Quantum Fields (original) (raw)

Theoretical and Mathematical Physics, 2011

Abstract

ABSTRACT This Chapter introduces main notions of quantum field theories (QFT). The commutation relations are imposed on operator functionals defined in terms of a relativistic inner product on a Cauchy surface. The product is uniquely defined for any two solutions to a field equation and is used for normalization of the solutions. The chosen way of quantization is more convenient in ‘mode by mode’ calculations of spectral functions in free QFT’s on non-trivial classical backgrounds, which is a central subject of this book. A distinctive feature of the exposition in this Chapter is that fairy general constructions are explained in details for fields which are important for physical applications (spins zero, one-half and one). The material also includes mode decompositions, Bose and Fermi statistics, creation and annihilation operators, Bogoliubov transformations, description of gauge theories. Much space is devoted to QFT’s on stationary classical backgrounds, relation to canonical quantization, the single-particle modes and eigenvalue problems which define the spectra of single-particle energies. Among the traditional issues are Green’s functions and calculations of averages. The Chapter ends with a brief introduction to a theory of quasinormal modes. These modes, although physically relevant, serve as an example of excitations which are not quantized.

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