Vittorio Gorini

Completely positive dynamical semigroups of N‐level systems

We establish the general form of the generator of a completely positive dynamical semigroup of an N‐level quantum system, and we apply the result to derive explicit inequalities among the physical parameters characterizing the Markovian evolution of a 2‐level system.

Properties of Quantum Markovian Master Equations

In this paper we give an essentially self-contained account of some general structural properties of the dynamics of quantum open Markovian systems. We review some recent results regarding the problem of the classification of quantum Markovian master equations and the limiting conditions under which the dynamical evolution of a quantum open system obeys an exact semigroup law (weak coupling limit and singular coupling limit). We discuss a general form of quantum detailed balance and its relation to thermal relaxation and to microreversibility.

Quantum detailed balance and KMS condition

A definition of detailed balance for quantum dynamical semigroups is given, and its close connection with the KMS condition is investigated.

Fundamental aspects of quantum theory

General Problems and Crucial Experiments.- Are Coherent States the Natural Language of Quantum Theory?.- The Classical Behaviour of Measuring Instruments.- A Continuous Superselection Rule as a Model of Classical Measuring Apparatus in Quantum Mechanics.- Quantum Interference Effect for Two Atoms Radiating a Single Photon.- Neutron Interferometry and Quantum Mechanics.- An Attempt at a Unified Description of Microscopic and Macroscopic Systems.- On the Quantum Theory of Continuous Measurements.- Conditional Expectations on Jordan Algebras.- Quantization and Stochastic Processes.- Stochastic Quantization of Gauge Fields and Constrained Systems.- Brownian Motion, Markov Cosurfaces, Higgs Fields.- Stochastic Description of Supersymmetric Fields with Values in a Manifold.- Quantum Stochastic Calculus in Fock space: A Review.- Quantum Markov Processes Driven by Bose Noise.- Stochastic Interpretation of Emission and Absorption of the Quantum of Action.- Probabilistic Expression for the Solution of the Dirac Equation in Fourier Space.- Are Dirac Electrons Faster than Light?.- Sojourn Times and First Hitting Times in Stochastic Mechanics.- Chaotic Behaviour in Quantum Mechanics.- Quantum Systems Periodically Perturbed in Time.- Chaotic Ionization of Highly-Excited Hydrogen Atoms by a Microwave Electric Field.- Possible Evidence of Deterministic Chaos for the Sinusoidally-Driven Weakly-Bound Electron.- Quantization of Non-Integrable Systems: the Hydrogen Atom in a Magnetic Field.- Quantum Ergodicity and Chaos.- Microscopic and Macroscopic Levels of Description New Techniques and Results.- Some Fundamental Properties of the Ground State of Atoms and Molecules.- Mass/Energy Gap Associated to Symmetry Breaking: A Generalized Goldstone Theorem for Long Range Interactions.- Embedding as a Description of the Relation Between Macro- and Microphysics.- Some Applications of Semigroups.- N-Level Systems Interacting with Bosons: Semiclassical Limits.- Bose-Einstein Condensation in Some Interacting Systems.- General Aspects of Gauge Theories.- Charge, Anomalies and Index Theory.- Adiabatic Phase Shifts for Neutrons and Photons.- The Gauge Principle in Modern Physics.- Infinite Dimensional Lie Algebras and Quantum Physics.- The Spins of Cyons and Dyons.- Beyond the Hall Effect: Pratical Engineering from Relativistic Quantum Field Theory.- Generalized Aharonov-Bohm Experiments with Neutrons.- Round-Table Discussion on the Aharonov-Bohm Effect.- The Aharonov-Bohm Effect is Real Physics not Ideal Physics.- Again About an Old Stuff: The Aharonov-Bohm Effect.- Against the Existence of the Aharonov-Bohm Effect.- Theories Without AB Effect Misrepresent the Dynamics of the Electromagnetic Field.- Some Cases of the Aharonov-Bohm Effect: Electron Scattering on Magnetic Strings.- Global Gauge Invariance in Two-Dimensional Quantum Mechanics.- Gravity in Quantum Mechanics.- Stability of Matter.- Ihe Gravitational Phase Transition.- Gravitational and Rotational Effects on Superconductors.- Quantum Fields on Manifolds: an Interplay Between Quantum Theory, Statistical Thermodynamics and General Relativity.- Quantum Field Theory in Gravitational Background.- Classical Scattering Theory on the Schwarzschild Metric and the Construction of Quantum Linear Fields on Black Holes.- The Stochastic versus the Euclidean Approach to Quantum Fields on a Static Space-Time.- Quantum Theory in Vector Bundles.- Anomalies and Their Cancellation.- Remarks about Metric Tensors on Fractal Structures.- Short Communications.- A New Gauge Without Any Ghost for Yang-Mills Theory.- Unitary Formalism for Time-Dependent Problems.- Finite Temperature Quantum Electrical Network Theory.- Two Remarks on the Physical Content of Stochastic Mechanics.- The LambShift for Resonances: Complex Dilations and Coupling Constant Thresholds in Relativistic Quantum Mechanics.- Energy Density and Roughening in the 3-D Ising Ferromagnet.- Bose-Einstein Condensation of Free Photons.- Variational Principle in Quantum Mechanics.- Geometric Quantum Mechanics.- Spectral Sum Rules for Confinement Potentials.- Metrology of Space-Time Dimension.- Chronological Disordering and the Absence of Correlations between Infinitely Separated States.- On Solutions of Quantum Stochastic Integral Equations.

RECIPROCITY PRINCIPLE AND THE LORENTZ TRANSFORMATIONS.

By using the principle of relativity, together with the customary assumptions concerning the nature of the space‐time manifold in special relativity, namely, space‐time homogeneity and isotropy of space, a simple but rigorous proof is given of the reciprocity relation for the relative motion of two inertial frames of reference, which is usually assumed as a postulate in the standard derivations of the Lorentz transformations without the principle of invariance of light velocity. A critical discussion is set forth of the question of eliminating the transformations with invariant imaginary velocity, which one unavoidably obtains together with the Lorentz transformations and the Galilean ones in adopting a procedure of this kind.

Resonances, scattering theory, and rigged Hilbert spaces

The problem of decaying states and resonances is examined within the framework of scattering theory in a rigged Hilbert space formalism. The stationary free,’’in,’’ and ’’out’’ eigenvectors of formal scattering theory, which have a rigorous setting in rigged Hilbert space, are considered to be analytic functions of the energy eigenvalue. The value of these analytic functions at any point of regularity, real or complex, is an eigenvector with eigenvalue equal to the position of the point. The poles of the eigenvector families give origin to other eigenvectors of the Hamiltonian: the singularities of the ’’out’’ eigenvector family are the same as those of the continued S matrix, so that resonances are seen as eigenvectors of the Hamiltonian with eigenvalue equal to their location in the complex energy plane. Cauchy theorem then provides for expansions in terms of ’’complete’’ sets of eigenvectors with complex eigenvalues of the Hamiltonian. Applying such expansions to the survival amplitude of a decaying state, one finds that resonances give discrete contributions with purely exponential time behavior; the background is of course present, but explicitly separated. The resolvent of the Hamiltonian, restricted to the nuclear space appearing in the rigged Hilbert space, can be continued across the absolutely continuous spectrum; the singularities of the continuation are the same as those of the ’’out’’ eigenvectors. The free, ’’in’’ and ’’out’’ eigenvectors with complex eigenvalues and those corresponding to resonances can be approximated by physical vectors in the Hilbert space, as plane waves can. The need for having some further physical information in addition to the specification of the total Hamiltonian is apparent in the proposed framework. The formalism is applied to the Lee–Friedrichs model and to the scattering of a spinless particle by a local central potential.

Quantum Zeno dynamics

The evolution of a quantum system undergoing very frequent measurements takes place in a proper subspace of the total Hilbert space (quantum Zeno effect). The dynamical properties of this evolution are investigated and several examples are considered.

N‐level systems in contact with a singular reservoir. II

We study a model of an N‐level atom coupled linearly to an infinite free boson bath whose time correlation functions are Gaussian. We prove that, in the limit when the decay time of the correlations of the bath goes to zero, the reduced dynamics of the atom is given by a completely positive dynamical semigroup. By varying the interaction parameters, any such semigroup can be obtained in the limit. We also discuss the formally analogous situation of an N‐level atom whose Hamiltonian contains an external fluctuating Gaussian stationary contribution.

Linear kinematical groups

We prove a theorem which states that in an (n+1)-dimensional space-time (n≧3) the only linear kinematical groups which are compatible with the isotropy of space are the Lorentz and Galilei groups. The special casesn=1 andn=2 are also briefly discussed.

Extreme affine transformations

We classify the extreme points of the compact convex set of affine maps of IRn which map into itself the closed unit ball. This work is a preliminary step towards solving the problem of finding the extreme points of the compact convex set of affine maps of theN×N density matrices (dynamical maps of anN-level system) and forn=3 furnishes the solution of the problem in the simplest case of a two-level system.