arXiv subjects

Jesús Clemente-Gallardo

Publications and source records attributed to Jesús Clemente-Gallardo.

5 recordsLinked to original sources

arXiv2025

Alternative tangent and cotangent structures and their physical applications

The conditions under which a given manifold $M$ may be given a tangent bundle or a cotangent bundle structure are analyzed. This is an important property arising in different contexts. For instance, in the study of integrability of a given dynamics the existence of alternative compatible structures is very relevant, as well as in the geometric approach to Classical Mechanics. On the other hand in the quantum-to-classical transition, a Weyl system plays an important role for it provides (within the so-called Weyl-Wigner formalism) a description of quantum mechanics on a (symplectic) phase-space $M$. A Lagrangian subspace $Q\subset M$ of the (linear) phase space determines thus a maximal set of pairwise commuting unitary operators, which is used to parametrize the quantum states. As the choice of this maximal Abelian set of observables is not unique, the different choices make the phase space to become diffeomorphic to different cotangent bundles $T^*Q$ corresponding to different choices for the base manifold (and hence the fibers). These motivating ideas are used to study how to define alternative tangent and/or cotangent bundle structures on a phase space.

math-ph↗

arXiv2024

Geometric flavours of Quantum Field theory on a Cauchy hypersurface. Part I: Gaussian analysis and other Mathematical aspects

In this series of papers we aim to provide a mathematically comprehensive framework to the Hamiltonian pictures of quantum field theory in curved spacetimes. Our final goal is to study the kinematics and the dynamics of the theory from the point of differential geometry in infinite dimensions.In this first part we introduce the tools of Gaussian analysis in infinite dimensional spaces of distributions. These spaces will serve the basis to understand the Schrödinger and Holomorphic pictures, over arbitrary Cauchy hypersurfaces, using tools of Hida-Malliavin calculus. Here the Wiener-Ito decomposition theorem provides the QFT particle interpretation. Special emphasis is done in the applications to quantization of these tools in the second part of this paper. We devote a section to introduce Hida test functions as a notion of second quantized test functions. We also analyze of the ingredients of classical field theory modeled as distributions paving the way for quantization procedures that will be analyzed in the second part of this series.

math-ph↗

arXiv2024

Geometric flavours of Quantum Field theory on a Cauchy hypersurface. Part II: Methods of quantization and evolution

In this series of papers we aim to provide a mathematically comprehensive framework to the Hamiltonian pictures of quantum field theory in curved spacetimes. Our final goal is to study the kinematics and the dynamics of the theory from the point of differential geometry in infinite dimensions. In this second part we use the tools of Gaussian analysis in infinite dimensional spaces introduced in the first part to describe rigorously the procedures of geometric quantization in the space of Cauchy data of a scalar theory. This leads us to discuss and establish relations between different pictures of QFT. We also apply these tools to describe the geometrization of the space of pure states of quantum field theory as a Khäler manifold. We use this to derive an evolution equation that preserves the geometric structure and avoids norm losses in the evolution. This leads us to a modification of the Schrödinger equation via a quantum connection that we discuss and exemplify in a simple case.

math-ph↗

arXiv2003

Motion on Lie groups and its applications in Control Theory

The usefulness in control theory of the geometric theory of motion on Lie groups and homogeneous spaces will be shown. We quickly review some recent results concerning two methods to deal with these systems, namely, a generalization of the method proposed by Wei and Norman for linear systems, and a reduction procedure. This last method allows us to reduce the equation on a Lie group $G$ to that on a subgroup $H$, provided a particular solution of an associated problem in $G/H$ is known. These methods are shown to be very appropriate to deal with control systems on Lie groups and homogeneous spaces, through the specific examples of the planar rigid body with two oscillators and the front-wheel driven kinematic car.

math-ph↗

arXiv1999

Connes' Tangent Groupoid and Strict Quantization

We address one of the open problems in quantization theory recently listed by Rieffel. By developping in detail Connes' tangent groupoid principle and using previous work by Landsman, we show how to construct a strict, flabby quantization, which is moreover an asymptotic morphism and satisfies the reality and traciality constraints, on any oriented Riemannian manifold. That construction generalizes the standard Moyal rule. The paper can be considered as an introduction to quantization theory from Connes' point of view.

math.DG↗