Table Of ContentPietro Giuseppe Fré, Alexander Fedotov
Groups and Manifolds
Also of Interest
Multivariable Calculus and Differential Geometry
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ISBN 978-3-11-036949-6, e-ISBN 978-3-11-036954-0
Tensors and Riemannian Geometry
With Applications to Differential Equations
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ISBN 978-3-11-037949-5, e-ISBN 978-3-11-037950-1
Symmetry
Through the Eyes of Old Masters
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ISBN 978-3-11-041705-0, e-ISBN: 978-3-11-041714-2
Series: De Gruyter Studies in Mathematical Physics
M. Efroimsky, L. Gamberg, D. Gitman, A. Lazarian, B. M. Smirnov
ISSN 2194-3532
Published titles in this series:
Vol. 42: Javier Roa: Regularization in Orbital Mechanics (2017)
Vol. 39: Vladimir K. Dobrev: Invariant Differential Operators 2 (2017)
Vol. 35: Vladimir K. Dobrev: Invariant Differential Operators 1 (2016)
Authors
Prof. Pietro Giuseppe Fré
Università degli Studi di Torino
Via P. Giuria 1
10125 Torino
Italy
[email protected]
Dr Alexander Fedotov
Department of Theoretical Physics
National Research Nuclear University MEPhI
Kashirskoe Shosse 31
Moscow 115409
Russia
[email protected]
ISBN 978-3-11-055119-8
e-ISBN (PDF) 978-3-11-055120-4
e-ISBN (EPUB) 978-3-11-055133-4
Library of Congress Cataloging-in-Publication Data
A CIP catalog record for this book has been applied for at the Library of Congress.
Bibliographic information published by the Deutsche Nationalbibliothek
The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed
bibliographic data are available on the Internet at http://dnb.dnb.de.
© 2018 Walter de Gruyter GmbH, Berlin/Boston
www.degruyter.com
This book is dedicated from the part of Pietro Frè to his beloved daughter Laura,
to his darling wife Olga and to his younger son Vladimir. On the part of
Alexander Fedotov it is dedicated to his darling wife Evgenya and to his beloved
daughters Ekaterina and Anna.
Preface
The present book has its initial basis in the course on Lie Algebras and Lie
Groups that one of us (P.F.) used to give at SISSA when he was Professor of
Theoretical Physics there in the years 1990–1996. The stimulus to revise and
update those lecture notes to the point of writing a self-contained book on such
topics came more recently when it was agreed that P.F., working several years in
Moscow as scientific attaché of the Italian Embassy, would give a course on
Groups and Symmetries at the National Research Nuclear University MEPhI of
Moscow. Further momentum the project gained from the association of the first
with the second author (A.F.) who conducted weekly sessions of exercises and
problems, attached to the Course.
A further motivation to write these introductory Lectures on Groups and
Manifolds for Physicists came by the imminent publication of other two more
advanced books written by the first author for Springer Verlag, namely:
1. Advances in Lie Algebras and Geometry from Supergravity.
2. A Conceptual History of Symmetry from Plato to Special Geometries.
Symmetry is the keyword underpinning our modern understanding of the
Fundamental Laws of Nature and it is also the central focus of most
mathematical disciplines, of which Geometry, in this respect, is at the top of the
priority list.
By the wording The Mathematics of Symmetry, every contemporary
Mathematician and Physicist understands Lie Algebras, Lie Groups, Coset
Manifolds and also Discrete and Finite Groups. There is no doubt that such
topics are classical and constitute the backbone in the education of students in
theoretical physics so that many excellent textbooks exist on the market that
have been written both by mathematicians and by physicists. Yet,
notwithstanding their established classical status, research continues in these
mathematical fields and many developments have taken place in the last twenty
years, several of them stimulated and directed by developments in
supersymmetric field theories, supergravity and superstrings. As the first author
advocates in the above quoted two books, the mathematical needs of the super-
world have promoted an extensive reframing and a deeper understanding of
many chapters in the Mathematics of Symmetry, leading to new conceptions that
usually are not reflected in the existing textbook literature. As their title suggest,
the above quoted two books deal with the task of presenting in a modern unified
way the Advances in Geometry and Lie Algebra Theory that are most relevant to
contemporary field theories, the supersymmetric ones having top priority.
Parallel aim in the writing was that of presenting the conceptual evolution of the
episteme from ancient times to the contemporary vision. Indeed, trying to
summarize the implications for the episteme of the last thirty-three years of
experimental physics, the first author arrived at the following conclusion.
Leaving apart the issue of quantization, that we can generically identify with
the functional path integral over classical configurations, we have, within our
contemporary physical paradigm, a rather simple and universal scheme of
interpretation of the Fundamental Interactions and of the Fundamental
Constituents of Matter based on the following few principles:
A) The categorical reference frame is provided by Field Theory defined by some
action A = L(Φ,∂Φ) where L(Φ,∂Φ) denotes some Lagrangian
M
∫
depending on a set of fields Φ(x).
B) All fundamental interactions are described by connections A on principal
fiber-bundles P(G,M) where G is a Lie group and the base manifold M is
some space-time in d = 4 or in higher dimensions.
C) All the fields Φ describing fundamental constituents are sections of vector
bundles B(G, V,M), associated with the principal one P(G,M) and determined
by the choice of suitable linear representations D(G) ∶ V → V of the structural
group G.
D) The spin zero particles, described by scalar fields ϕI , have the additional
feature of admitting non-linear interactions encoded in a scalar potential V
(ϕ) for whose choice general principles, supported by experimental
confirmation, have not yet been determined.
E) Gravitational interactions are special among the others and universal since
they deal with the tangent bundle TM →M to space-time. The relevant
connection is in this case the Levi-Civita connection (or some of its
generalizations with torsion) which is determined by a metric g on M.
In this framework that is certainly provisional, yet encompasses the whole of
Fundamental Physics as we currently understand it, the notions of Lie Group,
Lie Algebra, Fiber-Bundle and Connections are the central building blocks.
It follows that contemporary educational programmes in theoretical physics
should place the learning of these topics in an equally central position.
Furthermore, the approach to such a learning should be tuned to the modern
advanced visions discussed by the first author in the quoted two books.
The present textbook, which is introductory, streams from the motivations
illustrated above and encodes the following features that make it significantly
different from other existing textbooks:
1) The covered topics include both finite groups, Lie groups, Lie algebras and
as much as it is necessary of differential geometry to properly understand the
notions of manifold, fiber-bundle, coset manifold and connection on a
principal fiber-bundle extended to its associated bundles.
2) The exposition is pedagogical and oriented to physical applications. Proofs
are presented only for few basic theorems and the preference is given to those
that are constructive, leading to useful and applicable algorithms.
3) An extended collection of exercises is presented and solved that helps the
students in their process of progressive understanding of primary concepts.
The exercises are mainly focused on basic notions of algebra and finite group
theory. For Lie groups and manifolds we rather rely on a collection of
explicitly worked out examples of increasing complexity.
4) Special emphasis is given to the classification of simple Lie algebras in
terms of root systems and Dynkin diagrams, stressing the relation of the ADE
classification of simply laced Lie algebras with the ADE classification of
finite rotation groups.
5) As anticipated above, the root and weight lattices for Lie algebras are
discussed and utilized in several explicit examples of construction of linear
representations, illustrating the relation between the Dynkin language and that
of tensors, customary in physical theories.
6) A special attention is given to exceptional Lie algebras and to the explicit
construction of their fundamental and adjoint representations. Exceptional Lie
algebras, considered for long time as Old Curiosity Shop items proved instead
to be of fundamental relevance in modern theories of the world.
7) In the case of finite group theory, just as in the case of Lie group theory, the
teaching approach is based on in-depth case studies where all the advocated
conceptions are illustrated in explicit constructions. It goes without saying
that we chose cases corresponding to items of special relevance in
contemporary theoretical physics. For finite groups we utilize in particular the
crystallographic octahedral group and the second smallest simple group with
168 elements that is crystallographic in seven dimensions.
8) One distinctive feature of our presentation concerns the use of
MATHEMATICA codes for group-theoretical constructions. We illustrate
how all such constructions can be effectively dealt with by means of suitable
computer codes and a collection of dedicated MATHEMATICA NoteBooks is
illustrated in Appendix B. The actual NoteBooks can be downloaded by the
reader of this book from the open source site:
