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Author
Title Physics on Noncommutative Spacetimes
URL
Publication Date
Degree PhD
Degree Level doctoral
University/Publisher Syracuse University
Abstract 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 B<sub>xn</sub> 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 S<sup><sub>2</sub></sup>/F and S<sup>2</sup> in detail. We compute the spectrums of the spin 1 and spin 3/2 Dirac operators on S<sup><sub>2</sub></sup>/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 B<sub>xn</sub> 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.
Subjects/Keywords Fuzzy Sphere; Hopf Algebra; Noncommutative Geometry; Planck Scale Physics; Quantum Field Theory; Physics
Contributors Aiyalam P. Balachandran
Country of Publication us
Record ID oai:surface.syr.edu:phy_etd-1122
Repository syracuse-diss
Date Retrieved
Date Indexed 2019-10-07
Created Date 2012-05-01 07:00:00

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…probe physics at the Planck scale we need a probe whose Compton wavelength satisfies Mc ≤ lP (2) where lP is the Planck length given in Eq.(1) and M is mass of the probe. This implies that M≥ lP C (3) which upon…

…noncommutative parameter χ. The best bound obtained is χ 1024 TeV suggesting an energy beyond Planck scale. These come from forbidden processes occuring in neutrino experiments. This also suggests a further check on its validity. We also provide a brief survey of…

…helped describe the other known forces of nature. This can be looked at as the problem of length(or mass in Planck units) scales and our failure to understand how the large merges with the small. This is the problem of quantum gravity. This…

…problem occurs at the Planck length lP = G ≈ 1.616252(81) × 10−35 m c3 (1) where is the Planck’s constant, G is the universal gravitational constant and c is the speed of light. An important issue is the structure of spacetime at…

…argument for why spacetime at Planck scales become noncommutative was given by Doplicher, Fredenhagen and Roberts [11]. This argument combines Heisenberg’s uncertainty principle and Einstein’s theory of gravity. It goes as follows. In order to…

…substitution of values in SI units turns out to be the Planck mass, M ≈ mP = 2.17645 × 10−8 kg. The Schwarzschild radius for this mass is around 3.21 × 10−35 m which is of the order of lP . Thus such high masses in small volumes will cause black hole horizons…

…can produce black holes and black hole horizons will then limit spatial resolution suggesting ∆t ∆|x| ≥ lp2 , lP = a fundamental length scale. (4) The essential idea of what it means for spacetime or rather geometry to be noncommutative can…

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