Quantum field theory on curved backgrounds. I: The Euclidean functional integral
A mathematical construction of Euclidean quantum field theory on certain time-independent curved backgrounds is given. Instead of the algebraic techniques with Lorentzian signature used in the earlier approaches, the authors make use of Euclidean functional integral [\textit{J. Glimm}, and \textit{A. Jaffe}, Quantum Physics: a Functional Integral Point of View. 2nd ed. New York: Springer-Verlag (1987) (1981; Zbl 0461.46051] and Osterwalder-Schrader quantization [\textit{K. Osterwalder} and \textit{R. Schrader}, Commun. Math. Phys. 31, 83--112 (1973; Zbl 0274.46047); 42, 281--305 (1975; Zbl 0303.46034)]. Euclidean methods are known to be useful in the study of black holes, and a standard strategy for studying black hole (BH) thermodynamics is to analytically continue time in the BH metric [\textit{S. Carlip} and \textit{C. Teitelboim}, Phys. Rev. D (3) 51, No. 2, 622--631 (1995)], see also [\textit{M. B. Mensky}, Phys. Lett. A~314, 169--176 (2003; Zbl 1052.83523)], for a universal approach to general gravitational thermal effects). The reviewed paper implies a mathematical construction of scalar fields on any static, Euclidean BH background.
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