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In recent years topology has firmly established itself as an
important part of the physicist's mathematical arsenal. Topology
has profound relevance to quantum field theory-for example,
topological nontrivial solutions of the classical equa tions of
motion (solitons and instantons) allow the physicist to leave the
frame work of perturbation theory. The significance of topology has
increased even further with the development of string theory, which
uses very sharp topologi cal methods-both in the study of strings,
and in the pursuit of the transition to four-dimensional field
theories by means of spontaneous compactification. Im portant
applications of topology also occur in other areas of physics: the
study of defects in condensed media, of singularities in the
excitation spectrum of crystals, of the quantum Hall effect, and so
on. Nowadays, a working knowledge of the basic concepts of topology
is essential to quantum field theorists; there is no doubt that
tomorrow this will also be true for specialists in many other areas
of theoretical physics. The amount of topological information used
in the physics literature is very large. Most common is homotopy
theory. But other subjects also play an important role: homology
theory, fibration theory (and characteristic classes in
particular), and also branches of mathematics that are not directly
a part of topology, but which use topological methods in an
essential way: for example, the theory of indices of elliptic
operators and the theory of complex manifolds."
In recent years topology has firmly established itself as an
important part of the physicist's mathematical arsenal. It has many
applications, first of all in quantum field theory, but
increasingly also in other areas of physics. The main focus of this
book is on the results of quantum field theory that are obtained by
topological methods. Some aspects of the theory of condensed matter
are also discussed. Part I is an introduction to quantum field
theory: it discusses the basic Lagrangians used in the theory of
elementary particles. Part II is devoted to the applications of
topology to quantum field theory. Part III covers the necessary
mathematical background in summary form. The book is aimed at
physicists interested in applications of topology to physics and at
mathematicians wishing to familiarize themselves with quantum field
theory and the mathematical methods used in this field. It is
accessible to graduate students in physics and mathematics.
In recent years topology has firmly established itself as an
important part of the physicist's mathematical arsenal. Topology
has profound relevance to quantum field theory-for example,
topological nontrivial solutions of the classical equa tions of
motion (solitons and instantons) allow the physicist to leave the
frame work of perturbation theory. The significance of topology has
increased even further with the development of string theory, which
uses very sharp topologi cal methods-both in the study of strings,
and in the pursuit of the transition to four-dimensional field
theories by means of spontaneous compactification. Im portant
applications of topology also occur in other areas of physics: the
study of defects in condensed media, of singularities in the
excitation spectrum of crystals, of the quantum Hall effect, and so
on. Nowadays, a working knowledge of the basic concepts of topology
is essential to quantum field theorists; there is no doubt that
tomorrow this will also be true for specialists in many other areas
of theoretical physics. The amount of topological information used
in the physics literature is very large. Most common is homotopy
theory. But other subjects also play an important role: homology
theory, fibration theory (and characteristic classes in
particular), and also branches of mathematics that are not directly
a part of topology, but which use topological methods in an
essential way: for example, the theory of indices of elliptic
operators and the theory of complex manifolds."
In recent years topology has firmly established itself as an
important part of the physicist's mathematical arsenal. It has many
applications, first of all in quantum field theory, but
increasingly also in other areas of physics. The main focus of this
book is on the results of quantum field theory that are obtained by
topological methods. Some aspects of the theory of condensed matter
are also discussed. Part I is an introduction to quantum field
theory: it discusses the basic Lagrangians used in the theory of
elementary particles. Part II is devoted to the applications of
topology to quantum field theory. Part III covers the necessary
mathematical background in summary form. The book is aimed at
physicists interested in applications of topology to physics and at
mathematicians wishing to familiarize themselves with quantum field
theory and the mathematical methods used in this field. It is
accessible to graduate students in physics and mathematics.
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