Stochastic Methods and Computer Techniques in Quantum Dynamics

446 Pages·1984·23.042 MB·English

by  Prof. Dr. H. Mitter, L. Pittner (auth.), Prof. Dr. H. Mitter, L. Pittner (eds.)

Acta Physica Austriaca
Supplementum XXVI
Proceedings of the
XXIII. Internationale Universitatswochen fur Kernphysik 1984
der Karl-Franzens-U niversitat Graz
at Schladming (Steiermark, Austria)
February 20th-March 1st, 1984
Sponsored by
Bundesministerium fur Wissenschafi: und Forschung
Steiermarkische Landesregierung
International Centre for Theoretical Physics, Trieste
Sektion Industrie der Kammer der
Gewerblichen Wirtschafi: fur Steiermark
1984
Springer-Verlag
Wien New York
Stochastic Methods
and Computer Techniques
in Qyantum Dynamics
Edited by
H. Mitter and L. Pittner, Graz
With 37 Figures
1984
Springer-Verlag
Wien New York
Organizing Committee
Chairman
Prof Dr. H. Mitter
Institut rur Theoretische Physik
der Universitiit Graz
Committee Members
L. Pittner
L. Mathelitsch
W. Plessas
Secretary
Mrs. E. Neuhold
Miss E. Tandl
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ISSN 0065-1559
ISBN- 13:978-3-7091-8782-1 e-ISBN- 13:978-3-7091-8780-7
DOl: 10.1007/978-3-7091-8780-7
CONTENTS
MITTER, H., Preface .............•...............••••....
STREIT, L., Stochastic Processes - Quantum Physics •..... 3
STREATER, R.F., Quantum Stochastic Integrals ........•..• 53
ZOLLER, P., Stochastic Differential Equations .....•••..• 75
DEWITT-MORETTE, C., Feynman Path Integrals. From the
Prodistribution Definition to the Calculation of
Glory Scattering ..•...............•.........•.... 101
LOW, S.G., The Knife Edge Problem ..............•........ 171
BLANCHARD, Ph., Trapping for Newtonian Diffusion
Processes ........................................ 185
ALBEVERIO, S., H0EGH-KROHN, R., HOLDEN, H., Markov
Cosurfaces and Gauge Fields ...................... 211
ALBEVERIO, S., Non-Standard Analysis; Polymer Models,
Quantum Fields ...........•....................... 233
FROHLICH, J., Statistical Mechanics of Random Surfaces .. 255
SEILER, E., Stochastic Quantization and Gauge Fixing in
Gauge Theories ................................... 259
REBBI, C., Numerical Calculations in Quantum Field
Theories ......................................... 309
AREDE, T., The Heat Kernel on Riemannian Manifolds and
Lie Groups ....................................... 349
ATKINSON, D., Confinement in Continuum QCD .............. 361
BALOG, J., HRASKO, P., Fermion Boundary Condition and
e Angle ........••..••.............•.............. 365
BOLLE, D., ROEKAERTS, D., Semiclassical and High
Temperature Expansions for Systems with Magnetic
Field •...•.•.•.•••••.........•..........•..•..... 371
CONLON, J., The Ground State Energy of an Interacting
Bose Gas .................................................................................. 381
VI
EBERT, D., Functional Approach to a Superconductivity-
Type Quark-Model with Broken SU(4)-Syrnmetry ...... 389
GROSSE, H., Solitons in Solid State Physics ............. 393
HOEK, J., Normalization of Currents in Lattice QCD ...... 401
KIRSCHNER, R., Stochastic Quantization and Supersyrnmetry 409
HAAG, R., STEIN, U., NARNHOFER, H., Quantum Field Theory
in Gravitational Background ...................... 415
RIECKERS, A., The Josephson Potential as a Statistical
Phenomenon 423
ROSZKOWSKI, L., Generating Functional and Quantum
Stability in Field Theory Models with Solitons ... 427
RUMPF, H., Stochastic Quantization of the Linearized
Gravitational Field .............................. 435
SCHLERETH, H., Rotation-Invariant Regularization of
Quantum Chromodynamics in Strong Coupling 441
WAS, Z., Monte Carlo Simulation of the Process
+ - +-
e e +T T (Y ) •.................................... 447
PREFACE
This volume contains the written versions of lectures held
at the "23. Internationale Universit~tswochen fUr Kernphysik"
in Schladming, Austria, in February 1984. Once again the
generous support of our sponsors, the Austrian Ministry
of Science and Research, the Styrian Government and others,
had made it possible to organize this school. The aim of
the topics chosen for the meeting was to present different
aspects of stochastic methods and techniques. These
methods have opened up new ways to attack problems in a
broad field ranging from quantum mechanics to quantum field
theory. Thanks to the efforts of the lecturers it was
possible to take this development into account and show
relations to areas where stochastic methods have been
used for a long time. Due to limited space only short
manuscript versions of the many seminars presented could
be included. The lecture notes were reexamined by the authors
after the school and are now published in their final form.
It is a pleasure to thank all the lecturers for their
efforts which made it possible to speed up publication.
Thanks are also due to Mrs.Neuhold for her careful typing
of the notes.
H. Mitter
L. Pittner
Acta Physica Austriaca, Suppl. XXVI, 3-52 (1984)
© by Springer-Verlag 1984
STOCHASTIC PROCESSES - QUANTUM PHYSICS+
by
L. STREIT
Universitat Bielefeld
BiBoS
D-4800 Bielefeld. FR Germany
I. SOME INTRODUCTORY REHARKS
It is now about 20 years ago that I first spoke about
Wiener Integrals at one of the earliest Schladming Winter
Schools. As I was assembling my notes for those lectures
one of my senior colleagues remarked that Functional Inte
grals provided a nice reformulation of Quantum Theory, but
really there was hardly anything you could do with them that
one had not already done otherwise.
We have come a long way in those twenty years. - I am not
only thinking of the celebrated successes of Constructive
Quantum Field Theory - it is proper to mention here the
inspiration and insistence of K. Symanzik - but also such
"practical" results as that of E. Lieb on the dependence
of electronic binding energy on atomic separation, an old
conjecture from chemistry finally proven with the means
of functional integration.
+ Lecture given at the XXIII. Internationale Universitatswochel
fur Kernphysik,Schladming,Austria,February 20-March 1,1984.
4
Here in these lectures I shall not reach out to
sophististicated applications. Instead I have been asked
to give an elementary introduction to Stochastic Processes.
Hence I shall not proceed from Definitions to Theorems and
Proofs, but rather will use some concepts with which you
are well acquainted to acquaint you with some others that
are central to those parts of the mathematical theory which
playa role in quantum physics applications.
We shall start from simple quantum mechanics which
you all know, make a short tour through probability and
stochastic processes and finally return to the start: to
quantum dynamics in a form that may be less familiar to you,
i.e. in terms of Stochastic Processes.
We shall go on this excursion without attempting to
cover the whole field but on a rather narrow track like
a cross country skier, but I hope that the trail has been
so arranged that we shall pass by a couple of interesting
vistas.
What now are Stochastic Processes?
One author writes
"A Stochast~c Process may be loosely descr~bed as a system
running along in time and controlled by probabilistic laws."
But then:
What is NOT a Stochastic Process?
Examples: 1) The Dachstein?? (see however the illustrations
in [1])
2) The shape of Austria?? (see however Fig.3)
3) God?? (no comment)
Stochastic Processes are
o a vast field of research
o of great unifying power.
Some Examples,
to which we shall return in the course of these lectures:
5
-~-~~-
Quiet periods are found to be interrupted by bursts of
noise. These noise bursts form clusters, which form larger
clusters, which form larger clusters, ...
Q: Make a mathematical model!
For T = 0 the phase boundary is a straight line. (Fig.1)
Q: Make a mathematical model for T>O!(Fig.2)
Map of scale s,
1-D
= =
L L(s) L s
o
Q: Make a mathematical model!
oS
4. ~9~~_~9~~: win/lose 1
Q1: What happens to your capital?
A: You will surely lose it!
Q2: In what time?
Q1: Make a mathematical model!
Qn: What is the probability for the particle to wander from
Xo to x in time t?
-H t
A: pt(x,xo) e 0 o(x-xo)