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Book 8

Hagen Kleinert
Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets
4th edition

pp. 1-1547, World Scientific, Singapore 2006
Hardcover: ISBN 981-270-008-0, 138 USD
Paperback: ISBN 981-270-009-9, 38 USD

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This is the fourth, 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 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 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.

The powerful Feynman-Kleinert variational approach is explained and developed systematically into a variational perturbation expansion.
In contrast to ordinary perturbation expansions, divergencies are absent.
Instead, there is a uniform convergence 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.