Cirelli, Manià and Pizzocchero generalized quantum mechanics by Kähler geometry. Furthermore they proved that any unital C $^{*}$ -algebra is represented as a function algebra on the set of pure states with a noncommutative $*$ -product as an application. The ordinary quantum mechanics is regarded as a dynamical system of the projective Hilbert space ${\cal P}({\cal H})$ of a Hilbert space ${\cal H}$ . The space ${\cal P}({\cal H})$ is an infinite dimensional Kähler manifold of positive constant holomorphic sectional curvature. In general, such dynamical system is constructed for a general Kähler manifold of nonzero constant holomorphic sectional curvature $c$ . The Hilbert ball $B_{\cal H}$ is defined by the open unit ball in ${\cal H}$ and it is a Kähler manifold with $c<0$ . We introduce the quantum mechanics on $B_{\cal H}$ . As an application, we show the structure of the noncommutative function algebra on $B_{\cal H}$ .Cirelli, Manià and Pizzocchero generalized quantum mechanics by Kähler geometry. Furthermore they proved that any unital C$^{*}$-algebra is represented as a function algebra on the set of pure states with a noncommutative $*$-product as an application. The ordinary quantum mechanics is regarded as a dynamical system of the projective Hilbert space ${\cal P}({\cal H})$ of a Hilbert space ${\cal H}$. The space ${\cal P}({\cal H})$ is an infinite dimensional Kähler manifold of positive constant holomorphic sectional curvature. In general, such dynamical system is constructed for a general Kähler manifold of nonzero constant holomorphic sectional curvature $c$. The Hilbert ball $B_{\cal H}$ is defined by the open unit ball in ${\cal H}$ and it is a Kähler manifold with $c<0$. We introduce the quantum mechanics on $B_{\cal H}$. As an application, we show the structure of the noncommutative function algebra on $B_{\cal H}$.

arXiv:funct-an/9710002Oct 21, 19971 citation

From pure quantum states to quantum classical systems

Itai Carmeli

Jan 1, 20250 citations

Fibre bundle formulation of nonrelativistic quantum mechanics: I. Introduction. The evolution transport

Bozhidar Z Iliev

We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one and is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it the Hilbert space of a quantum system (from conventional quantum mechanics) is replaced with an appropriate Hilbert bundle of states and a pure state of the system is described by a lifting of paths or sections along paths in this bundle. The evolution of a pure state is determined through the bundle (analogue of the) Schrödinger equation. Now the dynamical variables and density operators are described via liftings of paths or morphisms along paths in suitable bundles. The mentioned quantities are connected by a number of relations derived in this paper. The present, first, part of this investigation is devoted to the introduction of basic concepts on which the fibre bundle approach to quantum mechanics rests. We show that the evolution of pure quantum mechanical states can be described as a suitable linear transport along paths, called evolution transport, of the state liftings in the Hilbert bundle of states of a considered quantum system.We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one and is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it the Hilbert space of a quantum system (from conventional quantum mechanics) is replaced with an appropriate Hilbert bundle of states and a pure state of the system is described by a lifting of paths or sections along paths in this bundle. The evolution of a pure state is determined through the bundle (analogue of the) Schrödinger equation. Now the dynamical variables and density operators are described via liftings of paths or morphisms along paths in suitable bundles. The mentioned quantities are connected by a number of relations derived in this paper. The present, first, part of this investigation is devoted to the introduction of basic concepts on which the fibre bundle approach to quantum mechanics rests. We show that the evolution of pure quantum mechanical states can be described as a suitable linear transport along paths, called evolution transport, of the state liftings in the Hilbert bundle of states of a considered quantum system.

arXiv:quant-ph/9803084Journal of Physics A Mathematical and GeneralMay 31, 200127 citations

Kähler quantization and entanglement

Tatyana Barron, Timothy Pollock

arXiv:1806.10226Reports on Mathematical PhysicsOct 1, 20173 citations

Entanglement of Mixed States in Kähler Quantization

Tatyana Barron, Alexander Kazachek

Springer proceedings in mathematics & statisticsJan 1, 20220 citations

Generalised Geometric Phase: Mathematical Aspects

Vivek M. Vyas

An operator generalisation of the notion of geometric phase has been recently proposed purely based on physical grounds. Here we provide a mathematical foundation for its existence, while uncovering new geometrical structures in quantum systems. While probing the average of any observable it is found that a quantum system exhibits different ray spaces and associated fibre bundle structures. The generalised geometric phase is understood as (an)holonomy of a connection over these fibre bundles. The underlying ray spaces in general are found to be pseudo-Kahler manifolds, and its symplectic structure gets manifests as the generalised geometric phase.An operator generalisation of the notion of geometric phase has been recently proposed purely based on physical grounds. Here we provide a mathematical foundation for its existence, while uncovering new geometrical structures in quantum systems. While probing the average of any observable it is found that a quantum system exhibits different ray spaces and associated fibre bundle structures. The generalised geometric phase is understood as (an)holonomy of a connection over these fibre bundles. The underlying ray spaces in general are found to be pseudo-Kahler manifolds, and its symplectic structure gets manifests as the generalised geometric phase.

arXiv:2312.14522Dec 22, 20230 citations

Cirelli, Manià and Pizzocchero generalized quantum mechanics by Kähler geometry. Furthermore they proved that any unital C

^{*}

-algebra is represented as a function algebra on the set of pure states with a noncommutative

*

-product as an application. The ordinary quantum mechanics is regarded as a dynamical system of the projective Hilbert space

{\cal P}({\cal H})

of a Hilbert space

{\cal H}

. The space

{\cal P}({\cal H})

is an infinite dimensional Kähler manifold of positive constant holomorphic sectional curvature. In general, such dynamical system is constructed for a general Kähler manifold of nonzero constant holomorphic sectional curvature

c

. The Hilbert ball

B_{\cal H}

is defined by the open unit ball in

{\cal H}

and it is a Kähler manifold with

c<0

. We introduce the quantum mechanics on

B_{\cal H}

. As an application, we show the structure of the noncommutative function algebra on

B_{\cal H}