arXiv subjects

David Martínez-Crespo

Publications and source records attributed to David Martínez-Crespo.

5 recordsLinked to original sources

arXiv2025

Entropy functionals and equilibrium states in mixed quantum-classical dynamics

The computational challenges posed by many-particle quantum systems are often overcome by mixed quantum-classical (MQC) models in which certain degrees of freedom are treated as classical while others are retained as quantum. One of the fundamental questions raised by this hybrid picture involves the characterization of the information associated to MQC systems. Based on the theory of dynamical invariants in Hamiltonian systems, here we propose a family of hybrid entropy functionals that consistently specialize to the usual Rényi and Shannon entropies. Upon considering the MQC Ehrenfest model for the dynamics of quantum and classical probabilities, we apply the hybrid Shannon entropy to characterize equilibrium configurations for simple Hamiltonians. The present construction also applies beyond Ehrenfest dynamics.

quant-ph↗

arXiv2025

Heisenberg dynamics of mixed quantum-classical systems

We consider the dynamics of interacting quantum and classical systems in the Heisenberg representation. Unlike the usual construction in standard quantum mechanics, mixed quantum-classical systems involve the interplay of unitary operators acting on the quantum observables and the Lagrangian trajectories sweeping the classical degrees of freedom. This interplay reflects an intricate structure which is made particularly challenging by the backreaction excerpted on the classical trajectories by the quantum degrees of freedom. While the backreaction is underestimated in the common Ehrenfest model, more recent methodologies succeed in capturing this important effect by resorting to Koopman wavefunctions in classical mechanics. Luckily, both Ehrenfest and Koopman models enjoy a variational framework which is exploited here to unfold the geometric structure underlying quantum-classical coupling. A special role is played by the action of the diffeomorphic Lagrangian paths on a non-Abelian pure-gauge potential which comprises statistical correlations. After presenting the treatment in the simple case of Ehrenfest dynamics, we move on to the Koopman model and present the role of the backreaction terms therein. Finally, we compare both models in the context of pure-dephasing systems.

physics.chem-ph↗

arXiv2025

Geometric Flavours of Quantum Field Theory on a Cauchy Hypersurface: Gaussian Analysis for the Hamiltonian Formalism and Applications to Cosmology

This thesis explores Quantum Field Theory (QFT) on curved spacetimes using a geometric Hamiltonian approach to the Schrödinger-like representation. In particular it studies the theory of the scalar field described through its configurations over a Cauchy hypersurface. It is focused on mathematical consistency based on analytic and geometric tools. The mathematical aspects of Gaussian integration theory in infinite-dimensional Topological Vector Spaces (TVS) are thoroughly reviewed. It also reviews the complex and holomorphic versions of important results and concepts of Gaussian integration. For example, the Wiener-Itô decomposition theorem or the definition of Hida test functions. The physical framework builds upon three interconnected levels: classical General Relativity (GR), Classical Statistical Field Theory (CSFT), and QFT. The work begins by extending the Koopman-van Hove (KvH) formalism of classical statistical mechanics to CSFT. This description is based upon prequantization theory. It reveals features inherent to both CSFT and QFT, that help delineate the genuine quantum features of a theory. Upon the prequantum program, the QFT of the scalar field is built mixing Geometric Quantization with the choice of Wick and Weyl orderings. Various quantum representations are introduced: the holomorphic, Schrödinger, field-momentum, and antiholomorphic. The relation among them is studied using integral transforms, including novel infinite-dimensional Fourier transforms. From a geometrical analysis, it is argued that a covariant time derivative that modifies the evolution equations should be added to the Schrödinger equation. This connection is unique and required by the geometrodynamical description ofthe coupling of QFT and GR. Finally, studying the free model on cosmological spacetimes it obtains particle creation effects on a dynamical equation.

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↗