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This book is a new edition of Volumes 3 and 4 of Walter Thirring's famous textbook on mathematical physics. The first part is devoted to quantum mechanics and especially to its applications to scattering theory, atoms and molecules. The second part deals with quantum statistical mechanics examining fundamental concepts like entropy, ergodicity and thermodynamic functions. The author builds on an axiomatic basis and uses tools from functional analysis: bounded and unbounded operators on Hilbert space, operator algebras etc. Mathematics is shown to explain the axioms in depth and to provide the right tool for testing numerical data in experiments.
This fourth edition of selecta of my work on the stability of
matter contains recent work on two topics that continue to
fascinate me: Quantum electrodynamics (QED) and the Bose gas. Three
papers have been added to Part VII on QED. As I mentioned in the
preface to the third edition, there must be a way to formulate a
non-perturbative QED, presumably with an ultraviolet cutoff, that
correctly describes low energy physics, i.e., ordinary matter and
its interaction with the electromagnetic field. The new paper
VII.5, which quantizes the results in V.9, shows that the
elementary no-pair version of relativistic QED (using the Dirac
operator) is unstable when many-body effects are taken into
account. Stability can be restored, however, if the Dirac operator
with the field, instead of the bare Dirac operator, is used to
define an electron. Thus, the notion of a bare electron without its
self-field is physically questionable."
Walter Thirring is the last offspring of an Austrian family of
scientists. In this moving narrative, he describes how he survived
the Nazi occupation and became instrumental in reconstructing
European science.Thirring is one of the last living physicists who
worked on the greatest discoveries and with the greatest scientists
of the 20th century. He recollects encounters with the old masters
like Einstein, Schroedinger, Heisenberg, Pauli and others as well
as his collaborations with the present stars like Murray Gell-Mann
and Elliott Lieb. The book presents the challenges faced when one
of the major paradigm shifts took place, namely, the shift away
from atomistic theory and Newtonian physics towards field theory
and quantum mechanics. Every step is presented in clear,
understandable language which reflects Thirring's extensive
experience in training the next generation. Additionally, Thirring
describes his fascinating and profound life experiences, growing up
under Nazi occupation, serving in the war, striving to establish
scientific excellence and in reaching out across the Iron Curtain.
A true Renaissance man, he concludes by discussing his love of
music, and it is clear that his passion for learning is only
matched by his passion for music, a sampling of which can be found
at .A work that inspires at every junction and is decisively
re-readable, Thirring's autobiography is assuredly a must-have for
anyone interested in science, physics and history.
The book addresses three major topics in mathematical physics: 1.
recent rigorous results in potential theory with appli- cations in
particle physics, 2. analyticity in quantum field theory and its
applica- tions, and 3. fundamentals and applications of the inverse
problem. In addition, the book contains some contributions on
questions of general interest in quantum field theory such as
nonperturbative solutions of quantum chromodynamics, bifurcation
theory applied to chiral symmetry, as well as exactly soluable
models. The volume closes with a brief review of geometric
approaches to particle physics and a phenomenological discussion of
Higgs interactions.
In the past decade the language and methods ofmodern differential
geometry have been increasingly used in theoretical physics. What
seemed extravagant when this book first appeared 12 years ago, as
lecture notes, is now a commonplace. This fact has strengthened my
belief that today students of theoretical physics have to learn
that language-and the sooner the better. Afterall, they willbe the
professors ofthe twenty-first century and it would be absurd if
they were to teach then the mathematics of the nineteenth century.
Thus for this new edition I did not change the mathematical
language. Apart from correcting some mistakes I have only added a
section on gauge theories. In the last decade it has become evident
that these theories describe fundamental interactions, and on the
classical level their structure is suffi cientlyclear to qualify
them for the minimum amount ofknowledge required by a theoretician.
It is with much regret that I had to refrain from in corporating
the interesting developments in Kaluza-Klein theories and in
cosmology, but I felt bound to my promise not to burden the
students with theoretical speculations for which there is no
experimental evidence. I am indebted to many people for suggestions
concerning this volume. In particular, P. Aichelburg, H. Rumpf and
H. Urbantke have contributed generously to corrections and
improvements. Finally, I would like to thank Dr. 1. Dahl-Jensen for
redoing some of the figures on the computer."
On the occasion of the 50th anniversary of the discovery of the
Schrodinger equation a small symposium was organized in Vienna. It
had mainly retrospective character, where after an appreciation of
Schrodinger's scientific achievements the results were collected
which one could extract from his equation. Of course not all the
developments which originated in Schrodingers dis coveries could be
included. Instead, it was attempted to present a review of the
established predictions which follow directly from his equation.
Despite the 50 years of its existence there are always new results
of this sort being found, especially because the necessary mathe
matical methods are being developed and become known to the
physicists slowly only now .. I want to take the opportunity here
to thank the lecturers for their efforts which they put into their
excellent talks and their written versions. With their help this
volume should become a useful document on the current mathematical
art in the treatment of the Schrodinger equation. Finally it is my
pleasant obligation to thank the Bundesministerium fUr Wissenschaft
und Forschung and the Kulturamt der Gemeinde Wien for their
financial support which made it possible to honor one of the great
Austrian scientists."
This volume combines the enlarged and corrected editions of both
volumes on classical physics of Thirring's famous course in
mathematical physics. With numerous examples and remarks
accompanying the text, it is suitable as a textbook for students in
physics, mathematics, and applied mathematics. The treatment of
classical dynamical systems uses analysis on manifolds to provide
the mathematical setting for discussions of Hamiltonian systems,
canonical transformations, constants of motion, and pertubation
theory. Problems discussed in considerable detail include:
nonrelativistic motion of particles and systems, relativistic
motion in electromagnetic and gravitational fields, and the
structure of black holes. The treatment of classical fields uses
the language of differenial geometry throughout, treating both
Maxwell's and Einstein's equations in a compact and clear fashion.
The book includes discussions of the electromagnetic field due to
known charge distributions and in the presence of conductors as
well as a new section on gauge theories. It discusses the solutions
of the Einstein equations for maximally symmetric spaces and spaces
with maximally symmetric submanifolds; it concludes by applying
these results to the life and death of stars.
The last decade has seen a considerable renaissance in the realm of
classical dynamical systems, and many things that may have appeared
mathematically overly sophisticated at the time of the first
appearance of this textbook have since become the everyday tools of
working physicists. This new edition is intended to take this
development into account. I have also tried to make the book more
readable and to eradicate errors. Since the first edition already
contained plenty of material for a one semester course, new
material was added only when some of the original could be dropped
or simplified. Even so, it was necessary to expand the chap ter
with the proof of the K-A-M Theorem to make allowances for the cur
rent trend in physics. This involved not only the use of more
refined mathe matical tools, but also a reevaluation of the word
"fundamental. " What was earlier dismissed as a grubby calculation
is now seen as the consequence of a deep principle. Even Kepler's
laws, which determine the radii of the planetary orbits, and which
used to be passed over in silence as mystical nonsense, seem to
point the way to a truth unattainable by superficial observation:
The ratios of the radii of Platonic solids to the radii of
inscribed Platonic solids are irrational, but satisfy algebraic
equations of lower order."
In this final volume I have tried to present the subject of
statistical mechanics in accordance with the basic principles of
the series. The effort again entailed following Gustav Mahler's
maxim, "Tradition = Schlamperei" (i.e., filth) and clearing away a
large portion of this tradition-laden area. The result is a book
with little in common with most other books on the subject. The
ordinary perturbation-theoretic calculations are not very useful in
this field. Those methods have never led to propositions of much
substance. Even when perturbation series, which for the most part
never converge, can be given some asymptotic meaning, it cannot be
determined how close the nth order approximation comes to the exact
result. Since analytic solutions of nontrivial problems are beyond
human capabilities, for better or worse we must settle for sharp
bounds on the quantities of interest, and can at most strive to
make the degree of accuracy satisfactory.
This book is a new edition of Volumes 3 and 4 of Walter Thirring
's famous textbook on mathematical physics. The first part is
devoted to quantum mechanics and especially to its applications to
scattering theory, atoms and molecules. The second part deals with
quantum statistical mechanics examining fundamental concepts like
entropy, ergodicity and thermodynamic functions.
Die Sprache und die Methoden der modernen Differentialgeometrie
sind in der vergangenen Dekade immer mehr in die theoretische
Physik eingedrungen. Was vor 15 Jahren, als das Buch zuerst als
Vorlesungsskriptum herauskam, noch extravagant erschien, ist heute
ein Gemeinplatz. Dies hat mich in der Ansicht gestarkt, dass die
Studenten der theoretischen Physik diese Sprache lernen mussen, je
eher desto besser. Schliesslich werden sie die Professoren des 21.
Jahrhunderts sein und es ware absurd, wurden sie dann die
Mathematik des 19. Jahrhunderts lehren. Daher habe ich in der neuen
Auflage auf dieser Symbolik beharrt, einige Fehler korrigiert und
ein Kapi- tel uber Eichtheorien hinzugefugt. Da es sich gezeigt
hat, dass sie die fundamentalen Wechselwirkungen beschreiben und
ihre Struktur zumindest auf dem klassischen Ni- veau hinreichend
klar ist, scheinen sie mir zur Minimalausrustung zu gehoeren, uber
die jeder Theoretiker verfugen muss. Mit Bedauern musste ich davon
Abstand nehmen, die neueren Entwicklungen der Kosmologie und
Kaluza-Klein-artige Theorien aufzu- nehmen, aber ich fuhlte mich an
mein ursprungliches Versprechen gebunden, den Studenten keine
theoretischen Spekulationen aufzuburden, fur die es keine sichere
experimentelle Evidenz gibt. Vielen Physikern bin ich fur Hinweise
bezuglich dieses Bandes sehr verpflichtet. Insbesondere P.
Aichelburg, H. Rumpf und vor allem H. Urbantke haben zahlreiche
Korrekturen und Verbesserungen angebracht. I. Dahl-Jensen sei dafur
gedankt, dass sie manche nach Gefuhl angefertigte Zeichnungen mit
dem Computer ins richtige Lot gebracht hat.
In den letzten Dekaden hat das Gebiet der klassischen dynamischen
Systeme eine beachtliche Renaissance erlebt, und manches, was beim
erst en Erscheinen dieses Kur- ses als mathematisch zu
hochgestochen erschien, ist heute Gemeingut der aktiven Physiker
geworden. Das Ziel der Neuauflage ist es, . dieser Entwicklung zu
dienen, indem ich versucht habe, das Buch leserfreundlicher zu
gestalten und Fehler auszu- merzen. Da schon die erste Auflage ffir
eine einsemestrige Vorlesung reichlich beladen war, wurde neues
Material nur in dem Mafie aufgenommen, als anderes weggelassen oder
vereinfacht werden konnte. Eine Erweiterung muf3te jedoch das
Kapitel mit dem Be- weis des KAM-Satzes erfahren, urn dem neuen
Trend in der Physik Rechnung zu tragen. Dieser besteht nicht nur in
der Verwendung feinerer mathematischer Hilfs- mittel, sondern auch
in einer Neubewertung des Wortes "fundamental". Was frfiher als
Schmutzeffekt abgetan wurde, erscheint heute als Folge eines
tieferen Prinzips. Ja so- gar diese Keplerschen Gesetze, welche die
Radien der Planetenbahnen bestimmen und die man als mystischen
Unsinn gerne verschwieg, scheinen in Richtung einer Wahrheit zu
deuten, die sich oberflachlicher Betrachtung verschlief3t:
SchachteluI). g vollkomme- ner platonischer Korper ffihrt zu
Verhaltnissen von Radien, die irrational sind, aber algebraischen
Gleichungen niederer Ordnung genfigen. Gerade solche
Irrationalzahlen lassen sich am schlechtesten durch rationale
approximieren, und Bahnen mit diesem Radiusverhaltnis sind
gegenfiber gegenseitigen Storungen am robustesten, da sie am
wenigsten unter Resonanzeffekten leiden. In letzter Zeit wurden
einige fiberraschende Resultate fiber chaotische Systeme gefunden,
doch hat ten deren Beweise leider den Rahmen dieses Buches
gesprengt und muf3ten unterbleiben.
In der Quantentheorie werden Observable durch Operatoren im
Hilbert-Raum dargestellt. Der dafA1/4r geeignete mathematische
Rahmen sind die Cx - Algebren, welche Matrizen und komplexe
Funktionen verallgemeinern. Allerdings benAtigt man in der Physik
auch unbeschrAnkte Operatoren, deren Problematik eigens untersucht
werden muA. Dementsprechend werden zunAchst mathematische Fragen
studiert und dann die Methoden auf atomare Systeme angewandt.
Obgleich man auAer dem Wasserstoffatom kaum explizit lAsbare
Probleme findet, lassen sich nicht nur allgemeine qualitative
Fragen, etwa bezA1/4glich des Energiespektrums und Streuverhaltens,
beantworten, sondern auch quantitativ kann man auch fA1/4r
kompliziertere Systeme fA1/4r meAbare GrAAen Schranken teils
befriedigender Genauigkeit finden. Inhaltsverzeichnis: Einleitung:
Die Struktur der Quantentheorie; GrAAenordnungen atomarer Systeme.-
Die mathematische Formulierung der Quantenmechanik: Lineare RAume;
Algebren; Darstellungen im Hilbertraum; Einparametrige Gruppen;
UnbeschrAnkte Operatoren und quadratische Formen.- Quantendynamik:
Das Weyl-System; Der Drehimpuls; Die Zeitentwicklung; Der Limes t;
StArungstheorie; StationAre Streutheorie.- Atomare Systeme: Das
Wasserstoffatom; Das H-Atom in AuAeren Feldern; Heliumartige Atome;
Streuung am einfachen Atom; Komplexe Atome; Kernbewegung und
einfache MolekA1/4le.
In der Quantentheorie werden Observable durch Operatoren im
Hilbert-Raum dargestellt. Der dafur geeignete mathematische Rahmen
sind die Cx - Algebren, welche Matrizen und komplexe Funktionen
verallgemeinern. Allerdings benotigt man in der Physik auch
unbeschrankte Operatoren, deren Problematik eigens untersucht
werden muss. Dementsprechend werden zunachst mathematische Fragen
studiert und dann die Methoden auf atomare Systeme angewandt.
Obgleich man ausser dem Wasserstoffatom kaum explizit losbare
Probleme findet, lassen sich nicht nur allgemeine qualitative
Fragen, etwa bezuglich des Energiespektrums und Streuverhaltens,
beantworten, sondern auch quantitativ kann man auch fur
kompliziertere Systeme fur messbare Grossen Schranken teils
befriedigender Genauigkeit finden. Inhaltsverzeichnis: Einleitung:
Die Struktur der Quantentheorie; Grossenordnungen atomarer
Systeme.- Die mathematische Formulierung der Quantenmechanik:
Lineare Raume; Algebren; Darstellungen im Hilbertraum;
Einparametrige Gruppen; Unbeschrankte Operatoren und quadratische
Formen.- Quantendynamik: Das Weyl-System; Der Drehimpuls; Die
Zeitentwicklung; Der Limes t; Storungstheorie; Stationare
Streutheorie.- Atomare Systeme: Das Wasserstoffatom; Das H-Atom in
ausseren Feldern; Heliumartige Atome; Streuung am einfachen Atom;
Komplexe Atome; Kernbewegung und einfache Molekule."
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