Related papers: Dynamics of geodesics, and Maass cusp forms
We study the macroscopic dynamics of fermion and quantum-spin systems with long-range, or mean-field, interactions, which turns out to be equivalent to an intricate combination of classical and short-range quantum dynamics. In this paper we…
General relativity allows solutions exhibiting closed timelike curves. Time travel generates paradoxes and quantum mechanics generalizations were proposed to solve those paradoxes. The implications of self-consistent interactions on acausal…
We study the macroscopic dynamical properties of fermion and quantum-spin systems with long-range, or mean-field, interactions. The results obtained are far beyond previous ones and require the development of a mathematical framework to…
The non-Hermitian quadratic oscillator studied by Swanson is one of the popular $PT$-symmetric model systems. Here a full classical description of its dynamics is derived using recently developed metriplectic flow equations, which combine…
Quantum mechanics is formulated as a geometric theory on a Hilbert manifold. Images of charts on the manifold are allowed to belong to arbitrary Hilbert spaces of functions including spaces of generalized functions. Tensor equations in this…
We consider surfaces embedded in a Riemannian manifold of arbitrary dimension and prove that many aspects of their differential geometry can be expressed in terms of a Poisson algebraic structure on the space of smooth functions of the…
These notes describe some links between the group $\mathrm{SL}_2(\mathbb{R})$, the Heisenberg group and hypercomplex numbers---complex, dual and double numbers. Relations between quantum and classical mechanics are clarified in this…
Building on parallels between geometric quantum mechanics and classical mechanics, we explore an alternative basis for quantum thermodynamics that exploits the differential geometry of the underlying state space. We develop both…
We compare classical and quantum dynamics of a particle in the de Sitter spacetimes with different topologies to show that the result of quantization strongly depends on global properties of a classical system. We present essentially…
In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schroedinger and Heisenberg frameworks from this perspective and discuss how the momentum map associated to the…
We discuss new approaches to fundamental problems of mathematics and mathematical physics such as mathematical foundation of quantum field theory, the Riemann hypothesis, and construction of noncommutative algebraic geometry.
We study the classical properties of a supersymmetric system which is often used as a model for supersymmetric quantum mechanics. It is found that the classical dynamics of the bosonic as well as the fermionic degrees of freedom is fully…
We present a class of thermodynamic systems with constant thermodynamic curvature which, within the context of geometric approaches of thermodynamics, can be interpreted as constant thermodynamic interaction among their components. In…
We present a picture of Lagrangean mechanics, free of some unnatural features (such as complete divergences). As a byproduct, a completely natural U(1)-bundle over the phase space appears. The correspondence between classical and quantum…
Classical thermodynamics contains familiar geometric relations associated with cyclic processes, most notably the identification of mechanical work with the area enclosed by a trajectory in the $(P,V)$ plane. We show that the area laws for…
A direct comparison of quantum and classical dynamical systems can be accomplished through the use of distribution functions. This is useful for both fundamental investigations such as the nature of the quantum-classical transition as well…
We give an overview of the interplay between the behavior of high energy eigenfunctions of the Laplacian on a compact Riemannian manifold and the dynamical properties of the geodesic flow on that manifold. This includes the Quantum…
The investigation of quantum-classical correspondence may lead to gain a deeper understanding of the classical limit of quantum theory. We develop a quantum formalism on the basis of a linear-invariant theorem, which gives an exact…
The quantum dynamics of a subset of interacting bosons in a subspace of fixed particle number is described in terms of symmetrized many-particle states. A suitable partial trace operation over the von Neumann equation of an $N$-particle…
Motivated by the construction of spectral manifolds in noncommutative geometry, we introduce a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of scalar fields. This commutation relation…