DIMENSIONAL REUCTIONin (1+2)-dimensionalCYLINDRICAL UNIVERSESandKLEIN-GORDON WAVESP. P. Fiziev,D.V. ShirkovBLThPh, JINR, DubnaTalk at the MI RAS SeimarMoscow , April 20, 2011
The basic problem of the standardapproach to quantum gravity is causedby the very classicalEinsten-Hilbert action in D = 1 + d :for dimension D > 2A New Idea:D. V. Sh., Particles and Nuclei (PEPAN), Lett. No 6 (162), 2010
Examples of 2-dim manifolds with variable geometries(surface of bottles)
Relation of the above (relativistic) examples withthe modern solid state physics of two-dimensional crystalsgraphene, fulerene,carbon nanotubes,carbon nanobuds,etc:
KG Equation for TEST PARTICLES:In the STATIC case we assume (at least locally)TheKleinGordonEquationonManifoldswithvariabletopologicaldimensionWe 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 Spacetime with Cylindrical SymmetryShape function:Standard anzatz:xSimple problems:The only nontrivial problem:Z-equationThe basicTheorem:Using proper changes of variables we can transform the Z-equation in the Schrodinger-like form:
Some Explicit ExamplesTwo Cylinders of Constant Radiuses R and r < R,Connected Continuously by a Part of Cone:The shapefunction:
ContinuousspectrumstatesExact Solutions and the limit *
The resonant states:M = 0S – matrixpolesA nontrivial dependence on the mass M:
Real forSchrodinger-type ODEwith potentialThe solution:
What was thereBEFORE the Big Bang?Is the Big Bang actuallya transition from aLOWER DIMENSIONALworld to theFOUR DIMENSIONALone???
DIMENSIONAL REDUCTION POINTSandDIMENSIONAL TRANSITION POINTSSymbolically:
=>Determinant:Field equations:Matter equations:
1.- expanding cylinder2.- static coneMOVIE3.- moving cone
=>=>omogeneous Monge(1784)-Ampere(1820) equation:General solution:General solution of the Cauchy problem:
Three classes of special solutionsinvolving one arbitrary functionwhereare arbitrary constants.
1. Creation and annihilation of pairs of DRPs is possible.2.3. ForRelative velocities:4. Forare zeros of
CONCLUDING REMARKS1. A signal, related to degree of freedom specificfor the higher-dim part does not penetrateinto the smaller-dim part,centrifugal force at the junction2. 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.
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 ???
?WearestilllookingfortheKEY !Thank Youfor your attention