Carroll, Robert - Mathematical Physics
PREFACE
A great deal of mathematics is used in studying physics, as is well known,
and it is my belief that a great deal of physics is used in developing math-
ematics (more than is perhaps realized). At one time it seemed convenient
(for me at least) to think of an equation physics = geometry, but one might
also make a case for physics = probability, or physics = recursion, etc.
It also seemed attractive at one time (to me) to think of the study of phy-
sics (and perhaps also mathematics) in the context of "recognizing God's
handiwork and praising it". But one can also ask of course whether God had
any choice in creation (cf. here [Pl] which deals with complexity, entropy,
information, recursive games, self-reproducing machines, etc.). It is also
perhaps fitting to think of relations between gods and civilizations (cf.
[Tul]). Frequently one makes mathematical models of a physical situation
and if the model is any good its mathematical study will lead to informa-
tion of use in physics.
physical intuition then so much the better; one will be looking then at phy-
sically interesting features and the mathematical questions asked and inves-
tigated will be enriched by the interaction with physics. Such an input can
also arise from numerical or computer study of a mathematical model; the
computational algorithmic thinking toward solvable numerical problems can
lead to theoretical insight into the model. One is of course advised not to
ask only those questions whose answers can be computed (but there may be
several schools of thought here as well).
We try to provide in general a rich selection of material and to indicate
as well current areas of interest and different points of view.
pecially interested in the interaction of ideas from apparently different
areas and their synthesis in the discovery process. In this direction we
also feel that the use of language is enriched by knowledge of other lan-
guages.
If this study can be directed or guided also by
We are es-
We try whenever possible to exhibit patterns and structure and will
vi ROBERT CARROLL
emphasize structure as providing a cradle for the nuturing of theory.
will give totally elementary introductions to many areas with complete de-
tails and will then continue to develop the themes in various ways at vari-
ous places in the book.
result that may be needed for illumination (with references) and no apology
seems necessary for omitting the proof.
times but the necessary details are usually there in the text or in the ap-
pendices,
in a way we have found personally instructive in learning and which we have
used effectively in teaching. For example in Chapter 2, 83-5, we develop a
number of structural formulas and results, in working out the necessary tech-
nical machinery as we go along, sometimes in a heuristic manner. In fact
we do not prove the abstract spectral theorem in Hilbert space for a self
adjoint operator as such (it is stated however in §2.2),and we do not give
an "axiomatic" treatment of spectral measures, projection operators, etc.
However we give the necessary formulas, details, and background to deal with
all these ideas and use the material in a way which amounts to proving e.g.
the spectral theorem after all. In fact in this way much more is done,in
that connections between various points of view are displayed as wel1,and
one sees the role of the various ingredients in practice.
needed is proved or sketched more or less completely so that the details
can be filled in in any case. The presentation thus may appear somewhat
disjointed at times but we have found it pedagogically more satisfactory
than a theorem-proof format and it has more meaning personally to proceed
in this way. In this spirit we have organized much material throughout the
book in a remark format (instead of theorem-proof) with the proofs of state-
ments indicated or carried out in the text, along with the general discus-
sion. Exercises are then interspersed throughout the text.
We have extracted material from many sources with ample references.
various ideas of proof or presentation, which we have found particularly
illuminating or stimulating, are hopefully conveyed to the reader. In or-
der to include enough material to justify a title as pretentious as "mathe-
matical physics" we have resorted to certain space saving devices (to mini-
mize the number of pages and the price).
goes on there are progressively fewer displayed formulas and we use the fol-
lowing substitute. There are 6 dark symbols, *, *, 0, by +, ., which are
used as display "indicators" in the text in the following order: *, A, 0,
We
Occasionally (but rarely) we will simply state a
The pace may appear to be fast at
Once beyond the first chapter some of the material is presented
What is actually
Thus
Thus in particular as the book
PREFACE vi i
6, 6, ., **, *A, ..., *., A*, AA, ..., Am, ..., .*, ..., .my ***, **A,...,
**my *A*, ... This tends to make the text rather dense at times but with a
little patience and practice this notation is quite efficient and useful.
There is a great deal on functional analysis in the book, probably enough
for a semester course in functional analysis, and most details are provided.
In particular the theory of distributions or generalized functions is devel-
oped in several ways.
chaos, black holes, index theory, superstrings, etc.) we do manage to touch
upon many topics of current interest (e.g. superconductivity, gauge field
theory, geometric quantization, Feynman integrals, quantum field theory, in-
verse problems, soliton theory, etc.), some of it in considerable detail
(e.g. inverse scattering and soliton theory).
many in terms of overall perspective) sections based on the author's work
and this should not be construed entirely as vanity (in particular it allows
us to develop considerable detail in areas which we know best).
ial in e.g. 51.6, 1.11, 2.6, 2.7 provides a good model for discussing cer-
tain areas of research and we have employed it successfully in lectures; the
theory of necessary ingredients such as spectral measures etc. is developed
as one goes along and this seems to make for meaningful pedagogy. In a sense
one of the main contributions of the book may involve Chapter 2 where a
rather full discussion of inverse scattering and elementary sol iton theory
is given. There are a number of new results and a lot of recent material.
We have not spent much time on physical derivations or the philosophy of
physics. This is a serious gap but one not possible to bridge under the im-
posed space limitations. It is very productive to link mathematical devel-
opment with physical reasoning. For example a nice complex of ideas revol-
ves around causality, hyperbol ic PDE, Fourier transforms and Pal ey-Wiener
ideas, scattering, triangularity of operators, etc. Similarly one has ide-
as of cohomology, gauge theory, currents, charges, etc. in field theory. We
feel the present era to be revolutionary in science and mathematics and
have tried to develop enough machinery to help the reader storm the barri-
cades. In the area of nonlinear PDE for example the methods of functional
analysis have reached a very hybrid abstract form,and we have preferred to
give a presentation of earlier versions of the theory,where there is more
contact with the original problems,and motivation is more visible. One can
emphasize here that it is wise to stay reasonably close to the source of
mathematical problems in physics in order to retain nourishment and vitality.
Although there are many omissions (nothing about
There are some (clearly too
The mater-
viii ROBERT CARROLL
Abstraction for itself is often attractive but we pursue this only in the
interest of nutrient structure. One should be free to use intuition, pic-
tures, analogy, etc. to develop the appropriate language for whatever phy-
sics is under consideration. The religion of embalming mathematics in axi-
omatic systems does not prove too profitable in mathematical physics (al-
though the reader will detect vestiges of a former flirtation with the Muse
of N. Bourbaki). The book makes very modest claims. We hope it can be use-
ful as a text, even a more or less introductory text, while serving as a
guide to some research areas of current interest.
sophisticated material with hopefully enough rigor to be believable and
enough heuristic content to stimulate further study.
The author would like to thank L. Nachbin for adding this book to the Notas
de Matematica series.
various people who made it possible to travel to conferences and give semi-
nar talks in the past 3 years while the book was being written; we mention
in particular L. Bragg, J. Dettman, J. Donaldson, A. Favini, T. Gill, R.
Gilbert, T. Kailath, E. Magenes, P. McCoy, C. Pucci, L. Raphael, F. Santosa,
and W. Zachary. I would also like to acknowledge relevant conversation dur-
ing this period with the above people as well as with (in particular) A.
Arosio, C. Baiocchi, M. Berger, M. Bernardi, A. Bruckstein, M. Cheney, D.
Colton, J. Cooper, S. Dolzycki, C. Foias, J. Goldstein, D. Isaacson, H.
Kaper, T. Kappeler, D. Kaup, M. Kon, I. Lasiecka, P. Lax, T. Mazumdar, J.
Neuberger, P. Newton, R. Newton, A. Pazy, H. Pollak, J. Rose, T. Seidman,
G. Strang, W. Strauss, W. Symes, P. Tondeur, G. Toth, and A. Yagle (with
apologies for omissions). Finally the book is dedicated to my wife Joan.
| Released: | 1988 |
| Publisher: | North-Holland |
| Page Count: | 411 |
| Format: | |
| Language: | English |
| ISBN-10: | 0444704434 |
| ISBN-13: | 9780444704436 |
PREFACE
A great deal of mathematics is used in studying physics, as is well known,
and it is my belief that a great deal of physics is used in developing math-
ematics (more than is perhaps realized). At one time it seemed convenient
(for me at least) to think of an equation physics = geometry, but one might
also make a case for physics = probability, or physics = recursion, etc.
It also seemed attractive at one time (to me) to think of the study of phy-
sics (and perhaps also mathematics) in the context of "recognizing God's
handiwork and praising it". But one can also ask of course whether God had
any choice in creation (cf. here [Pl] which deals with complexity, entropy,
information, recursive games, self-reproducing machines, etc.). It is also
perhaps fitting to think of relations between gods and civilizations (cf.
[Tul]). Frequently one makes mathematical models of a physical situation
and if the model is any good its mathematical study will lead to informa-
tion of use in physics.
physical intuition then so much the better; one will be looking then at phy-
sically interesting features and the mathematical questions asked and inves-
tigated will be enriched by the interaction with physics. Such an input can
also arise from numerical or computer study of a mathematical model; the
computational algorithmic thinking toward solvable numerical problems can
lead to theoretical insight into the model. One is of course advised not to
ask only those questions whose answers can be computed (but there may be
several schools of thought here as well).
We try to provide in general a rich selection of material and to indicate
as well current areas of interest and different points of view.
pecially interested in the interaction of ideas from apparently different
areas and their synthesis in the discovery process. In this direction we
also feel that the use of language is enriched by knowledge of other lan-
guages.
If this study can be directed or guided also by
We are es-
We try whenever possible to exhibit patterns and structure and will
vi ROBERT CARROLL
emphasize structure as providing a cradle for the nuturing of theory.
will give totally elementary introductions to many areas with complete de-
tails and will then continue to develop the themes in various ways at vari-
ous places in the book.
result that may be needed for illumination (with references) and no apology
seems necessary for omitting the proof.
times but the necessary details are usually there in the text or in the ap-
pendices,
in a way we have found personally instructive in learning and which we have
used effectively in teaching. For example in Chapter 2, 83-5, we develop a
number of structural formulas and results, in working out the necessary tech-
nical machinery as we go along, sometimes in a heuristic manner. In fact
we do not prove the abstract spectral theorem in Hilbert space for a self
adjoint operator as such (it is stated however in §2.2),and we do not give
an "axiomatic" treatment of spectral measures, projection operators, etc.
However we give the necessary formulas, details, and background to deal with
all these ideas and use the material in a way which amounts to proving e.g.
the spectral theorem after all. In fact in this way much more is done,in
that connections between various points of view are displayed as wel1,and
one sees the role of the various ingredients in practice.
needed is proved or sketched more or less completely so that the details
can be filled in in any case. The presentation thus may appear somewhat
disjointed at times but we have found it pedagogically more satisfactory
than a theorem-proof format and it has more meaning personally to proceed
in this way. In this spirit we have organized much material throughout the
book in a remark format (instead of theorem-proof) with the proofs of state-
ments indicated or carried out in the text, along with the general discus-
sion. Exercises are then interspersed throughout the text.
We have extracted material from many sources with ample references.
various ideas of proof or presentation, which we have found particularly
illuminating or stimulating, are hopefully conveyed to the reader. In or-
der to include enough material to justify a title as pretentious as "mathe-
matical physics" we have resorted to certain space saving devices (to mini-
mize the number of pages and the price).
goes on there are progressively fewer displayed formulas and we use the fol-
lowing substitute. There are 6 dark symbols, *, *, 0, by +, ., which are
used as display "indicators" in the text in the following order: *, A, 0,
We
Occasionally (but rarely) we will simply state a
The pace may appear to be fast at
Once beyond the first chapter some of the material is presented
What is actually
Thus
Thus in particular as the book
PREFACE vi i
6, 6, ., **, *A, ..., *., A*, AA, ..., Am, ..., .*, ..., .my ***, **A,...,
**my *A*, ... This tends to make the text rather dense at times but with a
little patience and practice this notation is quite efficient and useful.
There is a great deal on functional analysis in the book, probably enough
for a semester course in functional analysis, and most details are provided.
In particular the theory of distributions or generalized functions is devel-
oped in several ways.
chaos, black holes, index theory, superstrings, etc.) we do manage to touch
upon many topics of current interest (e.g. superconductivity, gauge field
theory, geometric quantization, Feynman integrals, quantum field theory, in-
verse problems, soliton theory, etc.), some of it in considerable detail
(e.g. inverse scattering and soliton theory).
many in terms of overall perspective) sections based on the author's work
and this should not be construed entirely as vanity (in particular it allows
us to develop considerable detail in areas which we know best).
ial in e.g. 51.6, 1.11, 2.6, 2.7 provides a good model for discussing cer-
tain areas of research and we have employed it successfully in lectures; the
theory of necessary ingredients such as spectral measures etc. is developed
as one goes along and this seems to make for meaningful pedagogy. In a sense
one of the main contributions of the book may involve Chapter 2 where a
rather full discussion of inverse scattering and elementary sol iton theory
is given. There are a number of new results and a lot of recent material.
We have not spent much time on physical derivations or the philosophy of
physics. This is a serious gap but one not possible to bridge under the im-
posed space limitations. It is very productive to link mathematical devel-
opment with physical reasoning. For example a nice complex of ideas revol-
ves around causality, hyperbol ic PDE, Fourier transforms and Pal ey-Wiener
ideas, scattering, triangularity of operators, etc. Similarly one has ide-
as of cohomology, gauge theory, currents, charges, etc. in field theory. We
feel the present era to be revolutionary in science and mathematics and
have tried to develop enough machinery to help the reader storm the barri-
cades. In the area of nonlinear PDE for example the methods of functional
analysis have reached a very hybrid abstract form,and we have preferred to
give a presentation of earlier versions of the theory,where there is more
contact with the original problems,and motivation is more visible. One can
emphasize here that it is wise to stay reasonably close to the source of
mathematical problems in physics in order to retain nourishment and vitality.
Although there are many omissions (nothing about
There are some (clearly too
The mater-
viii ROBERT CARROLL
Abstraction for itself is often attractive but we pursue this only in the
interest of nutrient structure. One should be free to use intuition, pic-
tures, analogy, etc. to develop the appropriate language for whatever phy-
sics is under consideration. The religion of embalming mathematics in axi-
omatic systems does not prove too profitable in mathematical physics (al-
though the reader will detect vestiges of a former flirtation with the Muse
of N. Bourbaki). The book makes very modest claims. We hope it can be use-
ful as a text, even a more or less introductory text, while serving as a
guide to some research areas of current interest.
sophisticated material with hopefully enough rigor to be believable and
enough heuristic content to stimulate further study.
The author would like to thank L. Nachbin for adding this book to the Notas
de Matematica series.
various people who made it possible to travel to conferences and give semi-
nar talks in the past 3 years while the book was being written; we mention
in particular L. Bragg, J. Dettman, J. Donaldson, A. Favini, T. Gill, R.
Gilbert, T. Kailath, E. Magenes, P. McCoy, C. Pucci, L. Raphael, F. Santosa,
and W. Zachary. I would also like to acknowledge relevant conversation dur-
ing this period with the above people as well as with (in particular) A.
Arosio, C. Baiocchi, M. Berger, M. Bernardi, A. Bruckstein, M. Cheney, D.
Colton, J. Cooper, S. Dolzycki, C. Foias, J. Goldstein, D. Isaacson, H.
Kaper, T. Kappeler, D. Kaup, M. Kon, I. Lasiecka, P. Lax, T. Mazumdar, J.
Neuberger, P. Newton, R. Newton, A. Pazy, H. Pollak, J. Rose, T. Seidman,
G. Strang, W. Strauss, W. Symes, P. Tondeur, G. Toth, and A. Yagle (with
apologies for omissions). Finally the book is dedicated to my wife Joan.
