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### H. Kleinert

## PATH INTEGRALS

###
in Quantum Mechanics, Statistics,

and Polymer Physics

World Scientific Publishing Co., Singapore 1995.

This is the second, 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.

### 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.
The present edition appeared first in German
under the title
**
*** PFADINTEGRALE*
* in Quantenmechanik, Statistik und Polymerphysik*,
**B.I.-Wissenschaftsverlag, Mannheim, 1993.
**

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