Related papers: Quantisation of a Family of Phase Spaces
When there is a family of complex structures on the phase space, parametrized by a set $S$, the prequantum Hilbert spaces produced by geometric quantization, using the half-form correction, also depends on these parameters. This way we…
Smooth composite bundles provide the adequate geometric description of classical mechanics with time-dependent parameters. We show that the Berry's phase phenomenon is described in terms of connections on composite Hilbert space bundles.
An operator-valued quantum phase space formula is constructed. The phase space formula of Quantum Mechanics provides a natural link between first and second quantization, thus contributing to the understanding of quantization problem. By…
Hamiltonian quantization of an integral compact symplectic manifold M depends on a choice of compatible almost complex structure J. For open sets U in the set of compatible almost complex structures and small enough values of Planck's…
Consider a fiber bundle in which the total space, the base space and the fiber are all symplectic manifolds. We study the relations between the quantization of these spaces. In particular, we discuss the geometric quantization of a vector…
We show that using the family of adapted K\"ahler polarizations of the phase space of a compact, simply connected, Riemannian symmetric space of rank-1, the obtained field $H^{corr}$ of quantum Hilbert spaces produced by geometric…
We define a quantum generalization of the algebra of functions over an associated vector bundle of a principal bundle. Here the role of a quantum principal bundle is played by a Hopf-Galois extension. Smash products of an algebra times a…
Constructing a classical mechanical system associated with a given quantum mechanical one, entails construction of a classical phase space and a corresponding Hamiltonian function from the available quantum structures and a notion of…
The phase space of a compact, irreducible, simply connected, Riemannian symmetric space admits a natural family of K\"ahler polarizations parametrized by the upper half plane $S$. Using this family, geometric quantization, including the…
We relate Berezin-Toeplitz quantization of higher rank vector bundles to quantum-classical hybrid systems and quantization in stages of symplectic fibrations. We apply this picture to the analysis and geometry of vector bundles, including…
Taking several statistical examples, in particular one involving a choice of experiment, as points of departure, and making symmetry assumptions, the link towards quantum theory developed in Helland (2005a,b) is surveyed and clarified. The…
We study the statistical mechanics of a general Hamiltonian system in the context of symplectic structure of the corresponding phase space. This covariant formalism reveals some interesting correspondences between properties of the phase…
Quantization identifies the cotangent bundle of projective space with the (non-Hermitian) rank-$1$ projections of a Hilbert space. We use this identification to study the natural geometric structures of these cotangent bundles and those of…
We present a classification of hamiltonian vector fields on multisymplectic and polysymplectic fiber bundles closely analogous to the one known for the corresponding dual jet bundles that appear in the multisymplectic and polysymplectic…
We quantise complex, infinite-dimensional projective space CP(H). We apply the result to quantise a complex, finite-dimensional, classical phase space C whose symplectic volume is infinite, by holomorphically embedding it into CP(H). The…
In these notes we construct a quantization functor, associating an Hilbert space H(V) to a finite dimensional symplectic vector space V over a finite field F_q. As a result, we obtain a canonical model for the Weil representation of the…
We consider various trace formulas for the cubic Schrodinger equation in the space of infinitely smooth functions subject to periodic boundary conditions. The formulas relate conventional integrals of motion to the periods of some Abelian…
Phase Space is the framework best suited for quantizing superintegrable systems--systems with more conserved quantities than degrees of freedom. In this quantization method, the symmetry algebras of the hamiltonian invariants are preserved…
In the context of Covariant Quantum Mechanics for a spin particle, we classify the ``quantum vector fields'', i.e. the projectable Hermitian vector fields of a complex bundle of complex dimension 2 over spacetime. Indeed, we prove that the…
We investigate the most general "phase space" of configurations, consisting of all possible ways of assigning elementary attributes, "energies", to elementary positions, "cells". We discuss how this space possesses structures that can be…