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R I E M A N N I A N M E T H O D IN Q U A N T U M F I E L D T H E O R Y ABOUT CURVED SPACE-TIME*) A. Uhlmann Department of Physics and NTZ, Karl-Marx-Universitiit, DDR-7010 Leipzig Starting from a given Lorentz metric g~k on a space-time manifold M and a tube T of world lines along which observers may move, we describe an algorithm to obtain the quantum theory of scalar particles of mass m. Let ei denote the 4-velocities of the world lines, and let V~, s real, be a family o f non-intersecting space-like hypersurfaces. We assume that the submanifolds V~ cover completely the tube T. Thus every point of T is crosssed by just one world line and by just one V,. The essential content of my talk is now in the following statement: One can reasonably associate to every V~ a Hilbert space F s of one-particle states which are realized by measurable function concentrated on V~. There is further defined a positive semidefinite operator H s acting on this Hilbert space Fs, and which is interpreted as the one-particle Hamiltonian at the instant Vs. Indeed, having at hand such a procedure one can do the following. One defines the Fock space ~ s to every F s and defines the appropriate Hamiltonian by second quantizing the operator Hs. With the wave operator A4 associated to 9~k one considers the wave equation ( - A 4 + m 2) ~o = 0 . I f u s Fs one solves this equation with the initial data ~0 = u and i(&p/On)= H,u on Vs. This transport the initial data to Vt with t =~ s, and this transport extends, obviously, to the associated Fock spaces. Generally, this transport from V~to Vt will not conserve the particle number, and the one-particle states of Vs will give rise to superpositions of many-particle states attached to Vr Hence the very problem is in constructing F s and H s. This will be achieved by associating to M, or only to the tube T, a Riemannian metric 9~k given by 9ik + 9ik = 2eiek (the signature of the Lorentz metric is + - - - ). Then the construction of F S and/-/s is achieved by solving (-/~4 + m 2 ) f = u on the set U Vt t<s *) Invited talk at the International Symposium "Selected Topics in Quantum Field Theory and Mathematical Physics", Bechyn6, Czechoslovakia, June 14--21, 1981. Czech. J. Phys. B 32 [1982] 573 A. Uhlmann: Riemannian method in quantum field theory . . . and several other manipulations dictated by experience in the Euclidean formulation of Quantum Field Theory & la Nelson, Guerra, Rosen, Simon, Hegerfeld, and others. They are described in [1] together with some references to previous work. Applying our procedure to stationary space-times it reproduces what everybody would suppose to hold in that case. But effective calculations are possible too for metrics of the Robertson-Walker type. Received 27. 10. 1981. Reference [1] Uhlmann A.: Czech. J. Phys. 11 (1981) 1249. 574 Czech. J. Phys. B 32 [1982]