The differential geometric structure of a Fréchet manifold of densities is developed, providing a geometrical framework for quantization related to Nelson’s stochastic mechanics. The Riemannian and symplectic structures of the density manifold are studied, and the Schrödinger equation is derived from a variational principle. By a theorem of Moser, the density manifold is an infinite dimensional homogeneous space, being the quotient of the group of diffeomorphisms of the underlying base manifold modulo the group of diffeomorphisms which preserve the Riemannian volume. From this structure and symplectic reduction, the quantization procedure is equivalent to Lie-Poisson equations on the dual of a semidirect product Lie algebra. A Poisson map is obtained between the dual of this Lie algebra and the underlying projective Hilbert space.
The differential geometric structure of a Fréchet manifold of densities is developed, providing a geometrical framework for quantization related to Nelson’s stochastic mechanics. The Riemannian and symplectic structures of the density manifold are studied, and the Schrödinger equation is derived from a variational principle. By a theorem of Moser, the density manifold is an infinite dimensional homogeneous space, being the quotient of the group of diffeomorphisms of the underlying base manifold modulo the group of diffeomorphisms which preserve the Riemannian volume. From this structure and symplectic reduction, the quantization procedure is equivalent to Lie-Poisson equations on the dual of a semidirect product Lie algebra. A Poisson map is obtained between the dual of this Lie algebra and the underlying projective Hilbert space.
Extended Schwinger's quantization procedure is used for constructing quantum mechanics on a manifold with a group structure. The considered manifold M is a homogeneous Riemannian space with the given action of isometry transformation group. Using the identification of M with the quotient space G/H, where H is the isotropy group of an arbitrary fixed point of M, we show that quantum mechanics on G/H possesses a gauge structure, described by the gauge potential that is the connection 1-form of the principal fiber bundle G(G/H,H). The coordinate representation of quantum mechanics and the procedure for selecting the physical sector of states are developed.
We show that any rule for quantization of functions on the phase space of a dynamical system is equivalent to a deformation of the Poisson bracket, in the sense of Lie algebras. All Schrödinger-type quantizations are equivalent mathematically but not physically. We examine whether a quantization rule exists in which the Poisson bracket of the free Hamiltonian and any function not more than quadratic in momentum gets quantized into the commutator of the quantized versions of these two observables. We show that if the (Riemannian) configuration manifold is irreducible, this condition uniquely specifies the quantization of these functions, in the cases when it can be satisfied. However, it can only be satisfied for systems whose configuration manifolds are one-dimensional, or have vanishing Ricci tensor, or are spaces of constant curvature. Thus the condition will not serve as a general physical principle which could be invoked to fix a quantization rule.
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 information-theoretic 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 information-theoretic discussion of the Chapter 3 we propose a non-relativistic, spin-less, non-linear quantum dynamical equation,with the Fisher information metric replaced by the Jensen-Shannon distance information. Furthermore, we show that the non-linear 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 three-qubit pure states with a specific (shifted) spectra of single-particle 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.
The symmetry properties of quantum variational principles are considered. Euler-Poincaré reduction theory is applied to the Dirac-Frenkel variational principle for Schrödinger's dynamics producing new variational principles for the different pictures of quantum mechanics (Schrödinger's, Heisenberg's and Dirac's) as well as for the Wigner-Moyal formulation. In addition, new variational principles for mixed states dynamics have been formulated. The already known geometric characterization of quantum mechanics on the complex projective space is shown to emerge naturally from the Euler-Poincaré variational principle. Semidirect-product structures are seen to produce new variational principles for Dirac's picture. \n \nOn the other hand, the variational and Hamiltonian approach to Ehrenfest expectation values dynamics is proposed. In the Lagrangian framework, the quantum variational principle for Schrödinger's dynamics is extended to account for both classical and quantum degrees of freedom. First, it is shown that the mean field model of any quantum mechanical system can be derived from a classical-quantum Euler-Poincaré Lagrangian on the direct sum Lie algebra of the Heisenberg and unitary groups. Then, the semidirect-product structure (named Ehrenfest group), is constructed using the displacement operator from the theory of coherent quantum states ( the unitary action of the Heisenberg group on the space of wavefunctions). New classical-quantum equations for Ehrenfest's expectation values dynamics are derived redefining the mean-field model Euler-Poincaré Lagrangian on the Lie algebra of the Ehrenfest group. In the Hamiltonian framework, first expectation values of the canonical observables are shown to be equivariant momentum maps for the unitary action of the Heisenberg group on quantum states. Then, the Hamiltonian structure for Ehrenfest's dynamics is shown to be Lie-Poisson for the Ehrenfest group. The variational formulation is then given a corresponding Hamiltonian structure. The classical-quantum Ehrenfest dynamics equations produce classical and quantum dynamics as special limit cases. \n \nIn the particular case of Gaussian states, expectation values couple to second order moments, so that GS are completely characterized by first and second moments. When the total energy is computed with respect to a Gaussian state, higher moments can be expressed in terms of the first two, so that the moment hierarchy closes for Gaussian states. Second moments are shown to be equivariant momentum maps for the action of the symplectic group on the space of Wigner functions. Eventually, Gaussian states are shown to possess a Lie-Poisson structure on the Jacobi group. This structure produces an energy-conserving variant of a class of Gaussian moment models that have appeared in the chemical physics literature.
Let (M,g) be a compact, connected and oriented Riemannian manifold. We denote D the space of smooth probability density functions on M. In this paper, we show that the Frechet manifold D is equipped with a Riemannian metric g^{D} and an affine connection \nabla^{D} which are infinite dimensional analogues of the Fisher metric and exponential connection in the context of information geometry. More precisely, we use Dombrowski's construction together with the couple (g^{D},\nabla^{D}) to get a (non-integrable) almost Hermitian structure on D, and we show that the corresponding fundamental 2-form is a symplectic form from which it is possible to recover the usual Schrodinger equation for a quantum particle living in M. These results echo a recent paper of the author where it is stressed that the Fisher metric and exponential connection are related (via Dombrowski's construction) to Kahler geometry and quantum mechanics in finite dimension.
Starting from the axioms of quantum mechanics as formalized by the systems of imprimitivity for homogeneous Riemannian manifolds, the classical theory is derived as a consequence, complete with its phase space realized as the space of pure classical states, a generalized version of the Wigner–Moyal correspondence rule, the Jordan and Lie algebra structures of functions on the cotangent bundle, given by point-wise multiplication and Poisson bracket, and the momentum map. A comparison is also given of the quantum and classical dynamics and equilibrium statistical mechanics of free particles on compact manifolds of constant negative curvature.