MSc., Faculty of Science, University of the Witwatersrand, 2011 / Anomalous dimensions are calculated for a certain class of operators in the restricted
Schur polynomial basis in the large N limit. A new computationally
simple form of the dilatation operator is derived and used in this dissertation.
The class of operators investigated have bare dimension of O(N). Thus the calculation
necessarily sums non-planar Feynmann diagrams as the planar approximation
has broken down for operators of this size. The operators investigated
have two long columns and the operators mix under the action of the dilatation
operator, however the mixing of operators having a different number of columns
is suppressed and can be neglected in the large N limit. The action of the one
loop dilatation operator is explicitly calculated for the cases where the operators
have two, three and four impurities and it is found that in a particular limit
the action of the one loop dilatation operator reduces to that of a discrete second
derivative. The lattice on which the discretised second derivative is defined
is provided by the Young tableaux itself. The one loop dilatation operator is
diagonalised numerically and produces a surprisingly simple linear spectrum,
with interesting degeneracies. The spectrum can be understood in terms of
a collection of harmonic oscillators. The frequencies of the oscillators are all
multiples of 8g2Y
M and can be related to the set of Young tableaux acted upon
by the dilatation operator. This equivalence to harmonic oscillators generalises
on previously found results in the BPS sector, and suggests that the system is
integrable. The work presented here is based primarily on research carried out
by R.de Mello Koch, V De Comarmond, and K. Jefferies in [1].
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:wits/oai:wiredspace.wits.ac.za:10539/10494 |
| Date | 07 October 2011 |
| Creators | De Comarmond, Vincent |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | application/pdf |
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