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An Excursion into Higher Dimensions

Osmo Pekonen. An Excursion into Higher Dimensions. String theory. a unified ” theory of everything ” f undamental idea: elementary particles in the Planck scale (10 -35 m) appear as tiny vibrating ” strings ”. My little paper , a long time ago …. The latest conference.

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An Excursion into Higher Dimensions

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  1. Osmo Pekonen An Excursion into Higher Dimensions

  2. Stringtheory • a unified ”theory of everything” • fundamental idea: elementaryparticles in the Planckscale (10-35 m) appear as tinyvibrating ”strings”

  3. My littlepaper, a long timeago…

  4. The latestconference

  5. Fundamentalforces of Nature • gravitation: Newton’s Principia (1687) • Einstein’s general relativity (1915) • Einstein’s field equation is a statement about the geometry of 4-dimensional spacetime.

  6. Standard Model of particlephysics Unification of the threeotherfundamentalforces: • electromagnetism (Maxwell’sequations~1864) • weakforce • strongforce Fundamentalconcept of unification: symmetrygroup • electromagnetism one foton  SU(1) • weak force3 weakbosonsW+, W−, ZSU(2) • strongforceeightgluons SU(3) Total number of dimensions of the unifiedsymmetrygroup G = SU(1)× SU(2) × SU(3) is 1 + 3 + 8 = 12.

  7. Mathematical framework of the Standard Model: Gaugetheory and Yang-Millsequations (1954) • C.N. Yang and Robert Mills: In a principalG-bundle minimize the Lagrangian to get the fieldequations

  8. Elementaryparticles of the Standard Model

  9. QuantumGravity: unificationbetweenGravitation and Standard Modelremains a mystery • Bold idea: perhapshigherdimensionsthan 4 cangivesomeadditionalleeway? • Pioneers of this idea: • Gunnar Nordström • Theodor Kaluza • Oskar Klein

  10. Bosonicstringtheory: propagatingstringstrace out 2 D worldsheets

  11. Feynman’squantizationrecipe: integrateover the ∞ D space of allpossiblepaths of an elementaryparticlefrom A to B

  12. Pointlikeparticlestrace out worldlineswhilestringlikeparticlestrace out worldsheets. What is the gain?

  13. The gain is that the space of worldsheets is parametrizedup to conformalequivalenceby a finitedimensionalspaceMg …which has 6 × genus – 6 dimensions and is known in classical mathematics as the Riemann Moduli Space, resp. its covering space the Teichmüller Moduli Space

  14. Thisleads to unifiedfieldtheory (BosonicStringTheory) in a spacetime of 26 dimensions

  15. Why 26 dimensions?

  16. Problem: Fermionsarenotinluded.Additional input: Supersymmetry.

  17. What is the gain of introducing SUSY? • The number of necessary dimensions of spacetime is lowered to a slightly more palatable D=10 in Superstring Theory. • Why dont we see the extra 6 dimensions? • They are supposed to be wrapped up at each point of 4-dimensional spacetime as a mathematical structure called Calabi-Yau space (for specialists: compact Kählerian 6-manifold with vanishing Ricci curvature).

  18. Calabi-Yauspaceattached to everypoint of ordinary 4-dimensional spacetime

  19. The symmetrygroup of SuperstringTheory is necessarily 496-dimensional …but there are only two relevant Lie groups of dimension 496: SO(32) and E8 × E8. Recall: the Standard Model only had 12 symmetries! The 248-dimensional E8 is a most beautiful mathematical object.

  20. New problem: therearetoomanydifferentsectors of stringtheory! • Are there 10 or 26 dimensions ? • Is the structure group SO(32) or E8 × E8 ? • There are myriads of different Calabi-Yau spaces. Which are the prefered ones? • Heterotic String Theory is a hybrid of Bosonic String Theory and Superstring Theory. • It was developed in 1985 by the ”Princeton string quartet” David Gross, Jeffrey Harvey, Emil Martinec and Ryan Rohm.

  21. The leadingtheoreticianaboutstrings is Edward Witten (born 1951)

  22. Witten is the father of M-theory. M-theory works in D=11 dimensions. At the expense of one extra dimension, it unifies the various 10-dimensional theories. There also exists D=12 dimensional F-theory due to Cumrun Vafa. Time also could be more than 1 dimensional.

  23. Dowe live in a holographicuniverse? In 1997 the Argentinian physicist Juan Maldacena discovered a new kind of duality AdS/CFT which works between a theory of gravity (AdS) and a lower dimensional gauge theory (CFT) without gravity.

  24. Is allthis science, orsomekind of dream? • There is no experimental evidence whatsoever either for the existence of SUSY or the validity of any form of String Theory. • However, String Theory has already left a durable trace in many fields of mathematics (Witten is a Fields medalist). • String Theory could also turn out to be utterly wrong… or perhaps we might never know. • So there is some room for fantasy!

  25. Thankyou for yourattention and welcome to the fantasy workshop!

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