Algebraic Methods In Statistical Mechanics And Quantum Field Theory Pdf

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On Quantum Statistical Mechanics: A Study Guide

We provide an introduction to a study of applications of noncommutative calculus to quantum statistical physics. Centered on noncommutative calculus, we describe the physical concepts and mathematical structures appearing in the analysis of large quantum systems and their consequences. These include the emergence of algebraic approach and the necessity of employment of infinite-dimensional structures.

As an illustration, a quantization of stochastic processes, new formalism for statistical mechanics, quantum field theory, and quantum correlations are discussed. In this paper, we will try to give an overview and road map to the area of quantum statistical mechanics without becoming too diverted by details.

In contrast, we put a strong emphasis on evolution of calculus which is used in the description of statistical mechanics. To make our exposition abundantly clear, we begin with a historical remark. Newton has given his principles for classical mechanics at the end of the 17th century. However, classical mechanics blossomed into a rich mathematical theory only in the second half of the 19th century. Consequently, in the second half of the 19th century, the principles of classical calculus were fully established.

This gave the opportunity to transform classical mechanics into a well-developed theory Lagrange, Hamilton, Liouville, etc. So with a mature theory of calculus available, it took a few more decades to obtain a fully fledged theory of classical mechanics.

Subsequently, classical statistical mechanics has appeared as a combined development of classical mechanics and probability theory. We will show that similar situation occurred also in the 20th century but in the context of quantum theory. He for the first time wrote a noncommutative derivation, a commutator. We recall that a derivation is a unary function satisfying the Leibniz product law. To see this, it is enough to note that a commutator satisfies the Leibniz rule!

It means that the basic relations of classical mechanics, should be replaced by where stands for the Poisson bracket, while denotes the commutator. But the quantization procedure begs two serious questions: 1 In which terms can relations 2 be represented? A brief answer to the first question says that relations 2 have no finite dimensional realization.

Although mathematical aspects of algebras of unbounded operators have been analyzed in many details see [ 11 — 13 ] , it is well known that formal calculations can be misleading see Section in [ 14 ].

Generally, it would seem that in quantum mechanics one can distinguish two schemes for a description of a physical system cf. The first one, just described, uses unbounded operators.

The second one uses bounded operators. The idea of introducing the norm topology on the set of observables was strongly advocated by Segal [ 16 ].

To argue in favor of this idea, one can say that in a laboratory a physicist deals with bounded functions of observables only! Here, we will argue that noncommutative integration theory offers the third scheme lying between the above discussed approaches. Consequently, as it will be described, one is getting a very well behaved -algebra of unbounded operators.

Moreover, bounded functions of self-adjoint elements of this algebra are elements of certain algebra of bounded operators. Turning to the second question, we should recall the so-called uniqueness theorem, attributed to von Neumann, Weyl, and Rellich. This theorem says that the answer to the second question takes into account the nature of the considered system. More precisely, a system will be called small if it has finite number of degrees of freedom. On the contrary, a system with an infinite number of degrees of freedom is called a large system.

The uniqueness theorem states that, for small systems, relations 2 , up to unitary equivalence, have a unique representation. We recall that the basis of that formalism is the pair where denotes all bounded linear operators on a separable, infinite-dimensional Hilbert space. In particular, density matrices describing quantum states form a convex generating subset of.

For large systems, the situation is very different. There are plenty of nonequivalent representations of relations 2 when the number of degrees of freedom is infinite. The crucial point to note here is that both statistical mechanics and field theory are par excellence theories of large systems! To give illustrative examples, we firstly mention problems associated with the Fock representation. The Fock representation was introduced in and subsequently fully elaborated by Cook in It is probably the best known scheme for a description of infinite quantum systems.

But, within this representation, one is able to describe only quasi-free systems. In other words, we cannot describe interacting particles. Furthermore, as was shown by van Hove in the fifties [ 18 , 19 ], see also subsection in [ 4 ] , there does not exist a nontrivial perturbation calculus within the Fock representation. Turning to the second example, we wish to discuss the quantum Gibbs Ansatz. The Gibbs Ansatz was designed to describe a classical canonical equilibrium state and, up to normalized constant, is given by.

We emphasize that this is the basic ingredient of classical statistical physics. The quantization of means that now is the Hamiltonian operator and to have a quantum state, we require that should be a trace class operator. But this is the case when, at least, necessary conditions are satisfied: has pure point spectrum with accumulation point at infinity. Unfortunately, even Hamiltonians of harmonic oscillators and the Hydrogen atom do not fulfill this requirement!

Consequently, we arrived at the conclusion that, in accordance with the second part of the non uniqueness theorem, one should take as a starting point algebraic structures which are different from. Before proceeding further, let us pause to describe briefly possible algebras other than which could be useful for a description of a quantum large system.

We start with the notion of -Banach algebra. It is a Banach space equipped with multiplication and involution. Both operations are continuous with respect to the topology induced by the norm. If , for , then is called commutative. When the norm satisfies the extra condition, , then such a -Banach algebra is called a -algebra and will be denoted by.

A von Neumann algebra is a concrete -algebra so for a Hilbert space which is closed with respect to the weak operator topology. The important point to note here is that every commutative von Neumann algebra is isomorphic to for some measure space and, conversely, for every -finite measure space , the -algebra is a von Neumann algebra. Here, stands for all essentially bounded functions on. Consequently, noncommutative von Neumann algebras provide nice starting point for the theory of noncommutative integration.

We complete this brief list of algebraic structures with the definition of -algebra. The multiplication in is composition of operators while the involution in is defined by , where is the usual Hilbert space adjoint.

In the thirties of the last century, von Neumann and Murray gave a classification of von Neumann algebras. To describe this classification, we, first of all, recall the definition of the center of the algebra : is called a factor if.

This decomposition is essentially unique. Therefore, to give the aforesaid classification of von Neumann algebras, one can restrict oneself to factors. One can distinguish three types of factors. The first type, denoted by I, consists of algebras of all linear bounded operators on a Hilbert space. If , then one gets , the algebra of matrices with complex entries. Such factors are denoted by.

These algebras are equipped with the canonical trace , that is, a partially defined positive, linear functional, such that for any. The second type, denoted by II, roughly speaking, consists of algebras such that their projections are of a specific type; more precisely, there are no minimal projections, but there are nonzero finite projections. Types I and II are called semifinite.

Such algebras have the important common property that they can be equipped with a trace. We emphasize that a given trace on a semifinite algebra can be different from the canonical one which was described for algebras of type I. Finally, there are also type III factors. The important property of these factors is that they cannot be equipped with a nontrivial trace see, e.

For a deeper discussion, we refer the reader to [ 6 ]. For a long time, type III algebras were, especially in mathematical physics, considered as exotic ones. But, in , this point of view was completely abandoned. In his work, Powers [ 21 ] was studying representations of uniformly hyperfinite algebras.

Such a model consists of an infinite number of sites, with the algebra associated with each site. Thus, local observables associated with a site are given by elements from. Local equilibrium at each site is given by a matrix of the form , where is the normalizing constant, and is the local Hamiltonian associated with a site.

Studying the thermodynamical limit of the above system, Powers has shown that, for , the equilibrium representations lead to type III of von Neumann algebras.

Moreover, if , one gets nonequivalent type III factors. The subsequent results obtained by Araki-Woods, Hugenholtz et al. We emphasize that this is in perfect harmony with the second part of the non uniqueness theorem; quantization of large systems leads to different algebras than!

As it was mentioned at the beginning, the precise description of limit and integral in classical calculus was steering the development of classical mechanics as well as statistical mechanics. Here, we wish to describe the analogous process but now for the quantum theory. In late thirties of the last century, von Neumann realized that noncommutative integration should play an essential role in quantum theory.

To start with, he proposed to carry out noncommutative integration by using tricky norms defined on matrix algebras see [ 23 ]. But the essential step was independently done by Segal [ 24 ] and Dixmier [ 25 ] in the early fifties. They generalized the concept of integration to much more general algebras. For semifinite von Neumann algebras, the theory of noncommutative integration was completed by Nelson in see [ 26 ]. It is very important to note that as a first step it was necessary to define the concept of noncommutative measurable operators quantum counterpart of measurable functions.

To this end, the concept of trace is necessary.

Methods of Quantum Field Theory in Statistical Physics

A scheme for constructing quantum mechanics not based on the Hilbert space and linear operators as primary elements of the theory is proposed. A particular variant of the algebraic approach is discussed. The elements of a noncommutative algebra i. The functionals are associated with the results of a single measurement. The ensembles of physical states are suggested for the role of quantum states in the standard quantum mechanics. It is shown that the mathematical formalism of the standard quantum mechanics can be fully recovered within this scheme. This is a preview of subscription content, access via your institution.

Algebraic methods in statistical mechanics and quantum field theory

Fewster, Kasia Rejzner. Publisher : arXiv. Description : We give a pedagogical introduction to algebraic quantum field theory AQFT , with the aim of explaining its key structures and features. Topics covered include: algebraic formulations of quantum theory and the GNS representation theorem, the appearance of unitarily inequivalent representations in QFT exemplified by the van Hove model , the main assumptions of AQFT and simple models thereof, the spectrum condition, etc.

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Algebraic Quantum Field Theory: An Introduction

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We provide an introduction to a study of applications of noncommutative calculus to quantum statistical physics. Centered on noncommutative calculus, we describe the physical concepts and mathematical structures appearing in the analysis of large quantum systems and their consequences. These include the emergence of algebraic approach and the necessity of employment of infinite-dimensional structures. As an illustration, a quantization of stochastic processes, new formalism for statistical mechanics, quantum field theory, and quantum correlations are discussed.

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Statistical Algebraic Approach to Quantum Mechanics

In the algebraic framework of quantum field theory we consider one parameter subgroups of lightlike translations. After establishing a few preliminary properties we prove a certain cluster property and then exhibit the close connection between such subgroups and a class of type III factors. A few applications of this connection are also discussed.

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    In particle physics , the history of quantum field theory starts with its creation by Paul Dirac , when he attempted to quantize the electromagnetic field in the late s.