Quantum Fluctuations and Bosonization

Abstract

Statistical physics is dealing with systems characterized by the fact that they have many degrees of freedom. One of the main problems consists of finding procedures for the extraction of the relevant physical quantities out of these extremely complex systems. We are faced with the problem of finding relevant reduction procedures which map the complex systems onto a simpler, tractable model at the price of introducing elements of uncertainty. Therefore probability theory is the natural mathematical tool in statistical physics. Since the early days of statistical physics, in classical (Newtonian) physical systems it is natural to model the observables by a collection of random variables acting on a probability space. Kolmogorovian probability techniques and results are the main tools in the development of classical statistical physics. A random variable is usually considered to be a measurable function with an expectation (state) given by an integral with respect to a suitable probability measure. Alternatively, a random variable can also be viewed as a multiplication operator by the associated function. Different random variables commute as multiplication operators. For this reason we speak of a commutative probabilistic model.