propagation of two logical vectors, ` and c, as computations are carried out.
  • Introduction to General Relativity

    Introduction to General Relativity

    INTRODUCTION TO GENERAL RELATIVITY Gerard 't Hooft Institute for Theoretical Physics Utrecht University and Spinoza Institute Postbox 80.195 3508 TD Utrecht, the Netherlands e-mail: [email protected] internet: http://www.phys.uu.nl/~thooft/ Version November 2010 1 Prologue General relativity is a beautiful scheme for describing the gravitational ¯eld and the equations it obeys. Nowadays this theory is often used as a prototype for other, more intricate constructions to describe forces between elementary particles or other branches of fundamental physics. This is why in an introduction to general relativity it is of importance to separate as clearly as possible the various ingredients that together give shape to this paradigm. After explaining the physical motivations we ¯rst introduce curved coordinates, then add to this the notion of an a±ne connection ¯eld and only as a later step add to that the metric ¯eld. One then sees clearly how space and time get more and more structure, until ¯nally all we have to do is deduce Einstein's ¯eld equations. These notes materialized when I was asked to present some lectures on General Rela- tivity. Small changes were made over the years. I decided to make them freely available on the web, via my home page. Some readers expressed their irritation over the fact that after 12 pages I switch notation: the i in the time components of vectors disappears, and the metric becomes the ¡ + + + metric. Why this \inconsistency" in the notation? There were two reasons for this. The transition is made where we proceed from special relativity to general relativity.
  • Transformations, Polynomial Fitting, and Interaction Terms

    Transformations, Polynomial Fitting, and Interaction Terms

    FEEG6017 lecture: Transformations, polynomial fitting, and interaction terms [email protected] The linearity assumption • Regression models are both powerful and useful. • But they assume that a predictor variable and an outcome variable are related linearly. • This assumption can be wrong in a variety of ways. The linearity assumption • Some real data showing a non- linear connection between life expectancy and doctors per million population. The linearity assumption • The Y and X variables here are clearly related, but the correlation coefficient is close to zero. • Linear regression would miss the relationship. The independence assumption • Simple regression models also assume that if a predictor variable affects the outcome variable, it does so in a way that is independent of all the other predictor variables. • The assumed linear relationship between Y and X1 is supposed to hold no matter what the value of X2 may be. The independence assumption • Suppose we're trying to predict happiness. • For men, happiness increases with years of marriage. For women, happiness decreases with years of marriage. • The relationship between happiness and time may be linear, but it would not be independent of sex. Can regression models deal with these problems? • Fortunately they can. • We deal with non-linearity by transforming the predictor variables, or by fitting a polynomial relationship instead of a straight line. • We deal with non-independence of predictors by including interaction terms in our models. Dealing with non-linearity • Transformation is the simplest method for dealing with this problem. • We work not with the raw values of X, but with some arbitrary function that transforms the X values such that the relationship between Y and f(X) is now (closer to) linear.
  • "Eternal" Questions in the XX-Century Theoretical Physics V

    "Eternal" Questions in the XX-Century Theoretical Physics V

    Philosophical roots of the "eternal" questions in the XX-century theoretical physics V. Ihnatovych Department of Philosophy, National Technical University of Ukraine “Kyiv Polytechnic Institute”, Kyiv, Ukraine e-mail: [email protected] Abstract The evolution of theoretical physics in the XX century differs significantly from that in XVII-XIX centuries. While continuous progress is observed for theoretical physics in XVII-XIX centuries, modern physics contains many questions that have not been resolved despite many decades of discussion. Based upon the analysis of works by the founders of the XX-century physics, the conclusion is made that the roots of the "eternal" questions by the XX-century theoretical physics lie in the philosophy used by its founders. The conclusion is made about the need to use the ideas of philosophy that guided C. Huygens, I. Newton, W. Thomson (Lord Kelvin), J. K. Maxwell, and the other great physicists of the XVII-XIX centuries, in all areas of theoretical physics. 1. Classical Physics The history of theoretical physics begins in 1687 with the work “Mathematical Principles of Natural Philosophy” by Isaac Newton. Even today, this work is an example of what a full and consistent outline of the physical theory should be. It contains everything necessary for such an outline – definition of basic concepts, the complete list of underlying laws, presentation of methods of theoretical research, rigorous proofs. In the eighteenth century, such great physicists and mathematicians as Euler, D'Alembert, Lagrange, Laplace and others developed mechanics, hydrodynamics, acoustics and nebular mechanics on the basis of the ideas of Newton's “Principles”.
  • The Universe Unveiled Given by Prof Carlo Contaldi

    The Universe Unveiled Given by Prof Carlo Contaldi

    Friends of Imperial Theoretical Physics We are delighted to announce that the first FITP event of 2015 will be a talk entitled The Universe Unveiled given by Prof Carlo Contaldi. The event is free and open to all but please register by visiting the Eventbrite website via http://tinyurl.com/fitptalk2015. Date: 29th April 2015 Venue: Lecture Theatre 1, Blackett Laboratory, Physics Department, ICL Time: 7-8pm followed by a reception in the level 8 Common room Speaker: Professor Carlo Contaldi The Universe Unveiled The past 25 years have seen our understanding of the Universe we live in being revolutionised by a series of stunning observational campaigns and theoretical advances. We now know the composition, age, geometry and evolutionary history of the Universe to an astonishing degree of precision. A surprising aspect of this journey of discovery is that it has revealed some profound conundrums that challenge the most basic tenets of fundamental physics. We still do not understand the nature of 95% of the matter and energy that seems to fill the Universe, we still do not know why or how the Universe came into being, and we have yet to build a consistent "theory of everything" that can describe the evolution of the Universe during the first few instances after the Big Bang. In this lecture I will review what we know about the Universe today and discuss the exciting experimental and theoretical advances happening in cosmology, including the controversy surrounding last year's BICEP2 "discovery". Biography of the speaker: Professor Contaldi gained his PhD in theoretical physics in 2000 at Imperial College working on theories describing the formation of structures in the universe.
  • Comparative Literature in Slovenia

    Comparative Literature in Slovenia

    CLCWeb: Comparative Literature and Culture ISSN 1481-4374 Purdue University Press ©Purdue University Volume 2 (2000) Issue 4 Article 11 Comparative Literature in Slovenia Kristof Jacek Kozak University of Alberta Follow this and additional works at: https://docs.lib.purdue.edu/clcweb Part of the Comparative Literature Commons, and the Critical and Cultural Studies Commons Dedicated to the dissemination of scholarly and professional information, Purdue University Press selects, develops, and distributes quality resources in several key subject areas for which its parent university is famous, including business, technology, health, veterinary medicine, and other selected disciplines in the humanities and sciences. CLCWeb: Comparative Literature and Culture, the peer-reviewed, full-text, and open-access learned journal in the humanities and social sciences, publishes new scholarship following tenets of the discipline of comparative literature and the field of cultural studies designated as "comparative cultural studies." Publications in the journal are indexed in the Annual Bibliography of English Language and Literature (Chadwyck-Healey), the Arts and Humanities Citation Index (Thomson Reuters ISI), the Humanities Index (Wilson), Humanities International Complete (EBSCO), the International Bibliography of the Modern Language Association of America, and Scopus (Elsevier). The journal is affiliated with the Purdue University Press monograph series of Books in Comparative Cultural Studies. Contact: <[email protected]> Recommended Citation Kozak, Kristof Jacek. "Comparative Literature in Slovenia." CLCWeb: Comparative Literature and Culture 2.4 (2000): <https://doi.org/10.7771/1481-4374.1094> This text has been double-blind peer reviewed by 2+1 experts in the field. The above text, published by Purdue University Press ©Purdue University, has been downloaded 2344 times as of 11/ 07/19.
  • Mathematical Foundations of Supersymmetry

    Mathematical Foundations of Supersymmetry

    MATHEMATICAL FOUNDATIONS OF SUPERSYMMETRY Lauren Caston and Rita Fioresi 1 Dipartimento di Matematica, Universit`adi Bologna Piazza di Porta S. Donato, 5 40126 Bologna, Italia e-mail: [email protected], fi[email protected] 1Investigation supported by the University of Bologna 1 Introduction Supersymmetry (SUSY) is the machinery mathematicians and physicists have developed to treat two types of elementary particles, bosons and fermions, on the same footing. Supergeometry is the geometric basis for supersymmetry; it was first discovered and studied by physicists Wess, Zumino [25], Salam and Strathde [20] (among others) in the early 1970’s. Today supergeometry plays an important role in high energy physics. The objects in super geometry generalize the concept of smooth manifolds and algebraic schemes to include anticommuting coordinates. As a result, we employ the techniques from algebraic geometry to study such objects, namely A. Grothendiek’s theory of schemes. Fermions include all of the material world; they are the building blocks of atoms. Fermions do not like each other. This is in essence the Pauli exclusion principle which states that two electrons cannot occupy the same quantum me- chanical state at the same time. Bosons, on the other hand, can occupy the same state at the same time. Instead of looking at equations which just describe either bosons or fermions separately, supersymmetry seeks out a description of both simultaneously. Tran- sitions between fermions and bosons require that we allow transformations be- tween the commuting and anticommuting coordinates. Such transitions are called supersymmetries. In classical Minkowski space, physicists classify elementary particles by their mass and spin.
  • Renormalization and Effective Field Theory

    Renormalization and Effective Field Theory

    Mathematical Surveys and Monographs Volume 170 Renormalization and Effective Field Theory Kevin Costello American Mathematical Society surv-170-costello-cov.indd 1 1/28/11 8:15 AM http://dx.doi.org/10.1090/surv/170 Renormalization and Effective Field Theory Mathematical Surveys and Monographs Volume 170 Renormalization and Effective Field Theory Kevin Costello American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair MichaelA.Singer Eric M. Friedlander Benjamin Sudakov MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 81T13, 81T15, 81T17, 81T18, 81T20, 81T70. The author was partially supported by NSF grant 0706954 and an Alfred P. Sloan Fellowship. For additional information and updates on this book, visit www.ams.org/bookpages/surv-170 Library of Congress Cataloging-in-Publication Data Costello, Kevin. Renormalization and effective fieldtheory/KevinCostello. p. cm. — (Mathematical surveys and monographs ; v. 170) Includes bibliographical references. ISBN 978-0-8218-5288-0 (alk. paper) 1. Renormalization (Physics) 2. Quantum field theory. I. Title. QC174.17.R46C67 2011 530.143—dc22 2010047463 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA.