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Nonperturbative methods in gauge theories
di Dmitri Antonov
edito da Pisa University Press
This textbook on nonperturbative phenomena stemmed from the lecture courses given for Ph.D. students at the Graduate Schools of the Universities of Pisa and Heidelberg, as well as at the Technical University of Lisbon, in 2009-2011. It describes in detail confinement and confining strings in the Georgi-Glashow model in three dimensions, as well as in the Abelian Higgs model in four dimensions. A separate part of the book is devoted to the Schwinger formulae, the world-line instantons, and the fermionic spin factor in the context of the world-line effective action. The book also illustrates some general analogies between quantum field theory and statistical mechanics, and provides an introduction to the Euclidean finite- temperature field theory, the deconfinement phase transitions, the renormalization group, as well as to such subjects as magnetic monopoles, Wilson loops, basics of nonperturbative quantum chromodynamics, anomalies in gauge theories, and
instantons in quantum mechanics.
Since, at x < b and x > a, ? falls off exponentially, for its normalization it suffices to integrate |?|2 over the interval b < x < a. Furthermore, since 1 ? x b dx?p? ? 4 is a rapidly varying function of x, it also suffices to use the following approximation:
cos2 ( 1 ? x b dx?p? ? 4 ) ? cos2 ( 1 ? x b dx?p? ? 4 )? = 1 2 .
The normalization condition then reads 1 = ? a b dx |?|2 c 2 2 ? a b dx p = c2 2 · T 2m .
In terms of the frequency of classical oscillations, ? = 2?/T , one obtains c = ? 2m? ? . (A1) Consider now the potential U(x) formed by two symmetric wells separated by a barrier . Had the barrier been impenetrable for the particle, the same energy levels E0 would be existing in both wells. These levels would then correspond to the motion of the particle in one of the two wells. Let us label the left well by number I, the right well by number II, and denote by ?0(x) the semiclassical wave function corresponding to the eigenenergy E0. A possibility of the underbarr...
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