Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006

Path Integralsin Quantum Mechanics, Statistics,Polymer Physics, and Financial MarketsPath Integralsin Quantum Mechanics, Statistics,Polymer Physics, and Financial MarketsHagen KleinertProfessor of PhysicsFreie Universität BerlinTo Annemarie and Hagen IINature alone knows what she wants.GoethePrefaceThe third edition of this book appeared in 2004 and was reprinted in the sameyear without improvements. The present fourth edition contains several extensions.Chapter 4 includes now semiclassical expansions of higher order.

Chapter 8 offersan additional path integral formulation of spinning particles whose action containsa vector field and a Wess-Zumino term. From this, the Landau-Lifshitz equationfor spin precession is derived which governs the behavior of quantum spin liquids.The path integral demonstrates that fermions can be described by Bose fields—thebasis of Skyrmion theories. A further new section introduces the Berry phase, auseful tool to explain many interesting physical phenomena. Chapter 10 gives moredetails on magnetic monopoles and multivalued fields. Another feature is new inthis edition: sections of a more technical nature are printed in smaller font size.They can well be omitted in a first reading of the book.Among the many people who spotted printing errors and helped me improvevarious text passages are Dr.

A. Chervyakov, Dr. A. Pelster, Dr. F. Nogueira, Dr.M. Weyrauch, Dr. H. Baur, Dr. T. Iguchi, V. Bezerra, D. Jahn, S. Overesch, andespecially Dr. Annemarie Kleinert.H. KleinertBerlin, June 2006viiviiiH. Kleinert, PATH INTEGRALSPreface to Third EditionThis third edition of the book improves and extends considerably the second editionof 1995:• Chapter 2 now contains a path integral representation of the scattering amplitude and new methods of calculating functional determinants for timedependent second-order differential operators. Most importantly, it introducesthe quantum field-theoretic definition of path integrals, based on perturbationexpansions around the trivial harmonic theory.• Chapter 3 presents more exactly solvable path integrals than in the previouseditions.

It also extends the Bender-Wu recursion relations for calculatingperturbation expansions to more general types of potentials.• Chapter 4 discusses now in detail the quasiclassical approximation to the scattering amplitude and Thomas-Fermi approximation to atoms.• Chapter 5 proves the convergence of variational perturbation theory. It alsodiscusses atoms in strong magnetic fields and the polaron problem.• Chapter 6 shows how to obtain the spectrum of systems with infinitely highwalls from perturbation expansions.• Chapter 7 offers a many-path treatment of Bose-Einstein condensation anddegenerate Fermi gases.• Chapter 10 develops the quantum theory of a particle in curved space, treatedbefore only in the time-sliced formalism, to perturbatively defined path integrals.

Their reparametrization invariance imposes severe constraints uponintegrals over products of distributions. We derive unique rules for evaluatingthese integrals, thus extending the linear space of distributions to a semigroup.• Chapter 15 offers a closed expression for the end-to-end distribution of stiffpolymers valid for all persistence lengths.• Chapter 18 derives the operator Langevin equation and the Fokker-Planckequation from the forward–backward path integral. The derivation in the literature was incomplete, and the gap was closed only recently by an elegantcalculation of the Jacobian functional determinant of a second-order differential operator with dissipation.ixx• Chapter 20 is completely new.

It introduces the reader into the applicationsof path integrals to the fascinating new field of econophysics.For a few years, the third edition has been freely available on the internet, andseveral readers have sent useful comments, for instance E. Babaev, H. Baur, B.Budnyj, Chen Li-ming, A.A. Drăgulescu, K. Glaum, I. Grigorenko, P.

Hollister, P.Jizba, B. Kastening, M. Krämer, W.-F. Lu, S. Mukhin, A. Pelster, C. Öcalır, M.B.Pinto, C. Schubert, S. Schmidt, R. Scalettar, C. Tangui, and M. van Vugt. Severalprinting errors were detected by T.S. Hatamian who had the idea of creating a discussion forum under the URL http://pub17.ezboard.com/fpathintegralsfrm7where readers ask questions.

Reported errors are corrected in the internet edition.When writing the new part of Chapter 2 on the path integral representation ofthe scattering amplitude I profited from discussions with R. Rosenfelder. In the newparts of Chapter 5 on polarons, many useful comments came from J.T. Devreese,F.M.

Peeters, and F. Brosens. In the new Chapter 20, I profited from discussionswith F. Nogueira, A.A. Drăgulescu, E. Eberlein, J. Kallsen, M. Schweizer, P. Bank,M. Tenney, and E.C. Chang.As in all my books, many printing errors were detected by my secretary S. Endriasand many improvements are due to my wife Annemarie without whose permanentencouragement this book would never have been finished.H. KleinertBerlin, August 2003H. Kleinert, PATH INTEGRALSPreface to Second EditionSince this book first appeared three years ago, a number of important developmentshave taken place calling for various extensions to the text.Chapter 4 now contains a discussion of the features of the semiclassical quantization which are relevant for multidimensional chaotic systems.Chapter 3 derives perturbation expansions in terms of Feynman graphs, whoseuse is customary in quantum field theory.

Correspondence is established withRayleigh-Schrödinger perturbation theory. Graphical expansions are used in Chapter 5 to extend the Feynman-Kleinert variational approach into a systematic variational perturbation theory. Analytically inaccessible path integrals can now beevaluated with arbitrary accuracy. In contrast to ordinary perturbation expansionswhich always diverge, the new expansions are convergent for all coupling strengths,including the strong-coupling limit.Chapter 10 contains now a new action principle which is necessary to derive thecorrect classical equations of motion in spaces with curvature and a certain class oftorsion (gradient torsion).Chapter 19 is new. It deals with relativistic path integrals, which were previouslydiscussed only briefly in two sections at the end of Chapter 15.

As an application,the path integral of the relativistic hydrogen atom is solved.Chapter 16 is extended by a theory of particles with fractional statistics (anyons),from which I develop a theory of polymer entanglement. For this I introduce nonabelian Chern-Simons fields and show their relationship with various knot polynomials (Jones, HOMFLY). The successful explanation of the fractional quantum Halleffect by anyon theory is discussed — also the failure to explain high-temperaturesuperconductivity via a Chern-Simons interaction.Chapter 17 offers a novel variational approach to tunneling amplitudes. It extends the semiclassical range of validity from high to low barriers.

As an application,I increase the range of validity of the currently used large-order perturbation theoryfar into the regime of low orders. This suggests a possibility of greatly improvingexisting resummation procedures for divergent perturbation series of quantum fieldtheories.The Index now also contains the names of authors cited in the text. This mayhelp the reader searching for topics associated with these names. Due to theirgreat number, it was impossible to cite all the authors who have made importantcontributions.

I apologize to all those who vainly search for their names.xixiiIn writing the new sections in Chapters 4 and 16, discussions with Dr. D. Wintgenand, in particular, Dr. A. Schakel have been extremely useful. I also thank ProfessorsG. Gerlich, P. Hänggi, H. Grabert, M. Roncadelli, as well as Dr. A. Pelster, andMr. R.

Karrlein for many relevant comments. Printing errors were corrected by mysecretary Ms. S. Endrias and by my editor Ms. Lim Feng Nee of World Scientific.Many improvements are due to my wife Annemarie.H. KleinertBerlin, December 1994H. Kleinert, PATH INTEGRALSPreface to First EditionThese are extended lecture notes of a course on path integrals which I delivered at theFreie Universität Berlin during winter 1989/1990. My interest in this subject datesback to 1972 when the late R. P. Feynman drew my attention to the unsolved pathintegral of the hydrogen atom. I was then spending my sabbatical year at Caltech,where Feynman told me during a discussion how embarrassed he was, not being ableto solve the path integral of this most fundamental quantum system.

In fact, this hadmade him quit teaching this subject in his course on quantum mechanics as he hadinitially done.1 Feynman challenged me: “Kleinert, you figured out all that grouptheoretic stuff of the hydrogen atom, why don’t you solve the path integral!” He wasreferring to my 1967 Ph.D. thesis2 where I had demonstrated that all dynamicalquestions on the hydrogen atom could be answered using only operations withina dynamical group O(4, 2). Indeed, in that work, the four-dimensional oscillatorplayed a crucial role and the missing steps to the solution of the path integral werelater found to be very few. After returning to Berlin, I forgot about the problem sinceI was busy applying path integrals in another context, developing a field-theoreticpassage from quark theories to a collective field theory of hadrons.3 Later, I carriedthese techniques over into condensed matter (superconductors, superfluid 3 He) andnuclear physics.

Path integrals have made it possible to build a unified field theoryof collective phenomena in quite different physical systems.4The hydrogen problem came up again in 1978 as I was teaching a course onquantum mechanics. To explain the concept of quantum fluctuations, I gave an introduction to path integrals. At the same time, a postdoc from Turkey, I. H. Duru,joined my group as a Humboldt fellow.

Since he was familiar with quantum mechanics, I suggested that we should try solving the path integral of the hydrogen atom.He quickly acquired the basic techniques, and soon we found the most importantingredient to the solution: The transformation of time in the path integral to a newpath-dependent pseudotime, combined with a transformation of the coordinates to1Quoting from the preface of the textbook by R.P.