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Doctor of Philosophy (PhD)


Aiyalam P. Balachandran


Fuzzy Sphere, Hopf Algebra, Noncommutative Geometry, Planck Scale Physics, Quantum Field Theory

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The structure of spacetime at the Planck scale remains a mystery to this date with a lot of insightful attempts to unravel this puzzle. One such attempt is the proposition of a `pointless' structure for spacetime at this scale. This is done by studying the geometry of the spacetime through a noncommutative algebra of functions defined on it. We call such spacetimes 'noncommutative spacetimes'. This dissertation probes physics on several such spacetimes. These include compact noncommutative spaces called fuzzy spaces and noncompact spacetimes. The compact examples we look at are the fuzzy sphere and the fuzzy Higg's manifold. The noncompact spacetimes we study are the Groenewold-Moyal plane and the Bxn plane.

A broad range of physical effects are studied on these exotic spacetimes. We study spin systems on the fuzzy sphere. The construction of Dirac and chirality operators for an arbitrary spin j is studied on both S2/F and S2 in detail. We compute the spectrums of the spin 1 and spin 3/2 Dirac operators on S2/F. These systems have novel thermodynamical properties which have no higher dimensional analogs, making them interesting models.

The fuzzy Higg's manifold is found to exhibit topology change, an important property for any theory attempting to quantize gravity. We study how this change occurs in the classical setting and how quantizing this manifold smoothens the classical conical singularity. We also show the construction of the star product on this manifold using coherent states on the noncommutative algebra describing this noncommutative space.

On the Moyal plane we develop the LSZ formulation of scalar quantum field theory. We compute scattering amplitudes and remark on renormalization of this theory. We show that the LSZ formalism is equivalent to the interaction representation formalism for computing scattering amplitudes on the Moyal plane. This result is true for on-shell Green's functions and fails to hold for off-shell Green's functions.

With the present technology available, there is a scarcity of experiments which directly involve the Planck scale. However there are interesting low and medium energy experiments which put bounds on the validity of established principles which are thought to be violated at the Planck scale. One such principle is the Pauli principle which is expected to be violated on noncommutative spacetimes. We introduce a noncommutative spacetime called the Bxn plane to show how transitions, not obeying the Pauli principle, occur in atomic systems. On confronting with the data from experiments, we place bounds on the noncommutative parameter.


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