Mathematical Physics - A Modern Introduction To It's Foundations.pdf - Programming - darekisap

Sadri Hassani

Mathematical Physics

A Modem Introduction to

Its Foundations

With 152 Figures

Springer

ODTlJ KU1"UPHANESt

M. E. T. U. liBRARY

Sadri Hassani

Department ofPhysics

Illinois State University

Normal, IL 61790

USA

hassani@entropy.phy.ilstu.edu

QC20

14394

c,2.

QC20 H394

METULIBRARY

.. 2

~themltlcoJ physicS: I modem

mllllllllllllllllllllllllllllll~11111111111111111111111111IIII

002m7S69

To my wife Sarah

and to my children

Dane Arash and Daisy Rita

336417

Libraryof Congress Cataloging-in-Publication Data

Hassani, Sadri.

Mathematical physics: a modem introductionits foundations /

Sadri Hassani.

p. em.

Includesbibliographical referencesand index.

ISBN 0-387-98579-4 (alk. paper)

1. Mathematical physics.

I. Title.

QC20.H394 1998

530.15--<1c21

98-24738

Printed on acid-freepaper.

permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY

10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use-in

connection with any form of informationstorage and retrieval, electronic adaptation, computer sctt-

ware, or by similar or dissimilarmethodology now known or hereafterdevelopedis forbidden.

The use of general descriptivenames, trade names, trademarks, etc., in this publication, even if the

formerare not especiallyidentified, is not to be taken as a sign that such names, as understoodby the

Trade Marks and Merchandise Marks Act, may accordinglybe used freely by anyone.

Productionmanagedby Karina Mikhli;manufacturing supervisedby ThomasKing.

Photocomposed copy preparedfrom the author's TeX files.

Printed and bound by HamiltonPrinting Co., Rensselaer, NY.

Printed in the United Statesof America.

9 8 7 6 5 4 3 (Correctedthird printing, 2002)

ISBN 0-387-98579-4

SPIN 10854281

Springer-Verlag New York Berlin: Heidelberg

A member. ofBertelsmannSpringer Science+Business Media GmbH

Preface

"Ich kann es nun einmal nicht lassen, in diesem Drama von

Mathematik und Physik---<lie sich im Dunkeln befrnchten,

aber von Angesicht zu Angesicht so geme einander verkennen

und verleugnen-die Rolle des (wie ich gentigsam erfuhr, oft

unerwiinschten) Boten zu spielen."

Hermann Weyl

It is said that mathematics is the language of Nature. If so, then physics is its

poetry. Nature started to whisper into our ears when Egyptians and Babylonians

were compelled to invent and use mathematics in their day-to-day activities. The

faint geomettic and arithmetical pidgin of over four thousand years ago, snitable

for rudimentary conversations with nature as applied to simple landscaping, has

turned into a sophisticated language in which the heart of matter is articulated.

The interplay between mathematics and physics needs no emphasis. What

may need to be emphasized is that mathematics is not merely a tool with which the

presentation of physics is facilitated, but the only medium in which physics can

survive. Just as language is the means by which humans can express their thoughts

and without which they lose their unique identity, mathematics is the only language

through which physics can express itself and without which it loses its identity.

And just as language is perfected due to its constant usage, mathematics develops

in the most dramatic way because of its usage in physics. The quotation by Weyl

above, an approximation to whose translation is "In this drama of mathematics

and physics-which fertilize each other in the dark, but which prefer to deny and

misconstrue each other face to face-I cannot, however, resist playing the role

ofa messenger, albeit, as I have abundantly learned, often an unwelcome one:'

PREFACE

is a perfect description of the natnral intimacy between what mathematicians and

physicists do, and the nnnatnral estrangement between the two camps. Some ofthe

most beantifnl mathematics has been motivated by physics (differential eqnations

by Newtonian mechanics, differential geometry by general relativity, and operator

theory by qnantnmmechanics), and some of the most fundamental physics has been

expressed in the most beantiful poetry of mathematics (mechanics in symplectic

geometry, and fundamental forces in Lie group theory).

I do uot want to give the impression that mathematics and physics cannot

develop independently. On the contrary, it is precisely the independence of each

discipline that reinforces not only itself, but the other discipline as well-just as the

stndy of the grammar of a language improves its usage and vice versa. However,

the most effective means by which the two camps can accomplish great success

is throngh an inteuse dialogue. Fortnnately, with the advent of gauge and string

theories ofparticle physics, such a dialogue has beenreestablished between physics

and mathematics after a relatively long lull.

Level and Philosophy of Presentation

This is a book for physics stndeuts interested in the mathematics they use. It

is also a book fur mathematics stndeuts who wish to see some of the abstract

ideas with which they are fantiliar come alive in an applied setting. The level of

preseutation is that of an advanced undergraduate or beginning graduate course (or

sequence of courses) traditionally called "Mathematical Methods of Physics" or

some variation thereof. Unlike most existing mathematical physics books intended

for the same audience, which are usually lexicographic collections of facts about

the diagonalization of matrices, tensor analysis, Legendre polynomials, contour

integration, etc., with little emphasis on formal and systematic development of

topics, this book attempts to strike a balance between formalism and application,

between the abstract and the concrete.

I have tried to include as mnch of the essential formalism as is necessary to

render the book optimally coherent and self-contained. This entails stating and

proving a large nnmber of theorems, propositions, lemmas, and corollaries. The

benefit of such an approach is that the stndent will recognize clearly both the power

and the limitation of a mathematical idea used in physics. There is a tendency on the

part ofthe uovice to universalize the mathematical methods and ideas eucountered

in physics courses because the limitations of these methods and ideas are not

clearly pointed out.

There is a great deal offreedom in the topics and the level of presentation that

instructors can choose from this book. My experience has showu that Parts I, TI,

Ill, Chapter 12, selected sections ofChapter 13, and selected sections or examples

of Chapter 19 (or a large snbset of all this) will be a reasonable course content for

advanced undergraduates. If one adds Chapters 14 and 20, as well as selected topics

from Chapters 21 and 22, one can design a course snitable for first-year graduate

PREFACE

vii

students. By judicious choice of topics from Parts VII and VIII, the instructor

can bring the content of the course to a more modern setting. Depending on the

sophistication ofthe students, this can be done either in the first year or the second

year of graduate school.

Features

To betler understand theorems, propositions, and so forth, students need to see

them in action. There are over 350 worked-out examples and over 850 problems

(many with detailed hints) in this book, providing a vast arena in which students

can watch the formalism unfold. The philosophy underlying this abundance can

be summarized as ''An example is worth a thousand words of explanation." Thus,

whenever a statement is intrinsically vague or hard to grasp, worked-out examples

and/or problems with hints are provided to clarify it. The inclusion of such a

large number of examples is the means by which the balance between formalism

and application has been achieved. However, although applications are essential

in understanding mathematical physics, they are only one side of the coin. The

theorems, propositions, lemmas, and corollaries, being highly condensed versions

of knowledge, are equally important.

A conspicuous feature of the book, which is not emphasized in other compa-

rable books, is the attempt to exhibit-as much as.it is useful and applicable-«

interrelationships among various topics covered. Thus, the underlying theme of a

vector space (which, in my opinion, is the most primitive concept at this level of

presentation) recurs throughout the book and alerts the reader to the connection

between various seemingly unrelated topics.

Another useful feature is the presentation of the historical setting in which

men and women of mathematics and physics worked. I have gone against the

trend of the "ahistoricism" of mathematicians and physicists by summarizing the

life stories of the people behind the ideas. Many a time, the anecdotes and the

historical circumstances in which a mathematical or physical idea takes form can

go a long way toward helping us understand and appreciate the idea, especially if

the interaction among-and the contributions of-all those having a share in the

creation of the idea is pointed out, and the historical continuity of the development

of the idea is emphasized.

To facilitate reference to them, all mathematical statements (definitions, theo-

rems, propositions, lemmas, corollaries, and examples) have been nnmbered con-

secutively within each section and are preceded by the section number. For exam-

ple, 4.2.9 Definition indicates the ninth mathematical statement (which happens

to be a definition) in Section 4.2. The end of a proof is marked by an empty square

D, and that of an example by a filled square III, placed at the right margin of each.

Finally, a comprehensive index, a large number of marginal notes, and many

explanatory underbraced and overbraced comments in equations facilitate the use

Mathematical Physics - A Modern Introduction To It's Foundations.pdf

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika: