Exceptional magic
Predrag Cvitanovic
Center for Chaos and Turbulence Studies
Niels Bohr Institute
Blegdamsvej 17,
DK-2100 Copenhagen O
Hovedfagområde Fysik | Tidsrum Forår 1997 |
Fagområde(r) Feltteori, matematik | Sted NBI Blegdamsvej |
Hold størelse 1 | |
Forudsætninger Gruppeteori, programmering i algebraisk sprog |
Bachelor project description
Field theory computations associate group theoretic factors
called casimirs with Feynman diagrams. There exists an easy technique for
evaluating such factors combinatorially, the diagrammatic ``birdtracks''
method. It requires no background (other than for motivation) of the quantum
field theory, only of the semisimple Lie algebras. Suprisingly, this method
leads naturally to an unusual construction of the five exceptional Lie
groups. The status of exceptional Lie groups in physics remains uncertain
- but at any given time some are believed to be candidates for some or
other mother of all theories, so exposure to them is probably not harmful.
Regardless of the import of exceptional Lie groups, in this project one
learns some generally useful tensor calculus that deals directly with the
building blocks of the theory of the invariants, Clebsch-Gordan series,
evaluation algorithms for group theoretic weights, etc.
The ``magic triangle'' for Lie algebras. The number in the lower left corner of each entry is the dimension of the defining representation.
The goal of the project is to check and extend certain group-theoretic calculations of ref. 1, in particular Chapter 17: E_8 family of invariance groups. This ``exceptional magic'' could be accessible to a mathematically motivated theoretical physics student. Parts of project are probably best done by means of Mathematica, Maple or some other algebraic language.
References