DIMENSIONAL REDUCTIONin (1+2)-dimensionalAXIAL UNIVERSES (AxU)andKLEIN-GORDON WAVESP. P. Fiziev,D.V. ShirkovBLThPh, JINR, DubnaTalk at the Seminar ofthe Faculty of PhysicsSofia University ,May 09, 2011
The basic problem of the standardapproach to quantum gravity is causedby the very classical Einstein-Hilbertaction in D = 1 + d :The same in theStandard ModelwithoutHiggs bosonfor dimension D > 2A New Idea:D. V. Shirkov, Particles and Nuclei (PEPAN), Lett. No 6 (162), 2010
Great Unification by dimensionalreduction instead of leptoquarks:
Greg (Grigoriy) Landsberg, Paris, July 2010
Relation of the above (relativistic) examples withthe modern solid state physics of two-dimensional crystalsgraphene, fulerene,carbon nanotubes,carbon nanobuds,etc: Nobel Prize in Physics 2010
Examples of 2-dim manifolds withvariable geometries(surfaces of bottles)D. V. Shirkov,,Good problem:Solvethe Klei n-Gordoneq.,the Dirac eq.the heat eq.,e.t.c.,on that kind ofvariable geometries.
KGEq for TEST PARTICLES:10 80baryons -EddingtonIn the STATIC case we assume (at least locally)numberTheKleinGordonEquationonManifoldswithvariabletopologicaldimensionWe consider the toy models in whichthe physical space is a continuousmerger :and THE TIME IS GLOBAL !Then we have local solutions:With common frequency:
Wave Equation in (1+2)-dim Static AxUShape function:Standard anzatz:Simple problems:The only nontrivial problem: Z-equation centrifugalThe basicTheorem:Using proper changes of variables we can transformthe Z-equation in the Schrodinger-like form:
Some Explicit ExamplesTwo Cylinders of Constant Radii R and r < R,Connected Continuously by a Part of Cone:The shapefunction:
ContinuousspectrumstatesExact Solutions and the limit *
The resonant states:M = 0S – matrix polesA nontrivial dependence on the mass M:
Real forSchrodinger-type ODEqwith potentialThe solution:
Expansion: 5% for 10 9 yearsWhat was thereBEFORE the Big Bang?Is the Big Bang actuallya transition from aLOWER DIMENSIONALworld to theFOURDIMENSIONALONE??? HST2011 New ideaP.F., D.Shirkov:
(1+2)-dim Time Dependent AxUInconsider
DIMENSIONAL REDUCTION POINTSandDIMENSIONAL TRANSITION POINTSSymbolically:
Einstein eqs for time dependent (1+2) –dim AxU=>Determinant:Field equations:Matter equations:
There vacuum solutions of Einstein eqs :1.- expanding cylinder2.- static coneMOVIE3.- moving coneLorentz transform
Solutions for the (1+2) AxU filled with “dust”=>=>omogeneous Monge(1784)-Ampere(1820) equation:Implicit general solution:=>Godograph of the velocity v(t,z)General solution of the Cauchy problem:
Three classes of special solutionsinvolving one arbitrary functionLorentz transformwhereare arbitrary constants.
1. Creation and annihilation of pairs of DRPs is possible.2.Lorentz transform3. Forzeros ofRelative velocities:4. For
Solutions of KGEq on (1+2)-dimAxU with positive termDiscretespectra:
Solutions to the KGEq on (1+2)-dimAxU filled with “dust”
CONCLUDING REMARKS1. A signal, related to degree of freedom specificfor the higher-dim part does not penetrateinto the smaller-dim part,the inertial forces at the junction.2. The specific spectrum of scalarexcitations characterizes the junctiongeometry. A new idea: To explain thebecause ofobserved particles spectra by geometry of thejunction between domains of the space-timewith different topological dimension.
3. The set of the dimensional reduction point -DPR may have complicated structure anddynamics:4. Critical behavior of the solutions for testparticles around DPR exist and depends bothon geometry and motion.
5. The parity violation, due to the asymmetryof space geometry could yield the CP- and T-violation.This gives a hope to discover a simple natural basis forexplanation of the real situation, concerningC, P, and T properties of the particles.6. Barion-antibarion asymmetry versusasymmetry of the space-time ???7. The Big Bang = ? = time transformation ofspace dimensions ???
?Weare stilllookingfor theMASTERKEY !Thank You foryour attention