Mathematical Structures in Physics Kolleg Studienstiftung Christoph Schweigert Hamburg University Department of Mathematics Section Algebra and Number Theory and Center for Mathematical Physics (as of 12.03.2016) Contents 0 Introduction 1 1 Newtonian mechanics 2 1.1 Galilei space, equations of motion . 2 1.2 Dynamics of Newtonian systems . 8 1.3 Examples . 10 2 Lagrangian mechanics 14 2.1 Variational calculus . 14 2.2 Systems with constraints . 24 2.3 Lagrangian systems: jet bundles as the kinematical setup . 28 2.4 Lagrangian dynamics . 43 2.5 Symmetries and Noether identities . 48 2.6 Natural geometry . 57 3 Classical field theories 61 3.1 Maxwell's equations . 61 3.2 Special relativity . 69 3.4 Electrodynamics as a gauge theory . 77 3.5 General relativity . 85 4 Hamiltonian mechanics 89 4.1 (Pre-) symplectic manifolds . 89 4.2 Poisson manifolds and Hamiltonian systems . 99 4.3 Time dependent Hamiltonian dynamics . 105 4.4 The Legendre transform . 108 i 5 Quantum mechanics 115 5.1 Deformations . 115 5.2 Kinematical framework for quantum mechanics: C∗-algebras and states . 119 5.3 Composite systems and Bell's inequality . 130 5.4 Dynamics of quantum mechanical systems . 132 5.5 Quantization . 141 5.6 Symmetries in quantum mechanics . 150 5.7 Examples of quantum mechanical systems . 159 5.8 Quantum statistical mechanics and KMS states . 168 5.9 Perturbation theory . 170 5.10 Path integral methods . 170 6 A glimpse to quantum field theory 171 A Differentiable Manifolds 174 A.1 Definition of differentiable manifolds . 174 A.2 Tangent vectors and differentiation . 177 A.3 Fibre bundles and Lie groups . 181 A.4 Vector fields and Lie algebras . 185 A.5 Differential forms and the de Rham complex . 188 A.6 Riemannian manifolds and the Hodge dual . 197 The current version of these notes can be found under http://www.math.uni-hamburg.de/home/schweigert/skripten/pskript.pdf as a pdf file. Please send comments and corrections to [email protected]! These notes are based on lectures delivered at the University of Hamburg in the fall terms 2007/2008, 2009/2010 and 2011/2012 as part of the master programs in physics and mathe- matical physics. A first version of these notes was taken in German by Linda Sass. I am grateful to the students in these years for their remarks and comments, in particular to Sion Chan-lang and Thomas Reichenb¨acher. References: References per Chapter: 1. Classical Mechanics and Lagrangian systems: Textbooks: • Vladimir I. Arnold: Mathematical methods of classical mechanics, Springer Graduate Text in Mathematics 60, Springer, New York, 1978. • Herbert Goldstein; Charles Poole and John Safko: Classical mechanics, Pearson/Addison Wesley, Upper Saddle River, 2002 Articles and monographs: ii • Anderson, Ian M.: Introduction to the variational bicomplex, in M. Gotay, J. Marsden and V. Moncrief (eds), Mathematical Aspects of Classical Field Theory, Contemporary Mathematics 132, Amer. Math. Soc., Providence, 1992, pp. 5173. http://www.math.usu. edu/~fg_mp/Publications/ComtempMath/IntroVariationalBicomplex.ps • You are also encouraged to go back to the classics: Emmy Noether: Invariante Variationsprobleme G¨ott.Nachr. 1918 235-257. English trans- lation at http://arxiv.org/abs/physics/0503066v1 • Peter Olver: Applications of Lie Groups to Differential Equations. Springer Graduate Texts in Mathematics 107, 1986 • Enrique G. Reyes: On covariant phase space and the variational bicomplex. Internat. J. Theoret. Phys. 43 (2004), no. 5, 1267{1286. http://www.springerlink.com/content/q151731w71l71q23/ • Ron Fulp and Jim Stasheff: Noether's variational theorem II and the BV formalism http://arxiv.org/abs/math/0204079 3. Electrodynamics and relativity: Textbooks: • Stephen Hawking and George Ellis: The large scale structure of space-time. Cambridge monographs on mathematical physics, 21st printing, 2008. • John David Jackson: Classical electrodynamics. 3rd edition, Wiley, 1999. • Bertel Laurent: Introduction to Spacetime. A First Course on Special Relativity. World Scientific Publishing Company, 1994 • Hartmann R¨omer, M. Forger: Elementare Feldtheorie. VCH, Weinheim 1993. http://www.freidok.uni-freiburg.de/volltexte/405/pdf/efeld1.pdf Articles and monographs: • Some of our treatment of electrodynamics follows the review: Friedrich W. Hehl: Maxwell's equations in Minkowski's world: their premetric generaliza- tion and the electromagnetic energy-momentum tensor http://arxiv.org/abs/0807.4249v1 4. Hamiltonian mechanics and symplectic geometry • Rolf Berndt: An introduction to symplectic geometry, American Mathematical Society, Providence, 2001 • Jean-Marie Souriau: Structure of dynamical systems. Birkh¨auser,Boston, 1997. http://www.jmsouriau.com/structure_des_systemes_dynamiques.htm 5. Quantum mechanics: Textbooks: • Ludwig D. Faddeev, O.A. Yakubovskii: Lectures on quantum mechanics for mathematics students. AMS Student Mathematical Library, Volume 47, 2009 iii • Albert Messiah: Quantum mechanics. Dover Publications, 1999. • Strocchi, F.: An introduction to the mathematical structure of quantum mechanics. Ad- vanced Series in mathematical physics 27, World Scientific, Singapur, 2005. • Jun John Sakurai: Modern Quantum Mechanics. Benjamin Cummings, 1985. • Leon A. Takhtajan: Quantum Mechanics for Mathematicians. Graduate Studies in Math- ematics, American Mathematical Society, 2008. Articles and monographs: • For C∗-algebras: Christian B¨arand Christian Becker: C∗-algebras, in: Quantum Field Theory on Curved Spacetimes, Lecture Notes in Physics, 786, Springer 2009. Also available at http://geometrie.math.uni-potsdam.de/documents/baer/QFT-ProceedingsCh01.pdf • For spectral theory: William Arveson: A Short Course on Spectral Theory. Springer Grad- uate Texts in Mathematics, 2002, Volume 209 • For quantization: Domenico Giulini: That Strange Procedure Called Quantisation. Lec- ture Notes in Physics 631, Springer, Berlin, 2003. http://de.arxiv.org/abs/quant-ph/0304202 Appendix A: Manifolds • Glen E. Bredon: Topology and geometry. Springer Graduate Texts in Mathematics 139, 1993. • Detlef Gromoll, Wilhelm Klingenberg and Wolfgang Meyer: Riemannsche Geometrie im Großen, Lecture Notes in Math. 55, Springer (1975). • Ib Madsen and Jørgen Tornehave: From Calculus to cohomology: de Rham cohomology and characteristic classes. Cambridge University Press 1997. • Raoul Bott and Loring W. Tu: Differential forms in algebraic topology. Springer Graduate Texts in Mathematics 82, New York, 1982. iv 0 Introduction Eugene Wigner writes in his famous article \The Unreasonable Effectiveness of Mathematics in the Natural Sciences"1 The miracle of the appropriateness of the language of mathematics for the formu- lation of the laws of physics is a wonderful gift which we neither understand nor deserve. It is one of the goals of this class to show at least few instances of this miracle. Our goal is twofold: the first goal is to show the appropriate mathematics to a student of physics, roughly familiar with all classes of theoretical physics except for quantum field theory. The second goal is to show to a student of mathematics some mature mathematical theories at work: to see how they describe aspects of our reality in the inanimate world - this is after what physics is about. Both goals in themselves are so ambitious that the course is bound to fail; hence it only matters that the course fails in such a way that students taking the course have a maximum benefit. I am not sure that the combination of these two goals necessarily increases the likelihood to fail. Indeed, the relation between developments in physics and in mathematics is a rather complicated one and there are more interrelation than one naively expects: Let me again quote Wigner: It is true, of course, that physics chooses certain mathematical concepts for the formulation of the laws of nature, and surely only a fraction of all mathematical concepts is used in physics. It is true also that the concepts which were chosen were not selected arbitrarily from a listing of mathematical terms but were developed, in many if not most cases, independently by the physicist and recognized then as having been conceived before by the mathematician. The course is based on rather personal choices. It does not at all pretend to show the only possible approach to mathematical physics, not even the most appropriate one. It emphasizes structural aspects and concepts and thereby prefers a general point of view to the example. The idea is to cover essentially those concepts of classical physics and quantum physics that are needed to understand quantum field theory. One should emphasize that classical mechanics and quantum mechanics cover an enormous range of aspects of our physical reality and that we do meet quite a few of the core achievements of theoretical physics. If we insist on preparing the reader to quantum field theory, then for the reason that quantum field theory is not only connected to many important recent developments in mathematics, but also to what one might consider the two main challenges of physics in our time: a unified description of all forces and particles in nature and an understanding of the collective behaviour of qunautm mechanical particles. I would like to highlight in the context the following aspects: • We try to set up a geometric setting for Lagrangian systems that allows to appreciate both theorems of Emmy Noether. Here, Einstein's comments on Noether's work might lead the path: 1Eugene Wigner, \The Unreasonable Effectiveness of Mathematics in the Natural Sciences," in: Communi- cations in Pure and Applied Mathematics, vol. 13, No. I (February 1960). 1 \Gestern erhielt ich von Fr. N¨othereine sehr interessante Arbeit ¨uber In- variantenbildung. Es imponiert mir, dass man diese Dinge von so allge- meinem Standpunkt ¨ubersehen kann. Es h¨atteden G¨ottingerFeldgrauen nicht geschadet, wenn sie zu Frl. N¨otherin die Schule geschickt worden w¨aren.Sie scheint ihr Handwerk gut zu verstehen!" 2 • We consistently use differential forms and a geometric approach. In this spirit, we also see electrodynamics with gauge potentials as an instance of differential cohomology. • We emphasize the role of observables. For this reason, we treat the Hilbert space as a derived concept in quantum mechanics.