Cornwell J.F. - Group theory in physics vol 1
Twenty years or so ago group theory could have been regarded mereLy as
providing 8 very valuable tool for the elucidation of the symmetry aspects
of physics problems, but recent developments l particu1arly in theoretical
high-energy physics have transfonned its roe, so that it now occupies a
crucial and indispensable poSition at the centre of the stage.. These develop-.
rnents have taken physicists increasingly deeper into the fascinati ng world
of the pure mathematicians. and have led to an ever-growing appreciation
of their achievements. That this recognition is in some respects rather
belated is to a large extent due to the unfortunate fact that much of modern
pure mathematics is written in a style that outsiders find difficult to
penetrate ConsequenUy one of the main objectives of these two volumes
(and particularly of the second) has been to help overcome this unnatural
barrier, and to present to theoretical physicists and others the relevant
mathematical developments in a form that should be easier to comprehend
and appreciate.
The main aim of these two volumes has been to provide a thorough and
self-contained account both of those parts of group theory that have been
found to be most usefu' and of their major applications to physical
problems. The treatment starts with the basic concepts and is carried right
through to some of the most significant and recent developments. The
areas of physics that appear include atomic physics, electronic energy
bands in solids, vibrations in molecules and solids, and the theory of
elementary particles. No prior knO'Medge of group theory is assumed and
fOr convenience various relevant a'gebraic concepts are summariLed in
Appendices A and B. It need hardly be said that the tit'e that has been chosen for these
volumes;l "Group Theory in Physics" ff does not imply that they contain
every application of group theoretical ideas to physics, nor that the
mathematical concepts contained within them are strictly restricted to those
of group theory. Some parts of physics" such as nuclear structure theoryc\'"
have had to be omitted completely. Moreover, even in those areas that have
been discussedI' a rigorous selection of topics has had to be made. This is
particularly so in applications to elementary-particle theory, where Iitera1ly
thousands of papers involving group theoretical techniques have been
written. On the other hand, the mathematical coverage goes outside the
strict confines of group theory itseif
for one is soon led to the study of lie
algebras, which l although related to Lie groups" are often developed by
mathematicians as a separate subject.
No apology should be needed for combining such diverse physical
applications in the same work", for it is a manifestation of the power of the
theory that it has such wide applicability. However" for the benefit of those
readers who may wish to concentrate on specific applications, the foUowing
list gives the relevant chapters:
(0 molecular vibrations: Chapters 1, 2J' and 4 to 7"
(ii) electronic energy bands in solids: Chapters 1
2j' 3 (Section 5 on'v), 4, 5J'
6J' 8 and 9f'
(iii) lattice vibrations in solids: Chapters 1, 2" and 4 to 9 1
(iv) atomic physics: Chapters 1 to 6 and 10 to 12 1
(v) elementary particles: Chapters 1 to 6. 10, 11, 12 (except Sections 6 to 8}1
and 13 to 19.
In the text the treatments of specific cases are frequently given under the
heading of uExamp1estJ'. The format is such that these are clearly disting-
uished from the main part of the text, the intention being to indicate that the
detailed analysis in the Example is not essential for the general understand-
ing of the rest of that section Or succeeding sections. Nevertheless" the
examples are important for two reasons.. Firstly, they give concrete realiza-
tions of the concepts that have just been introduced. Secondly. they
indicate how the concepts apply to certain physicaUy important groups or
algebras, thet"ebv allowing a :I'para11el
t treatment of a number of specific
cases. For instance, many of the properties of the groups SU(2) and SU(3)
are developed in a series of such Examples.
The proofs of theorems have been divided into three categories. First
there are t'lose that by virtue of the direct nature of their arguments assist
in the appreciation of the theorem. These are included in the main text.
Then there are proofs which are worth recording,. if onlv because it is
interesting to see to what extent they retain their validity when the
conditions of the theorem are changed slightly" as for example when a Lie
algebra is generalized to a Lie superalgebra.. The arguments involved in
these are usually less djrect
and so they have been relegated to appen-
dices. Finally there are proofs that are iust too long, or involve ideas that
have not been developed in these volumes.. For these aU that is given is a
reference Or references to works where they may be found..
In the second volume I have chosen to devote much more space to the
development of mathernatical techniques than to the treatment of specific
physical models, largely because the mathematics is likely to be more
durable. Consequently not all of the mathematical results that have been
derived are actually used explicitly in the models which are discussed in
detail. Indeed these models could be regarded as prototypes indicating
what can be achieved by this type of argument" rather than as definitive and
conclusive statements about the physical world. although the progress with
them and the agreement with experimental observations are extremefy
encouraging. The development of semi-simple lie algebras fol1ows the
classic approach of Cartan, which has the great advantage of being equally
applicable to a'i cases" and treatments that are valid only for restricted types
have been largely neglected. A considerable amount of useful data on
semi-simple lie algebras and groups has been presented in the Appen
dices... some of this having been specia1ly obtained by computer calculation.
I would like to thank Dr A.. Cant for his valuable comments on the first
drafts of certain chapters, and Miss L. M. McLean l Mrs J. Kubrycht and Mrs
N. Pacholek for the exceUence of their typing
Twenty years or so ago group theory could have been regarded mereLy as
providing 8 very valuable tool for the elucidation of the symmetry aspects
of physics problems, but recent developments l particu1arly in theoretical
high-energy physics have transfonned its roe, so that it now occupies a
crucial and indispensable poSition at the centre of the stage.. These develop-.
rnents have taken physicists increasingly deeper into the fascinati ng world
of the pure mathematicians. and have led to an ever-growing appreciation
of their achievements. That this recognition is in some respects rather
belated is to a large extent due to the unfortunate fact that much of modern
pure mathematics is written in a style that outsiders find difficult to
penetrate ConsequenUy one of the main objectives of these two volumes
(and particularly of the second) has been to help overcome this unnatural
barrier, and to present to theoretical physicists and others the relevant
mathematical developments in a form that should be easier to comprehend
and appreciate.
The main aim of these two volumes has been to provide a thorough and
self-contained account both of those parts of group theory that have been
found to be most usefu' and of their major applications to physical
problems. The treatment starts with the basic concepts and is carried right
through to some of the most significant and recent developments. The
areas of physics that appear include atomic physics, electronic energy
bands in solids, vibrations in molecules and solids, and the theory of
elementary particles. No prior knO'Medge of group theory is assumed and
fOr convenience various relevant a'gebraic concepts are summariLed in
Appendices A and B. It need hardly be said that the tit'e that has been chosen for these
volumes;l "Group Theory in Physics" ff does not imply that they contain
every application of group theoretical ideas to physics, nor that the
mathematical concepts contained within them are strictly restricted to those
of group theory. Some parts of physics" such as nuclear structure theoryc\'"
have had to be omitted completely. Moreover, even in those areas that have
been discussedI' a rigorous selection of topics has had to be made. This is
particularly so in applications to elementary-particle theory, where Iitera1ly
thousands of papers involving group theoretical techniques have been
written. On the other hand, the mathematical coverage goes outside the
strict confines of group theory itseif
for one is soon led to the study of lie
algebras, which l although related to Lie groups" are often developed by
mathematicians as a separate subject.
No apology should be needed for combining such diverse physical
applications in the same work", for it is a manifestation of the power of the
theory that it has such wide applicability. However" for the benefit of those
readers who may wish to concentrate on specific applications, the foUowing
list gives the relevant chapters:
(0 molecular vibrations: Chapters 1, 2J' and 4 to 7"
(ii) electronic energy bands in solids: Chapters 1
2j' 3 (Section 5 on'v), 4, 5J'
6J' 8 and 9f'
(iii) lattice vibrations in solids: Chapters 1, 2" and 4 to 9 1
(iv) atomic physics: Chapters 1 to 6 and 10 to 12 1
(v) elementary particles: Chapters 1 to 6. 10, 11, 12 (except Sections 6 to 8}1
and 13 to 19.
In the text the treatments of specific cases are frequently given under the
heading of uExamp1estJ'. The format is such that these are clearly disting-
uished from the main part of the text, the intention being to indicate that the
detailed analysis in the Example is not essential for the general understand-
ing of the rest of that section Or succeeding sections. Nevertheless" the
examples are important for two reasons.. Firstly, they give concrete realiza-
tions of the concepts that have just been introduced. Secondly. they
indicate how the concepts apply to certain physicaUy important groups or
algebras, thet"ebv allowing a :I'para11el
t treatment of a number of specific
cases. For instance, many of the properties of the groups SU(2) and SU(3)
are developed in a series of such Examples.
The proofs of theorems have been divided into three categories. First
there are t'lose that by virtue of the direct nature of their arguments assist
in the appreciation of the theorem. These are included in the main text.
Then there are proofs which are worth recording,. if onlv because it is
interesting to see to what extent they retain their validity when the
conditions of the theorem are changed slightly" as for example when a Lie
algebra is generalized to a Lie superalgebra.. The arguments involved in
these are usually less djrect
and so they have been relegated to appen-
dices. Finally there are proofs that are iust too long, or involve ideas that
have not been developed in these volumes.. For these aU that is given is a
reference Or references to works where they may be found..
In the second volume I have chosen to devote much more space to the
development of mathernatical techniques than to the treatment of specific
physical models, largely because the mathematics is likely to be more
durable. Consequently not all of the mathematical results that have been
derived are actually used explicitly in the models which are discussed in
detail. Indeed these models could be regarded as prototypes indicating
what can be achieved by this type of argument" rather than as definitive and
conclusive statements about the physical world. although the progress with
them and the agreement with experimental observations are extremefy
encouraging. The development of semi-simple lie algebras fol1ows the
classic approach of Cartan, which has the great advantage of being equally
applicable to a'i cases" and treatments that are valid only for restricted types
have been largely neglected. A considerable amount of useful data on
semi-simple lie algebras and groups has been presented in the Appen
dices... some of this having been specia1ly obtained by computer calculation.
I would like to thank Dr A.. Cant for his valuable comments on the first
drafts of certain chapters, and Miss L. M. McLean l Mrs J. Kubrycht and Mrs
N. Pacholek for the exceUence of their typing
