Modern Differential Geometry in Gauge Theories Maxwell Fields, Volume I

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2006

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2006

It is nowadays generally accepted that the theory of principal fiber bundles is the
appropriate mathematical framework for describing one of the most beautiful, as
well as important, physical theories, viz. the so-called gauge field theory, or gauge
theories, being, in effect, to quote, M. F. Atiyah, “physical theories of a geometrical
character.”
Now, in this context, a principal fibration is defined by the (local) gauge group
(or internal symmetry group) of the physical system (particle field) under consider-
ation. Yet, the particular physical system at issue is carried by, or lives on, a “space”
(vacuum) that in the classical case is usually a smooth (viz. C ∞ -) manifold. Within
our abstract framework, instead, this is, in general, an arbitrary topological space,
being also the base space of all the fiber spaces involved.
Accordingly, we do not use any notion of calculus (smoothness) in the classical
sense, though we can apply, most of the powerful machinery of the standard differ-
ential geometry, in particular, the theory of connections, characteristic classes, and
the like. However, all this is done abstractly, which constitutes an axiomatic treat-
ment of differential geometry in terms of sheaf theory and sheaf cohomology (see
A. Mallios [VS: Vols I, II]), while, as already noted, no calculus is used at all! So
the present study can be construed as a further application of that abstract (i.e., ax-
iomatic) point of view in the realm of gauge theories, given, as mentioned before,
the intimate connection of the latter theories with (differential) geometry.
Thus, working within the aforementioned abstract set-up, we essentially replace
all the previous fiber spaces (viz. principal and/or vector bundles) by the correspond-
ing sheaves of sections, the former being, of course, just our model (motivation),
while our study is otherwise, as has already explained above, quite abstract(!), that
is, axiomatic. Of course, in the classical case the two perspectives are certainly
mathematically speaking (categorically!) equivalent; however, the sheaf-theoretic
language, to which we also stick throughout the present treatment, is even in the
standard case, in common usage in the recent physics literature (cf., for instance, Yu.
I. Manin [1] or even S. A. Selesnick [1]). Thus, it proves that the same language is at
least physically more transparent, while, finally, being more practical. In addition,

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ebook

زبان

english

تاریخ انتشار

2006

نویسنده

Anastasios Mallios

شابک

ISBN-10 0817643788

20,000 تومان

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خرید کتاب Modern Differential Geometry in Gauge Theories Maxwell Fields, Volume I

Modern Differential Geometry in Gauge Theories Maxwell Fields, Volume I

گزارش تخلف

فرمت کتاب

ebook

زبان

english

تاریخ انتشار

2006

نویسنده

شابک

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