Symplectic techniques in geometric quantum mechanics and nonlinear quantum mechanics
In this thesis we study the roles played by symplectic geometry in quantum mechanics,in particular quantum dynamics and quantum information theory treated as two separate parts. The common ground for both parts is the geometrical formulation of quantum mechanics. In Chapter 2, we review the associa...
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myupmir.4145320160108T08:04:05Z Symplectic techniques in geometric quantum mechanics and nonlinear quantum mechanics 201306 Molladavoudi, Saeid In this thesis we study the roles played by symplectic geometry in quantum mechanics,in particular quantum dynamics and quantum information theory treated as two separate parts. The common ground for both parts is the geometrical formulation of quantum mechanics. In Chapter 2, we review the associated complex projective Hilbert space of quantum pure states, with symplectic and Riemannian structures and their roles in quantum dynamics and kinematics. In Chapter 3, we motivate the idea of informationtheoretic constraint on the differentiable manifold of probability distributions through the maximum uncertainty principle and the linear Schrodinger equation by introduction of the wave function. In Chapter 4, we review both regular and singular symplectic reduction of a symplectic manifold, which is acted upon properly and symplectically by a compact Lie group. Chapter 5 contains the author's original contributions to the first part of the thesis. In this chapter, by using the same informationtheoretic discussion of the Chapter 3 we propose a nonrelativistic, spinless, nonlinear quantum dynamical equation,with the Fisher information metric replaced by the JensenShannon distance information. Furthermore, we show that the nonlinear Schrodinger equation is in fact a Hamiltonian dynamics, namely it preserves the symplectic structure of the complex Hilbert space. The projected dynamics on the corresponding projective Hilbert space is derived and its properties are highlighted in further details. Chapter 6 contains the author's primary contributions to the second part of the thesis. In particular, by using the singular symplectic reduction of the Chapter 4 we explicitly construct the space of entanglement types of threequbit pure states with a specific (shifted) spectra of singleparticle reduced density matrices. Moreover, we obtain the image of the symplectic quotient under the induced Hilbert map, by using local unitary invariant polynomials. Then the symplectic structure on the principal stratum of the symplectic quotient is derived. Finally, it is discussed that other lower dimensional strata are relative equilibria on the original manifold and their stability properties are investigated under compact subgroups of the local unitary transformations. Symplectic geometry Quantum theory Symplectic manifolds 201306 Thesis http://psasir.upm.edu.my/id/eprint/41453/ http://psasir.upm.edu.my/id/eprint/41453/1/IPM%202013%206R.pdf application/pdf en public phd doctoral Universiti Putra Malaysia Symplectic geometry Quantum theory Symplectic manifolds 
institution 
Universiti Putra Malaysia 
collection 
PSAS Institutional Repository 
language 
English 
topic 
Symplectic geometry Quantum theory Symplectic manifolds 
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Symplectic geometry Quantum theory Symplectic manifolds Molladavoudi, Saeid Symplectic techniques in geometric quantum mechanics and nonlinear quantum mechanics 
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In this thesis we study the roles played by symplectic geometry in quantum mechanics,in particular quantum dynamics and quantum information theory treated as two
separate parts. The common ground for both parts is the geometrical formulation of quantum mechanics. In Chapter 2, we review the associated complex projective Hilbert space of quantum pure states, with symplectic and Riemannian structures and their roles in quantum dynamics and kinematics. In Chapter 3, we motivate the idea of informationtheoretic constraint on the differentiable manifold of probability distributions through the maximum uncertainty principle and the linear Schrodinger equation by introduction of the wave function.
In Chapter 4, we review both regular and singular symplectic reduction of a symplectic manifold, which is acted upon properly and symplectically by a compact Lie
group. Chapter 5 contains the author's original contributions to the first part of the thesis. In this chapter, by using the same informationtheoretic discussion of the Chapter 3 we propose a nonrelativistic, spinless, nonlinear quantum dynamical equation,with the Fisher information metric replaced by the JensenShannon distance information. Furthermore, we show that the nonlinear Schrodinger equation is in fact a Hamiltonian dynamics, namely it preserves the symplectic structure of the complex
Hilbert space. The projected dynamics on the corresponding projective Hilbert space is derived and its properties are highlighted in further details. Chapter 6 contains the author's primary contributions to the second part of the thesis. In particular, by using the singular symplectic reduction of the Chapter 4 we explicitly construct the space of entanglement types of threequbit pure states with a specific (shifted) spectra of singleparticle reduced density matrices. Moreover, we obtain the image of the symplectic quotient under the induced Hilbert map, by using local unitary invariant polynomials. Then the symplectic structure on the principal stratum of the symplectic quotient is derived. Finally, it is discussed that other lower dimensional strata are relative equilibria on the original manifold and their stability properties are investigated under compact subgroups of the local unitary transformations. 
format 
Thesis 
qualification_name 
Doctor of Philosophy (PhD.) 
qualification_level 
Doctorate 
author 
Molladavoudi, Saeid 
author_facet 
Molladavoudi, Saeid 
author_sort 
Molladavoudi, Saeid 
title 
Symplectic techniques in geometric quantum mechanics and nonlinear quantum mechanics 
title_short 
Symplectic techniques in geometric quantum mechanics and nonlinear quantum mechanics 
title_full 
Symplectic techniques in geometric quantum mechanics and nonlinear quantum mechanics 
title_fullStr 
Symplectic techniques in geometric quantum mechanics and nonlinear quantum mechanics 
title_full_unstemmed 
Symplectic techniques in geometric quantum mechanics and nonlinear quantum mechanics 
title_sort 
symplectic techniques in geometric quantum mechanics and nonlinear quantum mechanics 
granting_institution 
Universiti Putra Malaysia 
publishDate 
2013 
url 
http://psasir.upm.edu.my/id/eprint/41453/1/IPM%202013%206R.pdf 
_version_ 
1747811871425036288 