The Theory of Quantum Levy Processes

Various recent results on quantum L\'evy processes are presented. The first part provides an introduction to the theory of L\'evy processes on involutive bialgebras. The notion of independence used for these processes is tensor independence, which generalizes the notion of independence used in classical probability and corresponds to independent observables in quantum physics. In quantum probability there exist other notions of independence and L\'evy processes can also be defined for the five so-called universal independences. This is the topic of the second part. In particular, it is shown that boolean, monotone, and anti-monotone independence can be reduced to tensor independence. Finally, in the third part, several classes of quantum L\'evy processes of special interest are considered, e.g., L\'evy processes on real Lie algebras or Brownian motions on braided spaces. Several applications of these processes are also presented.Comment: Habilitation thesis EMAU Greifswald, 204 page

Patterson-Sullivan distributions for symmetric spaces of the noncompact type

We generalize parts of a special non-Euclidean calculus of pseudodifferential operators, which was invented by S. Zelditch for hyperbolic surfaces, to symmetric spaces $X=G/K$ of the noncompact type and their compact quotients $Y=\Gamma\backslash G/K$. We sometimes restrict our results to the case of rank one symmetric spcaes. The non-Euclidean setting extends the defintion of so-called Patterson-Sullivan distributions, which were first defined by N. Anantharaman and S. Zelditch for hyperbolic systems, in a natural way to arbitrary symmetric spaces of the noncompact type. We find an explicit intertwining operator mapping Patterson-Sullivan distributions into Wigner distributions. We study the important invariance and equivariance properties of these distributions. Finally, we describe asymptotic properties of these distributions

Admissible invariant canonical quantizations of classical mechanics

Wydział Fizyki: Zakład Fizyki MatematycznejW pracy rozwijana jest niezmiennicza procedura kwantowania klasycznych układów hamiltonowskich. Procedura ta bazuje na teorii kwantyzacji deformacyjnej, która została użyta do wprowadzenia kwantyzacji w dowolnych współrzędnych kanonicznych, jak również w sposób niezależny od układu współrzędnych. Wprowadzona została dwu-parametrowa rodzina kwantyzacji, która odtwarza większość rezultatów otrzymanych różnymi podejściami do kwantyzacji spotykanymi w literaturze. W dalszej części pracy skonstruowana została operatorowa reprezentacja dla ogólnej kwantyzacji i dowolnych współrzędnych kanonicznych. Wprowadzona została również bardzo ogólna rodzina uporządkowań operatorów położenia i pędu. Pokazane jest, że różnym kwantowaniom i współrzędnym kanonicznym odpowiadają różne porządki. Ten fakt pozwolił na konstrukcję operatorowej reprezentacji mechaniki kwantowej w spójny sposób, dla dowolnych współrzędnych kanonicznych, jak i w sposób niezależny od układu współrzędnych. Na zakończenie, używając rozwijanego formalizmu, wprowadzony został kwantowy analog klasycznych trajektorii na przestrzeni fazowej.In the thesis is developed an invariant quantization procedure of classical Hamiltonian mechanics. The procedure is based on a deformation quantization theory, which is used to introduce quantization in arbitrary canonical coordinates as well as in a coordinate independent way. A two-parameter family of quantizations is introduced, which reproduces most of the results received by different approaches to quantization found in the literature.The operator representation of quantum mechanics is constructed for a general quantization and arbitrary canonical coordinates. A very general family of orderings of operators of position and momentum is introduced. It is shown that for different quantizations and canonical coordinates correspond different orderings. This fact allowed to construct an operator representation of quantum mechanics in a consistent way for any canonical coordinates as well as in a coordinate independent way. Finally, using the developed formalism, a quantum analog of classical trajectories in phase space is introduced

A random matrix theory approach to complex networks

Si presenta un approccio matematico formale ai complex networks tramite l'uso della Random Matrix Theory (RMT). La legge del semicerchio di Wigner viene presentata come una generalizzazione del Teorema del Limite Centrale per determinati ensemble di matrici random. Sono presentati inoltre i principali metodi per calcolare la distribuzione spettrale delle matrici random e se ne sottolineano le differenze. Si è poi studiato come la RMT sia collegata alla Free Probability. Si è studiato come due tipi di grafi random apparentemente uguali, posseggono proprietà spettrali differenti analizzando le loro matrici di adiacenza. Da questa analisi si deducono alcune proprietà geometriche e topologiche dei grafi e si può analizzare la correlazione statistica tra i vertici. Si è poi costruito sul grafo un passeggiata aleatoria tramite catene di Markov, definendo la matrice di transizione del processo tramite la matrice di adiacenza del network opportunamente normalizzata. Infine si è mostrato come il comportamento dinamico della passeggiata aleatoria sia profondamente connesso con gli autovalori della matrice di transizione, e le principali relazioni sono mostrate

Spectral and dynamical properties of disordered and noisy quantum spin models

This thesis, divided into two parts, is concerned with the analysis of spectral and dynamical characteristics of certain quantum spin systems in the presence of either I) quenched disorder, or II) dynamical noise. In the first part, the quantum random energy model (QREM), a mean-field spin glass model with a many-body localisation transition, is studied. In Chapter 2, we attempt a diagrammatic perturbative analysis of the QREM from the ergodic side, proceeding by analogy to the single-particle theory of weak localisation. Whilst we are able to describe diffusion, the analogy breaks down and a description of the onset of localisation in terms of quantum corrections quickly becomes intractable. Some progress is possible by deriving a quantum kinetic equation, namely the relaxation of the one-spin reduced density matrix is determined, but this affords little insight and extension to two-spin quantities is difficult. We change our approach in Chapter 3, studying instead a stroboscopic version of the model using the formalism of quantum graphs. Here, an analytic evaluation of the form factor in the diagonal approximation is possible, which we find to be consistent with the universal random matrix theory (RMT) result in the ergodic regime. In Chapter 4, we replace the QREM’s transverse field with a random kinetic term and present a diagrammatic calculation of the average density of states, exact in the large-N limit, and interpret the result in terms of the addition of freely independent random variables. In the second part, we turn our attention to noisy quantum spins. Chapter 5 is concerned with noninteracting spins coupled to a common stochastic field; correlations arising from the common noise relax only due to the spins’ differing precession frequencies. Our key result is a mapping of the equation of motion of n-spin correlators onto the (integrable) non-Hermitian Richardson-Gaudin model, enabling exact calculation of the relaxation rate of correlations. The second problem, addressed in Chapter 6, is that of the dynamics of operator moments in a noisy Heisenberg model; qualitatively different behaviour is found depending on whether or not the noise conserves a component of spin. In the case of nonconserving noise, we report that the evolution of the second moment maps onto the Fredrickson-Andersen model – a kinetically constrained model originally introduced to describe the glass transition. This facilitates a rigorous study of operator spreading in a continuous-time model, providing a complementary viewpoint to recent investigations of random unitary circuits.EPSR