Ebook Quantum field theory III: Gauge theory - Part 1

Part 1 of ebook Quantum field theory III: Gauge theory provide readers with content about: the euclidean manifold as a paradigm; the euclidean space E3 (hilbert space and lie algebra structure); algebras and duality (tensor algebra, grassmann algebra, clifford algebra, lie algebra); representations of symmetries in mathematics and physics; the euclidean manifold E3; the lie group U(1) as a paradigm in harmonic analysis and geometry;...

Quantum Field Theory III: Gauge Theory Eberhard Zeidler Quantum Field Theory III: Gauge Theory A Bridge between Mathematicians and Physicists Eberhard Zeidler Max Planck Institute for Mathematics in the Sciences Inselstr 22-26 04103 Leipzig Germany ISBN 978-3-642-22420-1 e-ISBN 978-3-642-22421-8 DOI 10.1007/978-3-642-22421-8 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2006929535 Mathematics Subject Classification (2010): 35-XX, 47-XX, 49-XX, 51-XX, 55-XX, 81-XX, 82-XX © Springer-Verlag Berlin Heidelberg 2011 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 to prosecution under the German Copyright Law 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 Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) TO KRZYSZTOF MAURIN IN GRATITUDE Preface Sein Geist drang in die tiefsten Geheimnisse der Zahl, des Raumes und der Natur; er maß den Lauf der Gestirne, die Gestalt und die Kră afte der Erde; die Entwicklung der mathematischen Wissenschaft eines kommenden Jahrhunderts trug er in sich.1 Lines under the portrait of Carl Friedrich Gauss (1777–1855) in the German Museum in Munich Force equals curvature The basic principle of modern physics A theory is the more impressive, the simpler are its premises, the more distinct are the things it connects, and the broader is the range of applicability Albert Einstein (1879–1955) Textbooks should be attractive by showing the beauty of the subject Johann Wolfgang von Goethe (1749–1832) The present book is the third volume of a comprehensive introduction to the mathematical and physical aspects of modern quantum field theory which comprises the following six volumes: Volume Volume Volume Volume Volume Volume I: Basics in Mathematics and Physics II: Quantum Electrodynamics III: Gauge Theory IV: Quantum Mathematics V: The Physics of the Standard Model VI: Quantum Gravitation and String Theory It is our goal to build a bridge between mathematicians and physicists based on challenging questions concerning the fundamental forces in • the macrocosmos (the universe) and • the microcosmos (the world of elementary particles) His mind pierced the deepest secrets of numbers, space, and nature; he measured the orbits of the planets, the form and the forces of the earth; in his mind he carried the mathematical science of a coming century VII VIII Preface The six volumes address a broad audience of readers, including both undergraduate and graduate students, as well as experienced scientists who want to become familiar with quantum field theory, which is a fascinating topic in modern mathematics and physics, full of many crucial open questions For students of mathematics, detailed knowledge of the physical background helps to enliven mathematical subjects and to discover interesting interrelationships between quite different mathematical topics For students of physics, fairly advanced mathematical subjects are presented that go beyond the usual curriculum in physics The strategies and the structure of the six volumes are thoroughly discussed in the Prologue to Volume I In particular, we will try to help the reader to understand the basic ideas behind the technicalities In this connection, the famous ancient story of Ariadne’s thread is discussed in the Preface to Volume I: In terms of this story, we want to put the beginning of Ariadne’s thread in quantum field theory into the hands of the reader There are four fundamental forces in the universe, namely, • • • • gravitation, electromagnetic interaction (e.g., light), strong interaction (e.g., the binding force of the proton), weak interaction (e.g., radioactive decay) In modern physics, these four fundamental forces are described by • Einstein’s theory of general relativity (gravitation), and • the Standard Model in elementary particle physics (electromagnetic, strong, and weak interaction) The basic mathematical framework is provided by gauge theory: The main idea is to describe the four fundamental forces by the curvature of appropriate fiber bundles In this way, the universal principle force equals curvature is implemented There are many open questions: • A mathematically rigorous quantum field theory for the quantized version of the Standard Model in elementary particles has yet to be found • We not know how to combine gravitation with the Standard Model in elementary particle physics (the challenge of quantum gravitation) • Astrophysical observations show that 96 percent of the universe consists of both dark matter and dark energy However, both the physical structure and the mathematical description of dark matter and dark energy are unknown One of the greatest challenges of the human intellect is the discovery of a unified theory for the four fundamental forces in nature based on first principles in physics and rigorous mathematics In the present volume, we concentrate on the classical aspects of gauge theory related to curvature These have to be supplemented by the crucial, but elusive quantization procedure The quantization of the Maxwell–Dirac system leads to quantum electrodynamics (see Vol II) The quantization of both the full Standard Model in elementary particle physics and the quantization of gravitation will be studied in the volumes to come One cannot grasp modern physics without understanding gauge theory, which tells us that the fundamental interactions in nature are based on parallel transport, and in which forces are described by curvature, which measures the path-dependence of the parallel transport Preface IX Gauge theory is the result of a fascinating long-term development in both mathematics and physics Gauge transformations correspond to a change of potentials, and physical quantities measured in experiments are invariants under gauge transformations Let us briefly discuss this Gauss discovered that the curvature of a two-dimensional surface is an intrinsic property of the surface This means that the Gaussian curvature of the surface can be determined by using measurements on the surface (e.g., on the earth) without using the surrounding three-dimensional space The precise formulation is provided by Gauss’ theorema egregium (the egregious theorem) Bernhard Riemann (1826– ´ 1866) and Elie Cartan (1859–1951) formulated far-reaching generalizations of the theorema egregium which lie at the heart of • modern differential geometry (the curvature of general fiber bundles), and • modern physics (gauge theories) Interestingly enough, in this way, • Einstein’s theory of general relativity (the curvature of the four-dimensional space-time manifold), and • the Standard Model in elementary particle physics (the curvature of a specific fiber bundle with the symmetry group U (1) × SU (2) × SU (3)) can be traced back to Gauss’ theorema egregium In classical mechanics, a large class of forces can be described by the differentiation of potentials This simplifies the solution of Newton’s equation of motion and leads to the concept of potential energy together with energy conservation (for the sum of kinetic and potential energy) In the 1860s, Maxwell determined that the computation of electromagnetic fields can be substantially simplified by introducing potentials for both the electric and the magnetic field (the electromagnetic four-potential) Gauge theory generalizes this by describing forces (interactions) by the differentiation of generalized potentials (also called connections) The point is that gauge transformations change the generalized potentials, but not the essential physical effects Physical quantities, which can be measured in experiments, have to be invariant under gauge transformations Parallel to this physical situation, in mathematics the Riemann curvature tensor can be described by the differentiation of the Christoffel symbols (also called connection coefficients or geometric potentials) The notion of the Riemann curvature tensor was introduced by Riemann in order to generalize Gauss’ theorema egregium to higher dimensions In 1915, Einstein discovered that the Riemann curvature tensor of a four-dimensional space-time manifold can be used to describe gravitation in the framework of the theory of general relativity The basic idea of gauge theory is the transport of physical information along curves (also called parallel transport) This generalizes the parallel transport of vectors in the three-dimensional Euclidean space of our intuition In 1917, it was discovered by Levi-Civita that the study of curved manifolds in differential geometry can be based on the notion of parallel transport of tangent vectors (velocity vectors) 542 A Glance at Invariant Theory J Marsden and T Ratiu, Introduction to Mechanics and Symmetry, Springer, New York, 1999 E Binz and S Pods, The Geometry of Heisenberg Groups: With Applications in Signal Theory, Optics, Quantization, and Field Quantization, Amer Math Soc., Providence, Rhode Island, 2009 W Neutsch, Coordinates (in German), Spektrum, Heidelberg, 1995 (1350 pages) Concerning the Noether theorem and the energy-momentum tensor, we refer to: E Noether, Invariant variational problems, Gă ottinger Nachrichten, Math.phys Klasse 1918, 235–257 (in German) M Forger and H Ră omer, Currents and the energy-momentum tensor in classical field theory: a fresh look at an old problem, Annals of Physics 309 (2004), 306–389 Classification of the crystallographic groups: S Novikov and A Fomenko, Basic Elements of Differential Geometry and Topology, Kluwer, Dordrecht, 1987 Clifford Algebras and Spin Geometry S Lang, Algebra, Springer, New York, 2002 J Jost, Riemannian Geometry and Geometric Analysis, Springer, Berlin, 2008 J Moore, Lectures on Seiberg–Witten Invariants, Springer, Berlin, 1996 T Friedrich, Dirac Operators in Riemannian Geometry, Amer Math Soc., Providence, Rhode Island, 2000 M Atiyah, R Bott, and A Shapiro, Clifford modules, Topology (1964), 3–38 H Lawson and M Michelsohn, Spin Geometry, Princeton University Press, 1994 P Gilkey, Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem, CRC Press, Boca Raton, Florida, 1995 P Gilkey, The spectral geometry of Dirac and Laplace type, pp 289–326 In: Handbook of Global Analysis Edited by D Krupka and D Saunders, Elsevier, Amsterdam, 2008 N Berline, E Getzler, and M Vergne, Heat Kernels and Dirac Operators, Springer, New York, 1991 Applications to quantum field theory: B Fauser, A treatise on quantum Clifford algebras, postdoctoral thesis, University of Konstanz (Germany), 2002 Internet: http://arxiv.org/math.QA/0202059 B Fauser, On an easy transition from operator dynamics to generating functionals by Clifford algebras, J Math Phys 39 (1998), 4928–4947 Internet: http://arxiv.org/hep-th/9710186 B Fauser, On the relation of Manin’s quantum plane and quantum Clifford algebras, Czechosl J Physics 50(1) (2000), 1221–1228 Internet: http://arxiv.org/math.QA/0007137 8.16 Further Reading on Symmetry and Invariants 543 B Fauser, Clifford geometric quantization of inequivalent vacua, Math Meth Appl Sci 24 (2001), 885–912 Internet: http://arxiv.org/hep-th/9719947 B Fauser and R Ablamowicz, Clifford and Grassmann Hopf algebras via the BIGEBRA package for Maple, Computer Physics Communications 170(2) (2005), 115–130 Internet: http://arxiv.org/math-ph/0212032 Riemann Surfaces The theory of Riemann surfaces combines analysis, algebra, geometry, algebraic geometry, and number theory with each other in a beautiful way We recommend: M Waldschmidt, P Moussa, J Luck, and C Itzykson (Eds.), From Number Theory to Physics, Springer, New York, 1995 (collection of survey articles) L Ahlfors, Complex Analysis, McGraw Hill, 1966 (classic textbook) J Jost, Compact Riemann Surfaces: An Introduction to Contemporary Mathematics, Springer, Berlin, 1997 O Forster, Lectures on Riemann Surfaces, Springer, Berlin, 1981 R Narasimhan, Compact Riemann Surfaces Lectures given at the ETH Zurich, Birkhă auser, Basel, 1997 M Farkas and I Kra, Riemann Surfaces, Springer, New York, 1992 M Farkas and I Kra, Theta Constants, Riemann Surfaces and the Modular Group: An Introduction with Applications to Uniformization Theorems, Partition Identities and Combinatorial Number Theory, Amer Math Soc., Providence, Rhode Island, 2001 Conformal Field Theory and Infinite-Dimensional Lie Algebras H Kastrup, On the advancement of conformal transformations and their associated symmetries in geometry and theoretical physics, Ann Phys (Berlin) 17 (2008), 631–690 V Kac, Infinite-Dimensional Lie Algebras, Cambridge University Press, 1990 J Fuchs, Affine Lie Algebras and Quantum Groups: An Introduction with Applications in Conformal Field Theory, Cambridge University Press, 1992 P Di Francesco, P Mathieu, and D S´en´echal, Conformal Field Theory, Springer, New York, 1997 Supersymmetry As an introduction to supersymmetry including supersymmetric Riemann surfaces, we recommend: J Jost, Geometry and Physics (functorial approach to supersymmetry), Springer, Berlin, 2009 The supersymmetric version of the Standard Model in particle physics can be found in: 544 A Glance at Invariant Theory S Weinberg, Quantum Field Theory, Vol III, Cambridge University Press, 2000 W Hollik, E Kraus, M Roth, C Rupp, K Sibold, and D Stă ockinger, Renormalization of the minimal supersymmetric standard model, Nuclear Physics B 639 (2002), 3–65 D Bailin and A Love, Supersymmetric Gauge Field Theory and String Theory, Institute of Physics, Bristol, 1996 Furthermore, we recommend: J Lopuszanski, An Introduction to Symmetry and Supersymmetry in Quantum Field Theory, World Scientific, Singapore, 1991 M Chaichian and R Hagedorn, Symmetries in Quantum Mechanics: From Angular Momentum to Supersymmetry, Institute of Physics, Bristol, 1998 I Buchbinder and S Kuzenko, Ideas and Methods of Supersymmetry and Supergravity or A Walk Through Superspace, Institute of Physics, Bristol, 1995 A Khrennikov, Superanalysis, Kluwer, Dordrecht, 1999 J Wess and J Bagger, Supersymmetry and Supergravity, Princeton University Press, 1991 V Varadarajan, Supersymmetry for Mathematicians, Courant Lecture Notes, Amer Math Soc., Providence, Rhode Island, 2004 D Freed, Five Lectures on Supersymmetry, Amer Math Soc., Providence, Rhode Island, 1999 D Freed, D Morrison, and I Singer (Eds.), Quantum Field Theory, Supersymmetry, and Enumerative Geometry, Amer Math Soc., Providence, Rhode Island, 2006 P Deligne, E Witten et al (Eds.), Lectures on Quantum Field Theory: A Course for Mathematicians Given at the Institute for Advanced Study in Princeton, Vols 1, 2, Amer Math Soc., Providence, Rhode Island, 1999 V Guillemin and S Sternberg, Supersymmetry and Equivariant de Rham Theory, Springer, Berlin, 1999 S Bellucci, S Ferrara, and A Marrani, Supersymmetric Mechanics, Vol 1: Supersymmetry, Noncommutativity, and Matrix Models, Vol 2: The Attractor Mechanism and Space Time Singularities, Springer, Berlin, 2006 P Bin´etruy, Supersymmetry: Theory, Experiment, and Cosmology, Oxford University Press, 2006 P Srivasta, Supersymmetry, Superfields and Supergravity: An Introduction, Adam Hilger, Bristol, 1985 V Cort´es (Ed.), Handbook of Pseudo-Riemannian Geometry and Supersymmetry, European Mathematical Society, Zurich, 2010 Classic Monographs S Lie and F Engel, Theory of Transformation Groups, Vols 1–3, Teubner, Leipzig, 1888 Reprint: Chelsea Publ Company, 1970 (foundation of the local theory of Lie groups and Lie algebras) (in German) C Chevalley, Theory of Lie Groups, Princeton University Press, 1946 (15th printing, 1999) (foundation of the global theory) 8.16 Further Reading on Symmetry and Invariants H Weyl, The Theory of Groups and Quantum Mechanics, Dover, New York, 1931 (German edition: Springer, Berlin, 1929) H Weyl, The Classical Groups: Their Invariants and Representations, Princeton University Press, 1938 (2nd edition with supplement, 1946; 15th printing, 1997) H Weyl, Symmetry, Princeton University Press, 1952 E Wigner, Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra, Academic Press, New York, 1959 (German edition: Springer, Berlin, 1931) E Wigner, Symmetries and Reflections, Indiana University Press, Bloomington, 1970 L Pontryagin, Topological Groups, Gordon and Breach, 1966 (Russian edition: 1938) D Montgomery and L Zippin, Topological Transformation Groups, Interscience Publishers, New York, 1955 N Jacobson, Lie Algebras, Dover, New York, 1962 V Bargmann, Representations in Mathematics and Physics, Springer, Berlin, 1970 F Warner, Foundations of Differentiable Manifolds and Lie Groups, ScottForesman, Glenview, Illinois, 1971 J Serre, Linear Representations of Finite Groups, Springer, New York, 1977 J Serre, Lie Algebras and Lie Groups, Springer, Berlin, 1992 G Hochschild, Basic Theory of Algebraic Groups and Lie Algebras, Springer, New York, 1981 M Hammermesh, Group Theory and its Applications to Physical Problems, Dover, New York, 1989 S Lang, SL(2, R), Addison–Wesley, Reading, Massachusetts, 1975 V Varadarajan, Lie Groups, Lie Algebras, and Their Representations, Springer, New York, 1984 T Bră ocker and T tom Dieck, Representation Theory of Compact Lie Groups, Springer, Berlin, 1985 Locally Compact Groups E Wigner, On unitary representations of the inhomogeneous Lorentz group, Ann of Math 40 (1939), 149–204 V Bargmann, Irreducible representations of the Lorentz group, Ann of Math 48 (1947), 568–640 V Bargmann, On unitary ray representations of continuous groups, Ann of Math 59 (1954), 1–46 I Gelfand, R Minlos, and Ya Shapiro, Representations of the Rotation and Lorentz Groups and Their Applications, Pergamon Press, New York, 1963 Y Ohnuki, Unitary Representations of the Poincar´e Group and Relativistic Wave Equations, World Scientific, Singapore, 1987 545 546 A Glance at Invariant Theory A Knapp, Representation Theory of Semi-Simple Groups, Princeton University Press, 1986 A Knapp, Lie Groups, Lie Algebras, and Cohomology, Princeton University Press, 1988 A Knapp, Lie Groups Beyond an Introduction, Birkhă auser, Boston, 1996 M Stroppel, Locally Compact Groups, European Mathematical Society, Zurich, 2006 A lot of material can be found in: N Bourbaki, Lie Groups and Lie Algebras, Chaps 1–3, Springer, New York, 1989 N Bourbaki, Lie Groups and Lie Algebras, Chaps 4–6, Springer, New York, 2002 A Onishchik et al (Eds.), Lie Groups and Lie Algebras I–III, Encyclopedia of Mathematical Sciences, Springer, New York, 1993 Quantum Groups As an introduction, we recommend: C Kassel, M Rosso, and V Turaev, Quantum Groups and Knot Invariants, Soci´et´e Math´ematique de France, Paris, 1997 Furthermore, we recommend: A Klimyk and K Schmă udgen, Quantum Groups and Their Representations, Springer, Berlin, 1997 (many concrete examples) S Shnider and S Sternberg, Quantum Groups From Coalgebras to Drinfeld Algebras A Guided Tour, International Press, Boston, 1997 M Majid, Foundations of Quantum Group Theory, Cambridge University Press, 1995 T Timmermann, An Invitation to Quantum Groups and Duality: From Hopf Algebras to Multiplicative Unitaries and Beyond, European Mathematical Society, 2008 (compact and locally compact quantum groups; approach via operator algebras, generalization of Pontryagin duality) In addition, we recommend: Yu Manin, Topics in Noncommutative Geometry, Princeton University Press, 1991 S Woronowicz, Tannaka–Krein duality for compact matrix pseudogroups Twisted SU (N ) groups, Invent math 93 (1987), 35–76 S Woronowicz, Compact quantum groups Lectures given at ‘Les Houches 1995’, pp 845–884 See the next quotation A Connes, K Gaw¸edzki, and J Zinn-Justin (Eds.), Quantum Symmetries, Les Houches, 1995, North-Holland, Amsterdam, 1998 A Pressley, Quantum Groups and Lie Theory, Cambridge University Press, 2001 R Street, Quantum Groups: A Path to Current Algebra, Cambridge University Press, 2007 (theory of categories) The theory of quantum groups allows many applications in mathematics and physics: 8.16 Further Reading on Symmetry and Invariants 547 C Yang, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys Rev Lett 19 (1967), 1312– 1315 R Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, New York, 1982 L Faddeev, Integrable models in + 1-dimensional quantum field theory Lectures given at ‘Les Houches 1982’, pp 561–608 Edited by R Stora and B Zuber, North-Holland, Amsterdam, 1984 L Faddeev, How the algebraic Bethe ansatz works for integrable models Lectures given at ‘Les Houches 1995’, pp 149–220 Edited by A Connes et al., North-Holland, Amsterdam, 1998 V Turaev, Quantum Invariants of Knots and 3-Manifolds, de Gruyter, Berlin, 1994 H de Vega, Integrable Quantum Field Theories and Statistical Models: Yang–Baxter and Kac–Moody Algebras, World Scientific, Singapore, 2000 Tables For working with Lie groups and semisimple Lie algebras along with their representations, it is useful to use material summarized in tables This can be found in: P Atkins, M Child, and C Philips, Tables for Group Theory, Oxford University Press, 1978 B Slansky, Group theory for unified model building, Physics Reports 79(1) (1981), 1–128 N Bourbaki, Lie Groups and Lie Algebras, Chaps 4–6, Springer, New York, 2002 R Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications, Wiley, New York, 1974 B Simon, Representations of Finite and Compact Groups, Amer Math Soc., Providence, Rhode Island, 1996 G Ljubarskij, The Application of Group Theory in Physics, Pergamon Press, Oxford L Frappat, A Sciarinno, and P Sorba, Dictionary of Lie Algebras and Super Lie Algebras, Academic Press, New York, 2000 A Onishchik, Lectures on Real Semisimple Lie Algebras and Their Representations, European Mathematical Society, 2004 G Koster, J Dimmock, R Wheeler, and H Statz, Properties of the ThirtyTwo Point Groups, MIT Press, Cambridge, Massachusetts, 1969 W Neutsch, Coordinates: Theory and Applications, Spektrum, Heidelberg, 1350 pages (in German) For realizations of the exceptional Lie algebras and their Lie groups, we refer to Jacobson (1962) quoted on page 545 and to: N Jacobson, The Exceptional Lie Algebras, Mimeographed Lecture Notes, Yale University, New Haven, Connecticut, 1957 J Adams, Lectures on Exceptional Lie Groups, University Chicago Press, 1996 548 A Glance at Invariant Theory Differential Geometry and Gauge Theory Standard textbooks in modern differential geometry: S Kobayashi and K Nomizu, Foundations of Differential Geometry, Vols 1, 2, Wiley, New York, 1963 M Spivak, A Comprehensive Introduction to Differential Geometry, Vols 1–5, Publish or Perish, Boston, 1979 J Jost, Riemannian Geometry and Geometric Analysis, 5th edition, Springer, Berlin, 2008 Furthermore, we recommend: S Novikov and T Taimanov, Geometric Structures and Fields, Amer Math Soc., Providence, Rhode Island, 2006 B Dubrovin, A Fomenko, and S Novikov, Modern Geometry: Methods and Applications, Vols 1–3, Springer, New York, 1992 (including topological methods) Y Choquet-Bruhat, C DeWitt-Morette, and M Dillard-Bleick, Analysis, Manifolds, and Physics Vol 1: Basics; Vol 2: 92 Applications, Elsevier, Amsterdam, 1996 T Frankel, The Geometry of Physics, Cambridge University Press, 2004 B Felsager, Geometry, Particles, and Fields, Springer, New York, 1997 V Ivancevic and T Invancevic, Applied Differential Geometry: A Modern Introduction, World Scientific, Singapore, 2007 E Bick and F Steffen (Eds.), Topology and Geometry in Physics, Springer, Berlin, 2005 Fiber bundles and characteristic classes: J Milnor and J Stasheff, Characteristic Classes, Princeton University Press, 1974 D Husemoller, Fibre Bundles, Springer, New York, 1994 Mathematical approach to gauge theory: K Marathe, Topics in Physical Mathematics, Springer, London, 2010 M Henneaux and C Teitelboim, Quantization of Gauge Systems, Princeton University Press, 1993 G Naber, Topology, Geometry, and Gauge Fields, Springer, New York, 1997 Gauge theory, solitons, and the topology of manifolds: R Rajaraman, Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory, Elsevier, Amsterdam, 1987 A Kasmann, Glimpses of Soliton Theory: The Algebra and Geometry of Nonlinear Partial Differential Equations, Amer Math Soc., Providence, Rhode Island, 2011 D Freed and K Uhlenbeck, Instantons and Four-Manifolds, Springer, New York, 1984 Y Yang, Solitons in Field Theory and Nonlinear Analysis, Springer, New York, 2001 J Moore, Lectures on Seiberg–Witten Invariants, Springer, Berlin, 1996 8.16 Further Reading on Symmetry and Invariants 549 J Morgan, The Seiberg–Witten Equations and Applications to the Topology of Four-Manifolds, Princeton University Press, 1996 S Donaldson and P Kronheimer, The Geometry of Four-Manifolds, Oxford University Press, 1990 P Kronheimer and T Mirowka, Monopoles and Three-Manifolds, Cambridge University Press, 2007 S Donaldson, Floer Homology Groups, Cambridge University Press, 2002 M Atiyah, Collected Works, Vol V: Gauge Theories, Cambridge University Press, 2004 C Yang, Hermann Weyl’s contributions to physics, pp 7–21 In: Hermann Weyl (1885–1955), Springer, Berlin, 1985 Gauge theory in physics: C Taylor (Ed.), Gauge Theories in the Twentieth Century, World Scientific, Singapore, 2001 M Monastirsky, Topology of Gauge Fields and Condensed Matter, Plenum Press, New York, 1993 L Faddeev and A Slavnov, Gauge Fields, Benjamin, Reading, Massachusetts, 1980 A Das, Lectures on Quantum Field Theory, World Scientic, Singapore, 2008 M Bă ohm, A Denner, and H Joos, Gauge Theories of the Strong and Electroweak Interaction, Teubner, Stuttgart, 2001 T Kugo, Gauge Field Theory, Springer, Berlin, 1997 (translated from Japanese into German) Yu Makeenko, Methods of Contemporary Gauge Theory, Cambridge University Press, 2002 I Atchinson and A Hey, Gauge Theories in Particle Physics, Institute of Physics, Bristol, 1993 D Bailin and A Love, Introduction to Gauge Field Theory, Institute of Physics, Bristol, 1996 D Bailin and A Love, Supersymmetric Gauge Field Theory and String Theory, Institute of Physics, Bristol, 1996 B Zwiebach, A First Course in String Theory, Cambridge University Press, 2004 K Becker, M Becker, and J Schwarz, String Theory and M -Theory, Cambridge University Press, 2006 S Hollands, Renormalized Yang–Mills fields in curved spacetime, Rev Math Phys 20(9) (2008), 1033–1172 Internet: http://arxiv.org/0705.3340 P Langacker, The Standard Model and Beyond, CRC Press, Boca Raton, Florida, 2010 Supplementary material: http://www.sns.ias.edu/ pgl/SMB/ C Yang, Selected Papers, 1945–1980, Freeman, New York, 1983 Concerning the Standard Model in elementary particle physics, see also the references given on page 346 550 A Glance at Invariant Theory Number Theory Yu Manin and A Panchishkin, Introduction to Modern Number Theory, Encyclopedia of Mathematical Sciences, Vol 49, Springer, Berlin, 2005 (survey) ˇ Z Boreviˇc and I Safareviˇ c, Number Theory, Academic Press, New York, 1967 S Lang, Algebraic Number Theory, Springer, New York, 1986 T Apostol, Introduction to Analytic Number Theory, Springer, New York, 1986 T Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer, New York, 1990 M Waldschmidt, P Moussa, J Luck, and C Itzykson (Eds.), From Number Theory to Physics, Springer, New York, 1995 (survey articles) J Brunier, G van der Geer, G Harder, and D Zagier, The 1-2-3 of Modular Forms Lectures at a Summer School in Nordfjordeid, Norway, 2008, Springer, Heidelberg, 2009 (survey articles).46 K Kedlaya, p-adic Differential Equations, Cambridge University Press, 2010 History The mathematics of the nineteenth century strongly influenced the mathematics of the 20th century and the 21st century We recommend: F Klein, Vorlesungen u ăber die Entwicklung der Mathematik im 19 Jahrhundert, Vols 1, 2, Springer, Berlin, 1926 (in German) English edition: Development of Mathematics in the 19th Century, with a large appendix by R Hermann, Math Sci Press, New York, 1979 For the fascinating history of Lie groups, Lie algebras and algebraic groups, we recommend the essays written by Armand Borel (1923–2003) who worked at the Institute for Advanced Study in Princeton: A Borel, Essays in the History of Lie Groups and Algebraic Groups, History of Mathematics, Vol 21, Amer Math Soc., Providence, Rhode Island, 2001 We also refer to the following book review which contains a survey on the history of Lie groups and Lie algebras: V Varadarajan, Book review on “Lie Groups: An Approach through Invariants and Representations” by Claudio Procesi, Springer, New York, Bull Amer Math Soc 45(4) (2008), 661–674 Concerning the history of manifolds, we recommend: 46 In 1842 Sophus Lie was born in Nordfjordeide which is located at the Eidsfjord – a branch of the Nordfjord in Norway Sophus was the sixth child of a preacher For a long time, Lie lived in Germany He was a professor of mathematics at Leipzig University from 1886 until 1898 But all the time he was missing the beauty of his homeland Norway As a critically ill man, he returned to Norway in 1898 where he died in 1899 Problems 551 E Scholz, The concept of manifold, 1850–1950, pp 25–64 In: I James (Ed.), History of Topology, Oxford University Press, 1999 C Nash, Topology and physics – a historical essay, pp 359–416 In: I James (Ed.), History of Topology, Elsevier, Amsterdam, 1999 E Scholz (Ed.), Hermann Weyl’s ‘Space-Time-Matter’ and a General Introduction to his Scientic Work, Birkhă auser, Basel, 2001 J Lă utzen (Ed.), The Interaction between Mathematics, Physics, and Philosophy from 1850 to 1940, Kluwer, Dordrecht, 2004 D Flament et al (Eds.), G´eom´etrie au XXi`eme siecle (1930–2000): Histoire et horizons, Hermann, Paris, 2005 Problems 8.1 The correct index picture Consider the following equations: ij ij rj k ri • T ij + S ij = U ij , T i Aij = Tir Bkj , Tik + Ski = Ukr , T i + S i = Ui , kr • εijk v i w j ek , εijk v i wj ek , Tijij + Skr = U ab Vba , Aii + Bkil C kli = B j Cj , • B r Ars = Bk C ks , gij v i v j , gij v i vj Which of these equations not have the correct index picture (i.e., every additive term has the same free indices)? Solution: There are precisely five equations which not have the correct index picture (namely, number four, six, nine, and eleven) 8.2 Special transformation law (Lie derivative) Let v i and wi be tensorial families Use an explicit computation in order to show that vi ∂i wj − wi ∂i v j is again a tensorial family Solution: We have to show that v i ∂i w j − w i ∂i v j = In fact, it follows from v i = chain rule that vi Since ∂xi ∂xi ∂xj · (v i ∂i w j − w i ∂i v j ) ∂xj · v i and wj = ∂xj ∂xj (8.174) · w j together with the ∂wj ∂xi ∂wj ∂wj ∂ xj i j ∂xj i ∂wj = vi · = vi = vw + v i i i i ∂x ∂x ∂xi ∂xj ∂xj ∂xi ∂x ∂x ∂ xj ∂xi ∂xj v i wj = ∂ xj , ∂xj ∂xi we get (8.174) l 8.3 The Lie derivative Compute Lv T ij , Lv Tjk , and Lv Tjk i j i j i j Solution: From Lv (S T ) = (Lv S )T + S Lv T we get Lv (S i T j ) = (v s ∂s S i )T j − (S s ∂s v j )T j + S i (v s ∂s T j ) − S i T s ∂s v j ) = v s ∂s (S i T j ) − S s T j ∂s v i − S i T s ∂s v j Hence Lv T ij = v s ∂s T ij − T sj ∂s v i − T is ∂s v j Similarly, we get the following expressions: • Lv Tjk = v s ∂s Tjk − Tsj ∂s v s − T js ∂k v s , 552 A Glance at Invariant Theory l l l s • Lv Tjk = v s ∂s T l jk − Tsk ∂j v s − Tjs ∂k v s + Tjk ∂s v l jk l 8.4 The covariant partial derivative Compute ∇i T , ∇i Tjk , and ∇i Tjk Solution: A similar argument as in Problem 8.3 yields j r k r ∇i (S j T k ) = (∇i S j )T k + S j (∇i T k ) = (∂i S j + Γir S )T k + S j (∂i T k + Γir T ) j r k k j r S T + Γir S T = ∂i (S j T k ) + Γir j k jr Hence ∇i T jk = ∂i T jk + Γir T rk + Γir T Analogously, we get the following: s • ∇i Tjk = ∂i Tjk − Γijs Tsk − Γik Tjs , l l s l s l l s • ∇i Tjk = ∂i Tjk − Γij Tsk − Γik Tjs + Γis Tjk 8.5 Cartan’s magic formula – the brute force approach Use an explicit computation in order to prove Lv ω = iv (dω) + d(iv ω) (8.175) for the special case where ω = ωij dxi ∧ dxj with ωij = −ωji Solution: Note that • Lv ωij = v s ∂s ωij + ∂i v s · ωsj + ∂j v s · ωis , • Lv ωij = v s ∂s ωij + ∂i v s · ωsj − ∂j v s · ωsi , • Lv ω = Lv ωij · dxi ∧ dxj = (v s ∂s ωij + 2∂i v s · ωsj ) dxi ∧ dxj , • iv ω = 2v s ωsj dxj (by (8.64)), • d(iv ω) = (2∂i v s · ωsj + 2v s ∂i ωsj ) dxi ∧ dxj = (2∂i v s · ωsj − v s ∂i ωjs + v s ∂j ωis ) dxi ∧ dxj , • dω = ∂[s ωij] dxs ∧ dxi ∧ dxj , • iv (dω) = 3v s ∂[s ωij] dxi ∧ dxj = 3v s ∂[i ωjs] dxi ∧ dxj Recall that ∂[s ωij] denotes the antisymmetrization of ∂s ωij Using antisymmetry, we get the claim (8.175) 8.6 Cartan’s magic formula – the elegant index-free inductive approach Use the Leibniz rule (8.97) for the Lie derivative of differential forms on page 491 in order to prove (8.175) Solution: Let ω be a p-form Because of (8.97), the proof can be reduced to the special cases where p = and p = 1, by induction • p = : Use iv ω = • p = : The formula is true for ω := dxk In fact, using dd = and the commutation relation d(Lv μ) = Lv (dμ), we get (iv d + div )(dΘ) = d(iv dΘ) = d(Lv Θ) = Lv (dΘ) Finally, set Θ := xk 8.7 The special case of the Euclidean manifold E3 Set n = and gij := δij , as well as x1 := x, x2 := y, x3 := z Let U, u, v, w be smooth real-valued functions.47 Show that: • ∗1 = dx ∧ dy ∧ dz and ∗(dx ∧ dy ∧ dz) = 1; • ∗dx = dy ∧ dz and ∗(dy ∧ dz) = dx 47 On the Euclidean manifold E3 , one has not to distinguish between lower and upper indices For example, Tk = gkl T l = δkl T l = T k , and so on Problems 553 The remaining relations follow by using the cyclic permutation x ⇒ y ⇒ z ⇒ x In particular, ∗ ∗ ω = ω for all p-forms ω, p = 0, 1, 2, Furthermore, recalling d∗ ω = (−1)p ∗ d ∗ ω, show that: • • • • • • • dU = Ux dx + Uy dy + Uz dz and d∗ U = 0; d(udx+ vdy + wdz) = (vx − uy )dx∧dy +(wy − vz ) dy ∧dz +(uz − wx ) dz ∧dx; d(u dy ∧ dz + v dz ∧ dx + w dx ∧ dy) = (ux + vy + wz ) dx ∧ dy ∧ dz; d(U dx ∧ dy ∧ dz) = 0; d∗ (udx + vdy + wdz) = −ux − vy − wz ; d∗ (u dy∧dz+v dz∧dx+w dx∧dy) = (wy −vz ) dx+(uz −wx )dy+(vx −uy ) dz; d∗ (U dx ∧ dy ∧ dz) = −Ux dy ∧ dz − Uy dz ∧ dx − Uz dx ∧ dy These operations are closely related to: • v = ui + vj + wk and div v = ux + vy + wz ; • curl v = (wy −vz ) i+(uz −wx ) j+(vx −uy ) k, and grad U = Ux i+Uy j+Uz k Finally, recalling Δ := d∗ d + dd∗ , show that: • • • • ΔU = d∗ dU = − div grad U = −Uxx − Uyy − Uzz ; Δ(udx + vdy + wdz) = Δu · dx + Δv · dy + Δw · dz; Δ(u dy ∧dz +v dz ∧dx+w dx∧dy) = Δu·dy ∧dz +Δv ·dz ∧dx+Δw ·dx∧dy; Δ(U dx ∧ dy ∧ dz) = ΔU · dx ∧ dy ∧ dz Solution: If ω = ω 3! ijk dxi ∧ dxj ∧ dxk = ω123 dx ∧ dy ∧ dz, then ∗ω = ω εijk 3! ijk = ω123 Moreover, if ω = 12 ωij dxi ∧ dxj with ωij = −ωji, then ω = ω12 dx ∧ dy + ω31 dz ∧ dy + ω23 dy ∧ dz Hence ∗ω = 12 εijk ω ij · dxk = ω23 dx + ω31 dy + ω12 dz Moreover, if ω = udx + vdy + wdz, then dω = du ∧ dx + dv ∧ dy + dw ∧ dz Since du = ux dx + uy dy + uz dz, we get du ∧ dx = uy dy ∧ dx + uz dz ∧ dx In addition, dy ∧ dx = −dx ∧ dy Concerning ΔU , see the next problem in a more general setting 8.8 The Hodge codifferential and the Hodge Laplacian in n-dimensional Cartesian coordinates Choose a tensorial family gij with gij := δij , i, j = 1, , n for ωi1 ip dxi1 ∧ · · · ∧ dxip where ωi1 ip is an a fixed observer O Let ω = p! antisymmetric tensorial family Show that, for the observer O, we get δ ij ∂j ωii2 ip dxi2 ∧ · · · ∧ dxip , and (i) d∗ ω = − (p−1)! (ii) Δω = − p! δ ij ∂i ∂j ωi1 ip dxi1 ∧ · · · ∧ dxip Solution: Ad (i) Consider the special case ω = ωi dxi where p = Then: εii2 in ω i · dxi2 ∧ · · · ∧ dxin ; • ∗ω = (n−1)! εii2 in ∂k ω i ·dxk ∧dxi2 ∧· · ·∧dxin = ∂i ω i dx1 ∧dx2 ∧· · ·∧dxn ; • d(∗ω) = (n−1)! ∗ • d ω = − ∗ d(∗ω) = −∂ i ωi 554 A Glance at Invariant Theory Argue similarly if p = 2, , n Ad (ii) Again let ω = ωi dxi Then: • d∗ ω = −∂i ω i ; • dd∗ ω = −∂k ∂ i ωi dxk ; • dω = ∂k ωi dxk ∧ dxi = ∂[k ωi] dxk ∧ dxi ; • d∗ dω = −2∂ k ∂[k ωi] dxi = (−∂ k ∂k ωi + ∂ k ∂i ωk ) dxi ; • (dd∗ + d∗ d)ω = −∂ k ∂k ωi · dxi Argue similarly if p = 2, , n 8.9 Covariant partial derivative and Cartan derivative Prove Prop 8.21 on page 499 Solution: By definition of the covariant partial derivative, ∇i ωi1 ip = ∂i ωi1 ip − p X Γiis σ ωi1 iσ−1 siσ+1 ip σ=1 Antisymmetrization yields ∇[i ωi1 ip ] = ∂[i ωi1 ip ] − Altii1 ip p X Γiis σ ωi1 iσ−1 siσ+1 ip σ=1 Interchanging the indices i and iσ , the sign changes Hence ∇[i ωi1 ip ] = ∂[i ωi1 ip ] + Altii1 ip p X Γisσ i ωi1 iσ−1 siσ+1 ip σ=1 Summing up, we get 2∇[i ωi1 ip ] = 2∂[i ωi1 ip ] − Altii1 ip p X Tiis σ ωi1 iσ−1 siσ+1 ip , σ=1 − = by using 8.10 Proof of the determinant identity (8.167) on page 524 Solution: Consider the special case where n = By the Leibniz rule, we have the partial derivative ˛ ˛ ˛ ˛ ˛ ˛ ∂ ˛˛g11 g12 ˛˛ ˛˛∂k g11 ∂k g12 ˛˛ ˛˛ g11 g12 ˛˛ ˛ ˛=˛ ˛+˛ ˛ ∂xk ˛g21 g22 ˛ ˛ g21 g22 ˛ ˛∂k g21 ∂k g22 ˛ Γiis σ Γisσ i Tiis σ By the Laplace expansion formula, this is equal to ∂k g11 · A11 + ∂k g12 · A12 + ∂k g21 · A21 + ∂22 · A22 where Aij denotes the adjunct of the determinant g to the element gij By (1.14) on page 77, Aij = gg ji Since gij = gji , we also have g ij = g ji Summarizing, ∂k g = gg ij ∂k gij This is the claim if n = In the general case where n = 3, 4, , the proof proceeds analogously 8.11 The Hodge codifferential in terms of the covariant partial derivative Prove (8.157) on page 519 Hint: See Choquet–Bruhat et al., Analysis, Manifolds, and Physics, page 317, Vol 1, Elsevier, Amsterdam, 1996 8.12 Summary of important identities for differential forms Let ω, μ, ν be differential forms of degree p, q, r = 0, 1, , respectively, and let α, β be real numbers Prove some of the following formulas: Problems • • • • • • • • • • 555 (ω ∧ μ) ∧ ν = ω ∧ (μ ∧ ν) (associative law); ω ∧ μ = (−1)pq μ ∧ ω (supercommutative law);48 If q = r, then ω ∧ (αμ + βν) = αω ∧ μ + βω ∧ ν (distributive law); d(ω ∧ μ) = dω ∧ μ + (−1)p ω ∧ dμ (graded Leibniz rule); d(dω) = (Poincar´e’s cohomology rule); d∗ (d∗ ω) = (Hodge’s homology rule); Lv (dω) = d(Lv ω); Lv ω = iv (dω) + d(iv ω) (Cartan’s magic formula); dω(v, w) = Lv (ω(w)) − Lw (ω(v)) − ω([v, w]) (special Cartan–Lie formula); For p = 2, 3, , n, the general Lie–Cartan formula reads as follows: dω(v0 , v1 , , vp ) = p X ˆ i , , vp ) (−1)i Lvi ω(v0 , , v i=0 + X ˆ i, , v ˆ j , vp ) (−1)i+j ω([vi , vj ], v0 , , v i