Related papers: Quantum Probability, Renormalization and Infinite-…
The renormalization group is a tool that allows one to obtain a reduced description of systems with many degrees of freedom while preserving the relevant features. In the case of quantum systems, in particular, one-dimensional systems…
We deal with the general structure of (noncommutative) stochastic processes by using the standard techniques of Operator Algebras. Any stochastic process is associated to a state on a universal object, i.e. the free product $C^*$-algebra in…
We study a new class of infinite dimensional Lie algebras, which has important applications to the theory of integrable equations. The construction of these algebras is very similar to the one for automorphic functions and this motivates…
We develop a reformulation of the functional integral for bosons in terms of bilocal fields. Correlation functions correspond to quantum probabilities instead of probability amplitudes. Discrete and continuous global symmetries can be…
We discuss the consistency of the axioms which the definition of quantum Lie algebras is usually based on.
Quantum theory can be regarded as a non-commutative generalization of classical probability. From this point of view, one expects quantum dynamics to be analogous to classical conditional probabilities. In this paper, a variant of the…
We review the techniques used to renormalize quantum field theories at several loop orders. This includes the techniques to systematically extract the infinities in a Feynman integral and the implementation of the algorithm within computer…
The description of irreducible finite dimensional representations of finite dimensional solvable Lie superalgebras over complex numbers given by V.~Kac is refined. In reality these representations are not just induced from a polarization…
Methods for the reduction of the complexity of computational problems are presented, as well as their connections to renormalization, scaling, and irreversible statistical mechanics. Several statistically stationary cases are analyzed; for…
The article is devoted to some ``strange'' phenomena of representation theory and their interrelations. Cross-projective representations of pairs of anticommutative algebras, alloys, their universal envelopping Lie algebras and their…
The relation between renormalization and short distance singular divergencies in quantum field theory is studied. As a consequence a finite theory is presented. It is shown that these divergencies are originated by the multiplication of…
In this job, we will present a theory called Quantum Tomography that is the natural extension of the theory of detection of signals in classical telecommunications to Quantum Mechanics. This theory mainly consists in the reconstruction of a…
Quantum mechanics led to spectacular technological developments, discovery of new constituents of matter and new materials. However there is still no consensus on its interpretation and limitations. Some scientists and scientific writers…
The Lie algebras over the algebra of dual numbers are introduced and investigated.
For a certain class of Lie bialgebras $(A,A^*)$ the corresponding quantum universal enveloping algebras $U_q(A)$ are prooved to be equivalent to quantum groups Fun$_q(F^*)$, $F^*$ being the factor group for the dual group $G^*$. This…
Three extensions and reinterpretations of nonclassical probabilities are reviewed. (i) We propose to generalize the probability axiom of quantum mechanics to self-adjoint positive operators of trace one. Furthermore, we discuss the…
Self-normalized processes are basic to many probabilistic and statistical studies. They arise naturally in the the study of stochastic integrals, martingale inequalities and limit theorems, likelihood-based methods in hypothesis testing and…
Manifestly invariant renormalization scheme for supersymmetric gauge theories is proposed. This scheme is applied to supersymmetric quantum electrodynamics.
The renormalization method is specifically aimed at connecting theories describing physical processes at different length scales and thereby connecting different theories in the physical sciences. The renormalization method used today is…
Renormalization is a powerful technique in statistical physics to extract the large-scale behavior of interacting many-body models. These notes aim to give an introduction to perturbative methods that operate on the level of the stochastic…