Kroeger - Fractal Geometry in Quantum Mechanics, Field Th and Spin Systems.pdf

H. KroK ger / Physics Reports 323 (2000) 81}181 81
FRACTAL GEOMETRY IN
QUANTUM MECHANICS, FIELD THEORY
AND SPIN SYSTEMS
H. KROG GER
DeHpartement de Physique, UniversiteH Laval, QueHbec, QueH., Canada G1K 7P4
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
Physics Reports 323 (2000) 81}181
Fractal geometry in quantum mechanics, "eld theory
and spin systems
H. KroK ger
& & & &
Departement de Physique, Universite Laval, Quebec, Que., Canada G1K 7P4
Received May 1999; editor: J. Bagger
Contents
1. Introduction to fractal geometry 84 . Geometry of propagation of a relativistic
2. Fractal geometry in quantum mechanics 87 particle 117
. Brownian motion versus motion in . Non-local order parameters for
quantum mechanics 87 con"nement in lattice gauge theory 134
. Path integral quantization and fractal . Fractal geometry and critical behavior of
geometry of quantum paths 90 lattice "eld theories 147
. Numerical simulations for di!erent 5. Fractal geometry and quantum gravity 152
potentials 94 . Random surfaces and quantization of
. Can we measure experimentally the gravity 152
geometry of quantum mechanical . Fractal structure 154
propagation? 100 . Numerical results from lattice simulations.
3. Quantum physics on fractal space}time 108 Gravity coupled to matter: Ising model and
. SchroK dinger and Dirac equation viewed 3-state Potts model 160
from statistical mechanics 108 6. Fractal geometry and spin systems 161
. The principle of scale invariance 109 . Critical behavior of spin systems as
. Quantum physics on Cantor sets 109 a function of non-integer dimension of
4. Fractal geometry and quantum "eld space}time: Ising model 162
theory 110 . Geometry of critical clusters: Ising model 169
. Self-similarity and renormalization group . Fractal geometry of topological excitations:
equation 110 X> spin model 174
. Self-similarity and scale dependence. 7.