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About Modern Differential Geometry in Gauge Theories Yang–Mills Fields
<p>It is nowadays generally accepted that the theory of principal fiber bundles is the</p><p>appropriate mathematical framework for describing one of the most beautiful, as</p><p>well as important, physical theories, viz. the so-called gauge field theory, or gauge</p><p>theories, being in effect to quote M. F. Atiyah, “physical theories of a geometrical</p><p>character.”</p><p>Now, in this context, a principal fibration is defined by the (local) gauge group</p><p>(or internal symmetry group) of the physical system (particle field) under conside-</p><p>ration. Yet, the particular physical system at issue is carried by, or lives on, a “space”</p><p>(vacuum) that in the classical case is usually a smooth (viz. C ∞ -) manifold. Within</p><p>our abstract framework, instead, this is in general an arbitrary topological space,</p><p>being also the base space of all the fiber spaces involved.</p><p>Accordingly, we do not use any notion of calculus (smoothness) in the classical</p><p>sense, thoughwe can apply,most of the powerfulmachineryof the standarddifferen-</p><p>tial geometry, in particular, the theory of connections, characteristic classes, and the</p><p>like. However,all this is doneabstractly, whichconstitutes an axiomatic treatment of</p><p>differential geometry in terms of sheaf theory and sheaf cohomology (see A. Mallios</p><p>[VS: Vols I, II]), while, as already noted, no calculus is used at all! So the present</p><p>study can be construed as a further application of that abstract (i.e., axiomatic) point</p><p>of view in the realm of gauge theories, given, as mentioned before, the intimate</p><p>connection of the latter theories with (differential) geometry.</p><p>Thus, working within the aforementioned abstract set-up, we essentially replace</p><p>all the previous fiber spaces (viz. principaland/or vector bundles) by the correspond-</p><p>ing sheaves of sections, the former being, of course, just our model (motivation),</p><p>while our study is otherwise, as has already explained above, quite abstract(!), that</p><p>is, axiomatic. Of course, in the classical case the two perspectives are certainly</p><p>mathematically speaking (categorically!) equivalent; however, the sheaf-theoretic</p><p>language, to which we also stick throughout the present treatment, is even in the</p><p>standard case, in common usage in the recent physics literature (cf., for instance,</p><p>Yu.I.Manin[2]orevenS.A.Selesnick[1]).Thus,itprovesthatthesamelanguageis</p><p>at least physically more transparent, while finally being more practical. In addition,</p>
Detailed Information
| Author: | Anastasios Mallios |
|---|---|
| Publication Year: | 2010 |
| ISBN: | 817643796 |
| Pages: | 244 |
| Language: | other |
| File Size: | 2.2457 |
| Format: | |
| Price: | FREE |
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