arXiv subjects

Marco Falconi

Publications and source records attributed to Marco Falconi.

At least 19 recordsLinked to original sources

arXiv2026

Measuring a quantum system without problems

The process of measuring quantum observables has been plagued, since the inception of quantum mechanics, by the so-called measurement problem: it is impossible to read a definite outcome on a quantum scale. In its mathematical formulation, the problem takes the form of a no-go theorem, preventing the realization of quantum mechanical measurement schemes that comply both with the measurement and unitarity axioms. Building upon the idea of measurement that dates back to the early days of quantum mechanics, we overcome this century-old problem by proving the existence of a measurement scheme in which the probe (a quantized field) undergoes a quantifiable semiclassical transition, thus allowing, within error limits, for both an objective readout of the measuring device and the compliance with measurement axioms.

math-ph↗

arXiv2026

Wave Function Renormalization for Particle-Field Interactions

In this paper, we develop a wave function renormalization scheme for models of non-relativistic quantum particles interacting with a quantized relativistic field, in the Hamiltonian formalism of quantum field theory. We construct the interacting Hamilton operator, in its ground state representation, for a large class of particle-field interactions; hereby addressing a number of open problems related to both ultraviolet and infrared singularities in the spin-boson and Nelson models, where the infinite wave function renormalization plays a fundamental role.

math-ph↗

arXiv2025

Semiclassical limit of entropies and free energies

Entropy and free energy are central concepts in both statistical physics and information theory, with quantum and classical facets. In mathematics these concepts appear quite often in different contexts (dynamical systems, probability theory, von Neumann algebras, etc.). In this work, we study the von Neumann and Wehrl entropies from the point of view of semiclassical analysis. We first prove the semiclassical convergence of the von Neumann to the Wehrl entropy for quantum Gibbs states (thermal equilibrium), after a suitable renormalization has been taken into account. Then, we show that, in the same limit, the free energy functional defined with the Wehrl entropy $ \Gamma-$converges to its classical counterpart, so implying convergence of the minima and the associated minimizers.

math-ph↗

arXiv2025

Non-Trivial Renormalization of Spin-Boson Models with Supercritical Form Factors

In this paper we construct the non-trivial, renormalized Hamiltonian for a class of spin-boson models with supercritical form factors, including the one describing the Weisskopf-Wigner spontaneous emission. The renormalization is performed through both a self-energy and mass renormalization, in the so-called Hamiltonian formalism of constructive quantum field theory, implemented by a non-unitary dressing transformation. This solves the problem of triviality for unitarily-renormalized supercritical spin-boson models.

math-ph↗

arXiv2025

Ultraviolet Renormalization of the van Hove-Miyatake Model: an Algebraic and Hamiltonian Approach

In this short communication we discuss the ultraviolet renormalization of the van Hove-Miyatake scalar field, generated by any distributional source. An abstract algebraic approach, based on the study of a special class of ground states of the van Hove-Miyatake dynamical map is compared with an Hamiltonian renormalization that makes use of a non-unitary dressing transformation. The two approaches are proved to yield equivalent results.

math-ph↗

arXiv2024

Quasi-classical Limit of a Spin Coupled to a Reservoir

A spin (qubit) is in contact with a bosonic reservoir. The state of the reservoir contains a parameter {\varepsilon} interpolating between quantum and classical reservoir features. We derive the explicit expression for the time-dependent reduced spin density matrix, valid for all values of {\varepsilon} and for energy conserving interactions. We study decoherence and markovianity properties. Our main finding is that the spin decoherence is enhanced (full decoherence) when the spin is coupled to quantum reservoir states while it is dampened (partial decoherence) when coupled to classical reservoir states. The markovianity properties depend in a subtle way on the classicality parameter {\varepsilon} and on the finer details of the spin-reservoir interaction. We further examine scattering and periodicity properties for energy exchange interactions.

quant-ph↗

arXiv2024

Abstract semiclassical analysis of the van Hove model

In this paper we study the semiclassical limit $\hslash\to 0$ of a completely solvable model in quantum field theory: the van Hove model, describing a scalar field created and annihilated by an immovable source. Despite its simplicity, the van Hove model possesses many characterizing features of quantum fields, especially in the infrared region. In particular, the existence of non-Fock ground and equilibrium states in the presence of infrared singular sources makes a representation-independent algebraic approach of utmost importance. We make use of recent representation-independent techniques of infinite dimensional semiclassical analysis to establish the Bohr correspondence principle for the dynamics, equilibrium states, and long-time asymptotics in the van Hove model.

math-ph↗

arXiv2023

Convergence of states for polaron models in the classical limit

We consider the quasi-classical limit of Nelson-type regularized polaron models describing a particle interacting with a quantized bosonic field. We break translation-invariance by adding an attractive external potential decaying at infinity, acting on the particle. In the strong coupling limit where the field behaves classically we prove that the model's energy quasi-minimizers strongly converge to ground states of the limiting Pekar-like non-linear model. This holds for arbitrarily small external attractive potentials, hence this binding is fully due to the interaction with the bosonic field. We use a new approach to the construction of quasi-classical measures to revisit energy convergence, and a localization method in a concentration-compactness type argument to obtain convergence of states.

math.AP↗

arXiv2023

Renormalized Bogoliubov Theory for the Nelson Model

We consider the time evolution of the renormalized Nelson model, which describes $N$ bosons linearly coupled to a quantized scalar field, in the mean-field limit of many particles $N\gg 1$ with coupling constant proportional to $N^{-1/2}$. First, we show that initial states exhibiting Bose-Einstein condensation for the particles and approximating a coherent state for the quantum field retain their structure under the many-body time evolution. Concretely, the dynamics of the reduced densities are approximated by solutions of two coupled PDEs, the Schrödinger-Klein-Gordon equations. Second, we construct a renormalized Bogoliubov evolution that describes the quantum fluctuations around the Schrödinger-Klein-Gordon equations. This evolution is used to extend the approximation of the evolved many-body state to the full norm topology. In summary, we provide a comprehensive analysis of the Nelson model that reveals the role of renormalization in the mean-field Bogoliubov theory.

math-ph↗

arXiv2022

Quasi-Classical Spin-Boson Models

In this short note we study Spin-Boson Models from the Quasi-Classical standpoint. In the Quasi-Classical limit, the field becomes macroscopic while the particles it interacts with, they remain quantum. As a result, the field becomes a classical environment that drives the particle system with an explicit effective dynamics.

math-ph↗

arXiv2022

Towards a derivation of Classical ElectroDynamics of charges and fields from QED

The purpose of this article is twofold. On one hand, we rigorously derive the Newton--Maxwell equation in the Coulomb gauge from first principles of quantum electrodynamics in agreement with the formal Bohr's correspondence principle of quantum mechanics. On the other hand, we establish the global well-posedness of the Newton--Maxwell system on energy-spaces under weak assumptions on the charge distribution. Both results improve the state of the art, and are obtained by incorporating semiclassical and measure theoretical techniques. One of the novelties is the use of quantum propagation properties in order to build global solutions of the Newton--Maxwell equation.

math.AP↗

arXiv2022

Derivation of the Maxwell-Schrödinger Equations: A note on the infrared sector of the radiation field

We slightly extend prior results about the derivation of the Maxwell-Schrödinger equations from the bosonic Pauli-Fierz Hamiltonian. More concretely, we show that the findings from [SIAM J. Math. Anal. 52(5), 4900-4936 (2020)] about the coherence of the quantized electromagnetic field also hold for soft photons with small energies. This is achieved with the help of an estimate from [arXiv:2202.05015] which proves that the domain of the number of photon operator is invariant during the time evolution generated by the Pauli-Fierz Hamiltonian.

math-ph↗

arXiv2021

Semiclassical analysis of quantum asymptotic fields in the Yukawa theory

In this article, we study the asymptotic fields of the Yukawa particle-field model of quantum physics, in the semiclassical regime $\hslash\to 0$, with an interaction subject to an ultraviolet cutoff. We show that the transition amplitudes between final (respectively initial) states converge towards explicit quantities involving the outgoing (respectively incoming) wave operators of the nonlinear Schrödinger-Klein-Gordon (S-KG) equation. Thus, we rigorously link the scattering theory of the Yukawa model to that of the Schrödinger-Klein-Gordon equation. Moreover, we prove that the asymptotic vacuum states of the Yukawa model have a phase space concentration property around classical radiationless solutions. Under further assumptions, we show that the S-KG energy admits a unique minimizer modulo symmetries and identify exactly the semiclassical measure of Yukawa ground states. Some additional consequences of asymptotic completeness are also discussed, and some further open questions are raised.

math-ph↗

arXiv2021

Bogoliubov Dynamics and Higher-order Corrections for the Regularized Nelson Model

We study the time evolution of the Nelson model in a mean-field limit in which N non-relativistic bosons weakly couple (w.r.t. the particle number) to a positive or zero mass quantized scalar field. Our main result is the derivation of the Bogoliubov dynamics and higher-order corrections. More precisely, we prove the convergence of the approximate wave function to the many-body wave function in norm, with a convergence rate proportional to the number of corrections taken into account in the approximation. We prove an analogous result for the unitary propagator. As an application, we derive a simple system of PDEs describing the time evolution of the first- and second-order approximation to the one-particle reduced density matrices of the particles and the quantum field, respectively.

math-ph↗

arXiv2020

On some rigorous aspects of fragmented condensation

In this paper we discuss some aspects of fragmented condensation from a mathematical perspective. We first propose a simple way of characterizing finite fragmentation. Then, inspired by recent results of semiclassical analysis applied to bosonic systems with infinitely many degrees of freedom, we address the problem of persistence of fragmented condensation. We show that the latter occurs in interacting systems, in the mean-field regime, and in the limit of large gap of the one-body Hamiltonian.

cond-mat.quant-gas↗

arXiv2020

Ground State Properties in the Quasi-Classical Regime

We study the ground state energy and ground states of systems coupling non-relativistic quantum particles and force-carrying Bose fields, such as radiation, in the quasi-classical approximation. The latter is very useful whenever the force-carrying field has a very large number of excitations,and thus behaves in a semiclassical way, while the non-relativistic particles, on the other hand, retain their microscopic features. We prove that the ground state energy of the fully microscopic model converges to the one of a nonlinear quasi-classical functional depending on both the particles' wave function and the classical configuration of the field. Equivalently, this energy can be interpreted as the lowest energy of a Pekar-like functional with an effective nonlinear interaction for the particles only. If the particles are confined, the ground state of the microscopic system converges as well, to a probability measure concentrated on the set of minimizers of the quasi-classical energy.

math-ph↗

arXiv2020

The dilute Fermi gas via Bogoliubov theory

We study the ground state properties of interacting Fermi gases in the dilute regime, in three dimensions. We compute the ground state energy of the system, for positive interaction potentials. We recover a well-known expression for the ground state energy at second order in the particle density, which depends on the interaction potential only via its scattering length. The first proof of this result has been given by Lieb, Seiringer and Solovej. In this paper we give a new derivation of this formula, using a different method; it is inspired by Bogoliubov theory, and it makes use of the almost-bosonic nature of the low-energy excitations of the systems. With respect to previous work, our result applies to a more regular class of interaction potentials, but it comes with improved error estimates on the ground state energy asymptotics in the density.

math-ph↗

arXiv2018

Cylindrical Wigner measures

In this paper we study the semiclassical behavior of quantum states acting on the C*-algebra of canonical commutation relations, from a general perspective. The aim is to provide a unified and flexible approach to the semiclassical analysis of bosonic systems. We also give a detailed overview of possible applications of this approach to mathematical problems of both axiomatic relativistic quantum field theories and nonrelativistic many body systems. If the theory has infinitely many degrees of freedom, the set of Wigner measures, i.e. the classical counterpart of the set of quantum states, coincides with the set of all cylindrical measures acting on the algebraic dual of the space of test functions for the field, and this reveals a very rich semiclassical structure compared to the finite-dimensional case. We characterize the cylindrical Wigner measures and the \emph{a priori} properties they inherit from the corresponding quantum states.

math.FA↗