An Excursion into Higher Dimensions

Osmo Pekonen

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String theory

a unified ”theory of everything” fundamental idea: elementary

particles in the Planck scale (10-35 m) appear as tiny vibrating ”strings”

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My little paper, a long time ago…

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The latest conference

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Fundamental forces 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. 5

Standard Model of particle physics

Unification of the three other fundamental forces: electromagnetism (Maxwell’s equations

~1864) weak force strong forceFundamental concept of unification: symmetry group electromagnetism one foton SU(1) weak force3 weak bosons W+, W−,

ZSU(2) strong force eight gluons SU(3)Total number of dimensions of the unified symmetry group G = SU(1) × SU(2) × SU(3) is 1 + 3 + 8 = 12.

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Mathematical framework of the Standard Model: Gauge theory and Yang-Mills equations (1954)

C.N. Yang and Robert Mills:

In a principal G-bundleminimize the Lagrangian

to get the field equations

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Elementary particles of the Standard Model

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Quantum Gravity: unification between Gravitation and Standard Model remains a mystery

Bold idea: perhaps higher dimensions than 4 can give some additional leeway?

Pioneers of this idea:

Gunnar Nordström Theodor Kaluza Oskar Klein

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Bosonic string theory: propagating strings trace out 2 D worldsheets

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Feynman’s quantization recipe: integrate over the ∞ D space of all possible paths of an elementary particle from A to B

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Pointlike particles trace out worldlines while stringlike particles trace out worldsheets. What is the gain?

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The gain is that the space of worldsheets is parametrized up to conformal equivalence by a finite dimensional space Mg

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…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

This leads to unified field theory (Bosonic String Theory) in a spacetime of 26 dimensions

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Why 26 dimensions?

 

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Problem: Fermions are not inluded.Additional input: Supersymmetry.

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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).

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Calabi-Yau space attached to every point of ordinary 4-dimensional spacetime

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The symmetry group of Superstring Theory 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.

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New problem: there are too many different sectors of string theory!

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.

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The leading theoretician about strings is Edward Witten (born 1951)

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Witten is the father of M-theory.

M-theory works in D=11dimensions. 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.

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Do we live in a holographic universe?

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.

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Is all this science, or some kind 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!

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Thank you for your attention and welcome to the fantasy workshop!

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