Abstract
This Chapter introduces the notion of an effective action for a quantum field theory. The effective action is defined as a functional whose variations over classical backgrounds yield averages of operators (such as currents, stress-energy tensors and etc.). The arguments which lead to this definition are straightforward for non-interacting quantum fields. The effective action can then be defined by the Ray-Singer formula provided that one uses Laplace-type operators. This implies a ‘Euclidean’ formulation of the theory. To meet these requirements the analysis, after a short overview of statistical mechanics, is applied to finite-temperature field theories. The aim of the present Chapter is basically to show how the spectral geometry methods can be used to reproduce a number of known QFT results, usually derived with the help of Feynman diagrams. Among them are one-loop effective (Coleman-Weinberg) potential and beta functions in gauge theories. The material also includes the following topics: relation between the Euclidean effective action and partition functions, complex geometries, renormalization theory, properties of the effective action of gauge fields, Faddeev-Popov ghosts, and the asymptotic freedom in quantum chromodynamics.
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Fursaev, D., Vassilevich, D. (2011). Effective Action. In: Operators, Geometry and Quanta. Theoretical and Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0205-9_7
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DOI: https://doi.org/10.1007/978-94-007-0205-9_7
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