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 TMF, 2003, Volume 135, Number 1, Pages 82–94 (Mi tmf176)

Wave Equations in Riemannian Spaces

K. S. Mamaevaa, N. N. Trunovb

a St. Petersburg State University of Economics and Finance
b D. I. Mendeleev Institute for Metrology

Abstract: With regard to applications in quantum theory, we consider the classical wave equation involving the scalar curvature with an arbitrary coefficient $\xi$. General properties of this equation and its solutions are studied based on modern results in group analysis with the aim to fix a physically justified value of $\xi$. These properties depend essentially not only on the values of $\xi$ and the mass parameter but also on the type and dimension of the space. Form invariance and conformal invariance must be distinguished in general. A class of Lorentz spaces in which the massless equation satisfies the Huygens principle and its Green's function is free of a logarithmic singularity exists only for the conformal value of $\xi$. The same value of $\xi$ follows from other arguments and the relation to the known WKB transformation method that we establish.

Keywords: wave equation, curved space-time, conformal invariance, conformal transformation, Huygens principle

DOI: https://doi.org/10.4213/tmf176

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English version:
Theoretical and Mathematical Physics, 2003, 135:1, 520–530

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Revised: 13.05.2002

Citation: K. S. Mamaeva, N. N. Trunov, “Wave Equations in Riemannian Spaces”, TMF, 135:1 (2003), 82–94; Theoret. and Math. Phys., 135:1 (2003), 520–530

Citation in format AMSBIB
\Bibitem{MamTru03} \by K.~S.~Mamaeva, N.~N.~Trunov \paper Wave Equations in Riemannian Spaces \jour TMF \yr 2003 \vol 135 \issue 1 \pages 82--94 \mathnet{http://mi.mathnet.ru/tmf176} \crossref{https://doi.org/10.4213/tmf176} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1997652} \zmath{https://zbmath.org/?q=an:1178.58011} \transl \jour Theoret. and Math. Phys. \yr 2003 \vol 135 \issue 1 \pages 520--530 \crossref{https://doi.org/10.1023/A:1023235503054} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000183054500004} 

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• https://doi.org/10.4213/tmf176
• http://mi.mathnet.ru/eng/tmf/v135/i1/p82

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. N. N. Trunov, “A Class of Potentials for Which Exact Semiclassical Quantization Can Be Achieved”, Theoret. and Math. Phys., 138:3 (2004), 407–417
2. Yu. V. Pavlov, “Renormalization and Dimensional Regularization for a Scalar Field with Gauss–Bonnet-Type Coupling to Curvature”, Theoret. and Math. Phys., 140:2 (2004), 1095–1108
3. Pavlov, YV, “Space-Time Description of Scalar Particle Creation by a Homogeneous Isotropic Gravitational Field”, Gravitation & Cosmology, 14:4 (2008), 314
4. Lobashev, AA, “A universal effective quantum number for centrally symmetric problems”, Journal of Physics A-Mathematical and Theoretical, 42:34 (2009), 345202
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