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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References
Bogolyubov, N.N., Shirkov, D.V.: Introduction to the Theory of Quantized Fields. Intersci. Monogr. Phys. Astron. (1959)
Chandrasekhar, S.: The Mathematical Theory of Black Holes, 646 pp. Clarendon, Oxford (1992)
Chernikov, N.A., Tagirov, E.A.: Quantum theory of scalar fields in de Sitter space-time. Ann. Poincaré Phys. Theor. A 9, 109 (1968)
DeWitt, B.S.: Dynamical Theory of Groups and Fields. Gordon & Breach, New York (1965)
Esposito, G., Kamenshchik, A.Y., Pollifrone, G.: Euclidean Quantum Gravity on Manifolds with Boundary. Kluwer Academic, Dordrecht (1997)
Faddeev, L.D., Slavnov, A.A.: Gauge fields. Introduction to quantum theory. Front. Phys. 50, 1–232 (1980)
Fursaev, D.V.: Energy, Hamiltonian, Noether charge, and black holes. Phys. Rev. D 59, 064020 (1999). hep-th/9809049
Gitman, D.M., Tyutin, I.V.: Quantization of Fields with Constraints. Springer, Berlin (1990)
Hawking, S.W., Ellis, C.F.R.: The Large Scale Structure of Space-Time. Cambridge University Press, Cambridge (1973)
Hod, S.: Bohr’s correspondence principle and the area spectrum of quantum black holes. Phys. Rev. Lett. 81, 4293 (1998). gr-qc/9812002
Kokkotas, K.D., Schmidt, B.G.: Quasi-normal modes of stars and black holes. Living Rev. Relativ. 2, 2 (1999). gr-qc/9909058
Konoplya, R.A., Zhidenko, A.: Quasinormal modes of black holes: from astrophysics to string theory (2011). 1102.4014
Peskin, M.E., Schroeder, D.V.: An Introduction to Quantum Field Theory. Addison-Wesley, Reading (1995)
Schutz, B.F.: Geometrical Methods of Mathematical Physics. Cambridge University Press, Cambridge (1982)
Vassilevich, D.V.: QED on curved background and on manifolds with boundaries: unitarity versus covariance. Phys. Rev. D 52, 999–1010 (1995). gr-qc/9411036
Weinberg, S.: The Quantum Theory of Fields. Vol. 1: Foundations. Cambridge University Press, Cambridge (1995)
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Fursaev, D., Vassilevich, D. (2011). Quantum Fields. In: Operators, Geometry and Quanta. Theoretical and Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0205-9_2
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DOI: https://doi.org/10.1007/978-94-007-0205-9_2
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