Research
MAIN RESEARCH LINES
ANYONS AND QUANTUM PARTICLES WITH MAGNETIC FIELDS
(D. Cafiero, M.Correggi, D. Fermi)
Particles obeying to exotic fractional statistics (anyons) are known to be forbidden by the postulates of quantum mechanics in the usual three-dimensional world. However, as observed in fractional quantum Hall physics, two-dimensional quasi-particles, i.e., excitations of certain low-dimensional quantum systems, may carry a fractional charge and behave as anyons.
In a suitable representation (magnetic gauge), anyonic particles are described as conventional bosons carrying a singular magnetic flux of Aharonov-Bohm (AB) type in terms. The investigation of the well-posedness of such magnetic Schrödinger operators (as suitable self-adjoint realizations) as well as their spectral and scattering properties is highly non-trivial, even in the non-interacting case. We plan to address such questions in different physical settings.
CONSTRUCTIVE LATTICE QUANTUM FIELD THEORY
(M. Falconi)
The contemporary point of view in high-energy physics is that quantum field theories are only valid as effective theories, emerging at observable time scales from some underlying (and yet unknown) theory. It is therefore natural to consider theories on a discrete spacetime (in the form of a lattice), where the lattice spacing is small enough that the theory would give accurate predictions even at high-energy scales (e.g., the LHC scale). The presence of a discrete spacetime has both advantages - since the finite lattice spacing acts as an ultraviolet cutoff preserving many crucial symmetries of the system - and disadvantages - for example, it is unclear whether the knowledge of correlations functions in Euclidean spacetime would be sufficient to reconstruct the physical Minkowskian theory.
My aim, in collaboration with A. Giuliani (Roma Tre) and V. Mastropietro (UniMi), is to put lattice theories, and especially lattice gauge theories, on solid mathematical foundations. The natural starting point, focus of an ongoing collaboration, is Quantum ElectroDynamics on the lattice (LQED), that we are studying perturbatively using renormalization group methods. Further extensions to the non-perturbative regime, and to the ElectroWeak theory (LEW) are planned. Another topic of interest is the formulation of Wightman/Osterwalder-Schrader-like axioms that could encompass gauge theories on the lattice in a good way, and then to investigate the validity of the reconstruction and Wick-rotation theorems in this context.
DECOHERENCE AND MARKOVIANITY IN OPEN QUANTUM SYSTEMS
(M.Correggi, M. Falconi, M. Fantechi)
In considering quasi-classical systems of quantized particles interacting with semiclassical radiation, the quantum particles become an open system subjected to the action of the semiclassical radiation acting as a thermodynamic environment, itself evolving freely or being frozen in time.
We are studying the behavior of such quasi-classical open "small" quantum system, for its interaction with the environment is explicit, albeit involving operations that alter the unitary nature of an isolated quantum evolution. In particular, we are interested in understanding whether the small system exhibits decoherence, and when it does how such decoherence "interacts" with the semiclassical scaling of the environment. We are also interested in Markovianity: is the effective evolution of the small system still Markovian? If it is not (as general considerations suggest), could we measure its non-Markovianity, and compare it with similar results in different scaling regimes?
(M. Correggi, M. Moscolari)
To appear.
KINETIC THEORY OF GASES AND ITS APPLICATIONS TO RAREFIED GAS DYNAMICS
(M. Cantoni, S. Lorenzani)
The subject of rarefied gas dynamics can be defined as the study of gas flows in which the average value of the distance between two subsequent collisions of a molecule (the so-called mean free path) is not negligible in comparison with a length typical of the structure of the flow being considered. The field is thus seen to be one that intrinsically requires the use of statistical ideas typical of the kinetic theory of gases as embodied in the integro-differential equation proposed by Boltzmann in 1872 and bearing his name.
Our current research lines on this topic focus on the following main items.
Application of analytical and semi-analytical techniques to obtain approximate solutions of the Boltzmann equation (variational methods, integral equations, elementary solutions).
Investigation of gas rarefaction effects in Micro-Electro-Mechanical Systems (MEMS) devices. The micromachinery fabrication techniques have become more and more mature in the last few decades. In particular, the microelectromechanical system (MEMS) devices (inertial sensors, accelerometers, gyroscopes) developed rapidly and found many applications in microelectronics, medicine, biology, optics and other high technology fields. Micro- and nanodevices are often operated in gaseous environments (typically air), and thus their performances are affected by the gas around them. Since the smallest characteristic length of MEMS is comparable with (or smaller than) the mean free path of the gas molecules, the traditional computational fluid dynamics methods, based on the Euler or the Navier-Stokes equations, fail in predicting the flows related to these devices. Therefore, an accurate analysis of such microfluidic systems requires the solution of the Boltzmann equation.
MANY-BODY QUANTUM SYSTEMS
(A. Calignano, M.Correggi)
To appear.
(M.Correggi, F. Perani)
To appear.
(M. Cantoni, M.Correggi, S. Lorenzani)
To appear.
(A. Olgiati)
To appear.
MATHEMATICAL ASPECTS OF SOLID STATE PHYSICS
(M. Moscolari)
To appear.
QUANTUM FIELDS IN CURVED MANIFOLDS
(F. Belgiorno)
Quantum field theory (QFT) in curved spacetime is a generalization of standard QFT to the case of curved manifolds like the ones of General Relativity. Special attention has been devoted to quantum effects on black hole backgrounds and to analogous phenomena in condensed matter systems. The most interesting phenomenon is represented by black hole evaporation, discovered by S.W.Hawking in 1974. Hawking predicted that black holes are not black: as a consequence of quantum mechanical effects, they radiate. The event horizon generates pairs of quanta: one particle of each pair emerges into the external region whereas its partner falls into the singularity. The Hawking effect, in spite of its theoretical interest, cannot be measured for astrophysical black holes. Analogue gravity is a recent branch of physics whose original aim is to reproduce the kinematical conditions which are at the root of the Hawking effect and to confirm it experimentally in condensed matter systems (like e.g. water, Bose-Einstein condensates, dielectric media). In particular, in 1981 W.Unruh showed that there is a mathematical analogy between the behavior of classical and quantum fields in the vicinity of black hole horizons and sound waves in tran-sonic fluid flows and raised the possibility of doing experiments with these sonic analogues. Sonic analogs are based on the observation that sound waves in flowing fluids are (under appropriate conditions) governed by the same wave equation as a scalar field in a curved space-time. The acoustic horizon, which occurs if the velocity of the fluid exceeds the speed of sound within the liquid, acts on sound waves exactly as a black hole horizon does on, for example, scalar waves. Still, the presence of dispersion plays a fundamental role in the associated physics in actual lab systems, where nontrivial dispersion relations occur.
Our recent theoretical investigations were focused on the analogous Hawking effect in dielectric media, and our analysis involved quantum electrodynamics in dielectric media in presence of a traveling dielectric perturbation (which in experiments can be generated by a strong laser pulse). A blocking horizon is present under suitable conditions. A covariant version of the Hopfield model has been set up in order to take into account dispersion, and thermality of the emitted radiation has been analytically shown to be still present. We also studied solitonic solutions in presence of a nonlinear quartic term in the polarization field.
A general mathematical framework for a rigorous and complete analysis of the scattering process associated with the analogous Hawking effect has been provided: a general master equation which allows to study in an unified way many models occurring in the physical literature (Corley model, dielectrics, BEC, water) has been introduced. The horizon appears as a turning point of a fourth order ordinary differential equation which is obtained by separation of variables. Thermality has been shown in all the cases. Further developments focus also on the so-called subcritical case.
Further research lines:
Quantum effects are taken into account in the case of charged and rotating black holes. In particular, the loss of electric charge by an electrically charged black hole is related to the so-called Schwinger effect (1951) involving pair creation of charged particles by a sufficiently intense electric field.
SEMI- AND QUASI- CLASSICAL QUANTUM FIELD THEORY
(M.Correggi, M. Falconi, M. Fantechi, T. Pistillo)
In many physical observations, the interaction between matter and light can be understood by taking into account the quantum nature of matter, but neglecting the one of light. These mixed quantum-classical systems shall however emerge as an effective description of a microscopic, purely quantum, theory. Our effort is devoted towards the foundation of solid mathematical grounds to this fact.
More precisely, we study the spectral and dynamical properties of systems of quantum particles in interaction with a quantized force-carrying field, in a scaling regime in which the field is semiclassical. In the limit, the system partially decouples: the quantum particles are driven by the now classical force field, thus behaving as an open system, while the field itself evolves freely, thus acting as an environment. The ground state energy and minimizing sequences of the microscopic system converge to their respective counterparts in a nonlinear coupled quantum-classical minimization problem. The analysis is carried out using tools of semiclassical analysis for infinite-dimensional systems, that we developed specifically for this context, building on the seminal work by Z. Ammari and F. Nier.
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Another aspect of this line of research aims at obtaining a better understanding of classical electrodynamics, by considering it as the semiclassical limit of quantum electrodynamics. As it is well-known in physics, classical electrodynamics is ill-posed: point charges lead to instability of matter and the collapse of electrons into nuclei, while extended charges with small radii also lead to unphysical behavior such as runaway and non-causal solutions.
Our aim is to take advantage of the regularizing nature of quantum electrodynamics to define, through Bohr's correspondence principle, a well-posed classical electrodynamics rid of unphysical features for small - and hopefully point-like - charges. A first result in this direction has been obtained by M.F., together with Z. Ammari and F. Hiroshima, by proving well-posedness of the electrodynamics of extended and smooth enough charges by lifting the problem to QED and then exploiting the correspondence principle. Such well-posedness was unknown by purely classical means, and thus paves the way to interesting perspectives concerning the problem above.
SMOLUCHOWSKI KINETIC EQUATION AND ITS APPLICATIONS TO BIOMEDICAL RESEARCH
(S. Lorenzani)
The original system of evolution equations proposed by Smoluchowski (1917) was introduced to describe the binary coagulation of colloidal particles moving according to Brownian motions, and several additional physical processes have been subsequently incorporated into the model (fragmentation, condensation, influence of external fields). In our research, the Smoluchowski equation has been considered to model the aggregation and diffusion of β-amyloid peptide (Aβ) in the cerebral tissue, a process thought to be associated with the development of Alzheimer's disease. Starting from a model defined at a microscopic scale (the size of a single neuron), we have derived, using the homogenization theory, a macromodel in order to describe at the macroscopic level the effects of production and agglomeration of the Aβ in the brain affected by Alzheimer's disease.
SUPERFLUIDITY AND SUPERCONDUCTIVITY
(M.Correggi)
(M.Correggi, M. Falconi)