Goncalves de Assis, P. E (2009). NonHermitian Hamiltonians in Field Theory. (Doctoral thesis, City University London)

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Abstract
This thesis is centred around the role of nonHermitian Hamiltonians in Physics both at the quantum and classical levels. In our investigations of twolevel models we demonstrate [1] the phenomenon of fast transitions developed in the PT symmetric quantum brachistochrone problem may in fact be attributed to the nonHermiticity of evolution operator used, rather than to its invariance under PT operation. Transition probabilities are calculated for Hamiltonians which explicitly violate PT symmetry. When it comes to Hilbert spaces of infinite dimension, starting with nonHermitian Hamiltonians expressed as linear and quadratic combinations of the generators of the su(1; 1) Lie algebra, we construct [2] Hermitian partners in the same similarity class. Alongside, metrics with respect to which the original Hamiltonians are Hermitian are also constructed, allowing to assign meaning to a large class of nonHermitian Hamiltonians possessing real spectra. The finding of exact results to establish the physical acceptability of other nonHermitian models may be pursued by other means, especially if the system of interest cannot be expressed in terms of Lie algebraic elements. We also employ [3] a representation of the canonical commutation relations for position and momentum operators in terms of realvalued functions and a noncommutative product rule of differential form. Besides exact solutions, we also compute in a perturbative fashion metrics and isospectral partners for systems of physical interest. Classically, our efforts were concentrated on integrable models presenting PT  symmetry. Because the latter can also establish the reality of energies in classical systems described by Hamiltonian functions, we search for new families of nonlinear differential equations for which the presence of hidden symmetries allows one to assemble exact solutions. We use [4] the Painleve test to check whether deformations of integrable systems preserve integrability. Moreover we compare [5] integrable deformed models, which are thus likely to possess soliton solutions, to a broader class of systems presenting compacton solutions. Finally we study [6] the pole structure of certain real valued nonlinear integrable systems and establish that they behave as interacting particles whose motion can be extended to the complex plane in a PT symmetric way.
Item Type:  Thesis (Doctoral) 

Additional Information:  © 2009 Paulo Assis. Published as "NonHermitian Hamiltonians in Field Theory: PTsymmetry and applications", VDM Verlag Dr. Müller (10 Nov 2010), Saarbrücken, Germany, 9783639294309. 
Subjects:  Q Science > QC Physics 
Divisions:  City University London PhD theses School of Engineering & Mathematical Sciences > Department of Mathematical Science 
URI:  http://openaccess.city.ac.uk/id/eprint/2118 
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