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This is the blurb as it appears on the cover of the book.

H. Kleinert

PATH INTEGRALS

in Quantum Mechanics, Statistics, Polymer Physics,
and Financial Markets

World Scientific Publishing Co., Singapore 2003.

This is the third, significantly expanded edition of the comprehensive textbook of 1990 on the theory and applications of path integrals. It is the first book to explicitly solve path integrals of a wide variety of nontrivial quantum-mechanical systems, in particular of the hydrogen atom. The solutions have become possible by two major advances. The first is a new euclidean path integral formula which increases the restricted range of applicability of Feynman's famous formula to include singular attractive 1/r- and 1/r^2-potentials. The second is a simple quantum equivalence principle governing the transformation of euclidean path integrals to spaces with curvature and torsion which leads to time-sliced path integrals which are manifestly invariant under coordinate transformations.

In addition to the time-sliced definition, we give a perturbative definition to path integrals which makes them invariant under coordinate transformations. A consitent implementation of this property leads to an extension of the theory of generlaized functions by defining uniquely integrals over products of distributions.

The powerful Feynman-Kleinert variational approach is explained and developed systematically into a variational perturbation theory which, in contrast to ordinary perturbation theory, produces convergent expansions. The convergence is uniform from weak to strong couplings, opening a way to precise approximate evaluations of analytically unsolvable path integrals.

Tunneling processes are treated in detail. The results are used to determine the lifetime of supercurrents, the stability of metastable thermodynamic phases, and the large-order behavior of perturbation expansions. A new variational treatment extends the range of validity of previous tunneling theories from large to small barriers. A corresponding extension of large-order perturbation theory now also applies to small orders.

Special attention is devoted to path integrals with topological restrictions. These are relevant to the understanding of the statistical properties of elementary particles and the entanglement phenomena in polymer physics and biophysics. The Chern-Simons theory of particles with fractional statistics ( anyons) is introduced and applied to explain the fractional quantum Hall effect.

The relevance of path integral to financial markets is discussed, and improvements of the famous Black-Scholes formula for option prices are developed which account for the fact that large market fluctuation occur much more frequently than in Gaussian distributions.

ABOUT THE AUTHOR

Hagen Kleinert is Professor of Physics at the Freie Universität Berlin, Germany. As a visiting scientist, he has spent extended periods of time at CERN, the European Organization for Nuclear Research in Geneva; at the California Institute of Technology in Pasadena; at the Universities of California in Berkeley, Santa Barbara, and San Diego; at the Los Alamos National Laboratories in New Mexico; and at Princeton University, New Jersey. He has made numerous contributions to our understanding of particle physics, mathematical physics, condensed matter physics, chemical physics, and nuclear physics. His two-volume book Gauge Fields in Condensed Matter, published by World Scientific, develops a new quantum field-theory of phase transitions on the basis of disorder fields. Such fields have since become a powerful tool to investigate the statistical properties of fluctuating line-like excitations in various many-body systems such as superfluids, superconductors, and crystals. His book Critical Poperties of Phi^4-Theories, coauthored with V. Schulte-Frohlinde, presents a thorough introduction to the field of critical phenomena and develops powerful resummation methods for the divergent perturbation expansions of quantum field theories, resulting in the most accurate critical exponents so far.



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