Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets
World Scientific, 2004 - 1468 pages
This is the third, significantly 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 perturbationexpansions. 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 Chem-Simons theory of particles with fractional statistics (anyohs) 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.
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Path Integrals Elementary Properties and Simple Solutions
Appendix 2A Derivation of BakerCampbellHausdorff and Magnus For
External Sources Correlations and Perturbation Theory
Semiclassical Time Evolution Amplitude
Variational Perturbation Theory
Path Integrals with Topological Constraints
Many Particle Orbits Statistics and Second Quantization
Spaces with Curvature and Torsion
Schrödinger Equation in General MetricAffine Spaces
New Path Integral Formula for Singular Potentials
Path Integral of Coulomb System
Solution of Further Path Integrals by the DuruKleinert Method
Path Integrals in Polymer Physics
Polymers and Particle Orbits in Multiply Connected Spaces
Path Integrals in Spherical Coordinates
FixedEnergy Amplitude and Wave Functions