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Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, by Hagen Kleinert
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This is the fourth, expanded edition of the comprehensive textbook published in 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 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/r2 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 that are manifestly invariant under coordinate transformations. In addition to the time-sliced definition, the author gives a perturbative definition of path integrals which makes them invariant under coordinate transformations. A consistent implementation of this property leads to an extension of the theory of generalized 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 also applies now 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 integrals to financial markets is discussed, and improvements of the famous Black-Scholes formula for option prices are given which account for the fact that large market fluctuations occur much more frequently than in the commonly used Gaussian distributions. The author's other book on 'Critical Properties of Theories' gives a thorough introduction to the field of critical phenomena and develops new powerful resummation techniques for the extraction of physical results from the divergent perturbation expansions.
- Sales Rank: #4178695 in Books
- Published on: 2009-05-18
- Original language: English
- Number of items: 1
- Dimensions: 9.10" h x 2.20" w x 6.40" l, 4.40 pounds
- Binding: Hardcover
- 1624 pages
Review
Nobody possibly is better suited to write a book about path integrals than Hagen Kleinert. -- Physics in Canada
The best single book on the subject: the depth of knowledge and exposition is representative of a true master scholar. -- Australian Physics
The book of Hagen Kleinert became very popular and serves as an encyclopedia on path integral (PI) methods. --Zentralblatt MATH
From the Inside Flap
This is the fifth, expanded edition of the comprehensive textbook published in 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 the hydrogen atom. The solutions have been made possible by two major advances. The first is a new euclidean path integral formula which increases the restricted range of applicability of Feynman's time-sliced formula to include singular attractive 1/r- and 1/r2-potentials. The second is a new nonholonomic mapping principle carrying physical laws in flat spacetime to spacetimes with curvature and torsion, which leads to time-sliced path integrals that are manifestly invariant under coordinate transformations.
In addition to the time-sliced definition, the author gives a perturbative, coordinate-independent definition of path integrals, which makes them invariant under coordinate transformations. A consistent implementation of this property leads to an extension of the theory of generalized functions by defining uniquely products of distributions.
The powerful FeynmanKleinert variational approach is explained and developed systematically into a variational perturbation theory which, in contrast to ordinary perturbation theory, produces convergent results. The convergence is uniform from weak to strong couplings, opening a way to precise evaluations of analytically unsolvable path integrals in the strong-coupling regime where they describe critical phenomena.
Tunneling processes are treated in detail, with applications to the lifetimes of supercurrents, the stability of metastable thermodynamic phases, and thelarge-order behavior of perturbation expansions. A variational treatment extends the range of validity to small barriers. A corresponding extension of the large-order perturbation theory now also applies to small orders.
Special attention is devoted to path integrals with topological restrictions needed to understand the statistical properties of elementary particles and the entanglement phenomena in polymer physics and biophysics. The ChernSimons theory of particles with fractional statistics (anyons) is introduced and applied to explain the fractional quantum Hall effect.
The relevance of path integrals to financial markets is discussed, and improvements of the famous BlackScholes formula for option prices are developed which account for the fact, recently experienced in the world markets, that large fluctuations occur much more frequently than in Gaussian distributions.
Most helpful customer reviews
1 of 1 people found the following review helpful.
Remarkable... and gigantic
By Joseph
Though I haven't read very much of this enormous tome, I like the idea of reading it (some day). Kleinert has been extremely thorough here, and his expository style is excellent. One word of caution: if you're mathematically oriented and are looking for a rigorous foundation of functional integration of the kind used in Feynman's approach to quantum mechanics, you won't find it here. This book concerns itself more with explaining and displaying the many uses of path integration, than with the foundations of the subject. That said, I highly recommend this fascinating book.
2 of 2 people found the following review helpful.
Very good book.
By Francisco Ortiz Arango
This is an excellent book for people who already know a little of the issue of path integrals, but for someone Neophyte be difficult to understand.
Its applications are excellent.
0 of 1 people found the following review helpful.
A Bible for applications of path integrals
By Weidong SU
This is absolutely a great work for therectical physicists in current time who are interested in various aspects of applications of an important tool of mathematical physics.
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