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TMF, 2011, Volume 167, Number 2, Pages 323–336 (Mi tmf6643)  

This article is cited in 9 scientific papers (total in 9 papers)

Solutions of the Klein–Gordon equation on manifolds with variable geometry including dimensional reduction

P. P. Fizievab, D. V. Shirkova

a Joint Institute for Nuclear Research, Dubna, Moscow Oblast, Russia
b Sofia University St. Kliment Ohridski, Sofia, Bulgaria

Abstract: We develop the recent proposal to use dimensional reduction from the four-dimensional space–time $(D=1+3)$ to the variant with a smaller number of space dimensions $D=1+d$, $d<3$, at sufficiently small distances to construct a renormalizable quantum field theory. We study the Klein–Gordon equation with a few toy examples (“educational toys”) of a space–time with a variable spatial geometry including a transition to a dimensional reduction. The examples considered contain a combination of two regions with a simple geometry (two-dimensional cylindrical surfaces with different radii) connected by a transition region. The new technique for transforming the study of solutions of the Klein–Gordon problem on a space with variable geometry into solution of a one-dimensional stationary Schrödinger-type equation with potential generated by this variation is useful. We draw the following conclusions: $(1)$ The signal related to the degree of freedom specific to the higher-dimensional part does not penetrate into the smaller-dimensional part because of an inertial force inevitably arising in the transition region (this is the centrifugal force in our models). $(2)$ The specific spectrum of scalar excitations resembles the spectrum of real particles; it reflects the geometry of the transition region and represents its “fingerprints”. $(3)$ The parity violation due to the asymmetric character of the construction of our models could be related to the CP symmetry violation.

Keywords: dimensional reduction, space with variable geometry, Klein–Gordon equation, spectrum of scalar excitations, CP symmetry violation


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English version:
Theoretical and Mathematical Physics, 2011, 167:2, 680–691

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Received: 19.12.2010

Citation: P. P. Fiziev, D. V. Shirkov, “Solutions of the Klein–Gordon equation on manifolds with variable geometry including dimensional reduction”, TMF, 167:2 (2011), 323–336; Theoret. and Math. Phys., 167:2 (2011), 680–691

Citation in format AMSBIB
\by P.~P.~Fiziev, D.~V.~Shirkov
\paper Solutions of the~Klein--Gordon equation on manifolds with variable geometry including dimensional reduction
\jour TMF
\yr 2011
\vol 167
\issue 2
\pages 323--336
\jour Theoret. and Math. Phys.
\yr 2011
\vol 167
\issue 2
\pages 680--691

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    This publication is cited in the following articles:
    1. Obukhov Yu.N., Silenko A.J., Teryaev O.V., “Dirac fermions in strong gravitational fields”, Phys. Rev. D, 84:2 (2011), 024025, 6 pp.  crossref  mathscinet  adsnasa  isi  elib  scopus
    2. D. V. Shirkov, “Imagery of symmetry in current physics”, Theoret. and Math. Phys., 170:2 (2012), 239–248  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    3. Fiziev P.P., Shirkov D.V., “The (2+1)-dimensional axial universes-solutions to the Einstein equations, dimensional reduction points and Klein-Fock-Gordon waves”, J. Phys. A, 45:5 (2012), 055205, 15 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    4. Silenko A.J., “Momentum and spin dynamics of Dirac particles at effective dimensional reduction”, International Conference on Theoretical Physics: Dubna-Nano 2012, J. Phys.: Conf. Ser., 393, 2012, 012034  crossref  adsnasa  isi  scopus
    5. Stoica O.C., “Metric Dimensional Reduction At Singularities With Implications To Quantum Gravity”, Ann. Phys., 347 (2014), 74–91  crossref  mathscinet  zmath  adsnasa  isi  scopus
    6. Silenko A.J., Teryaev O.V., “Spin Effects and Compactification”, Phys. Rev. D, 89:4 (2014), 041501  crossref  mathscinet  adsnasa  isi  scopus
    7. Stoica O.C., “The Geometry of Black Hole Singularities”, Adv. High. Energy Phys., 2014, 907518  crossref  mathscinet  isi  scopus
    8. Tarloyan A.S., Ishkhanyan T.A., Ishkhanyan A.M., “Four five-parametric and five four-parametric independent confluent Heun potentials for the stationary Klein-Gordon equation”, Ann. Phys.-Berlin, 528:3-4 (2016), 264–271  crossref  zmath  isi  scopus
    9. Stoica O.C., “Revisiting the Black Hole Entropy and the Information Paradox”, Adv. High. Energy Phys., 2018, 4130417  crossref  mathscinet  isi  scopus
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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