quantum operator algebras in quantum field theories

0.1 Introduction

This is a topic entry that introduces quantum operator algebrasPlanetmathPlanetmathPlanetmath and presents concisely the important roles they play in quantum field theories.

Definition 0.1.

Quantum operator algebras (QOA) in quantum field theories are defined as the algebrasMathworldPlanetmathPlanetmathPlanetmath of observable operators, and as such, they are also related to the von Neumann algebraMathworldPlanetmathPlanetmathPlanetmath; quantum operators are usually defined on Hilbert spacesMathworldPlanetmath, or in some QFTs on Hilbert space bundles or other similarPlanetmathPlanetmath families of spaces.

Remark 0.1.

RepresentationsPlanetmathPlanetmath of Banach *-algebras (that are defined on Hilbert spaces) are closely related to C* -algebra representations which provide a useful approach to defining quantum space-timesPlanetmathPlanetmath.

0.2 Quantum operator algebras in quantum field theories: QOA Role in QFTs

Important examples of quantum operators are: the Hamiltonian operatorPlanetmathPlanetmath (or Schrödinger operator), the position and momentum operators, Casimir operatorsMathworldPlanetmath, unitary operators and spin operators. The observable operators are also self-adjointPlanetmathPlanetmath. More general operators were recently defined, such as Prigogine’s superoperators.

Another development in quantum theoriesPlanetmathPlanetmath was the introduction of Frechét nuclear spaces or ‘rigged’ Hilbert spaces (Hilbert bundles). The following sectionsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath define several types of quantum operator algebras that provide the foundation of modern quantum field theories in mathematical physics.

0.2.1 Quantum groups; quantum operator algebras and related symmetries.

Quantum theories adopted a new lease of life post 1955 when von Neumann beautifully re-formulated quantum mechanics (QM) and quantum theories (QT) in the mathematically rigorous context of Hilbert spaces and operator algebras defined over such spaces. From a current physics perspective, von Neumann’ s approach to quantum mechanics has however done much more: it has not only paved the way to expanding the role of symmetryPlanetmathPlanetmathPlanetmath in physics, as for example with the Wigner-Eckhart theorem and its applications, but also revealed the fundamental importance in quantum physics of the state space geometry of quantum operator algebras.

0.3 Basic mathematical definitions in QOA:

0.3.1 Von Neumann algebra

Let denote a complex (separable) Hilbert space. A von Neumann algebra 𝒜 acting on is a subset of the algebra of all bounded operatorsMathworldPlanetmathPlanetmath () such that:

  • (i) 𝒜 is closed underPlanetmathPlanetmath the adjointPlanetmathPlanetmathPlanetmath operation (with the adjoint of an element T denoted by T*).

  • (ii) 𝒜 equals its bicommutant, namely:

    𝒜={A():B(),C𝒜,(BC=CB)(AB=BA)}. (0.1)

If one calls a commutant of a set 𝒜 the special set of bounded operators on () which commute with all elements in 𝒜, then this second condition implies that the commutant of the commutant of 𝒜 is again the set 𝒜.

On the other hand, a von Neumann algebra 𝒜 inherits a unital subalgebraPlanetmathPlanetmathPlanetmath from (), and according to the first condition in its definition 𝒜, it does indeed inherit a *-subalgebra structureMathworldPlanetmath as further explained in the next section on C* -algebras. Furthermore, one also has available a notable bicommutant theorem which states that: “𝒜 is a von Neumann algebra if and only if A is a *-subalgebra of L(H), closed for the smallest topology defined by continuous maps (ξ,η)(Aξ,η) for all <Aξ,η)> where <.,.> denotes the inner productMathworldPlanetmath defined on H ”.

For a well-presented treatment of the geometry of the state spaces of quantum operator algebras, the reader is referred to Aflsen and Schultz (2003; [AS2k3]).

0.3.2 Hopf algebra

First, a unital associative algebra consists of a linear space A together with two linear maps:

m :AAA,(multiplication) (0.2)
η :A,(unity)

satisfying the conditions

m(m𝟏) =m(𝟏m) (0.3)
m(𝟏η) =m(η𝟏)=id.

This first condition can be seen in terms of a commuting diagram :

AAAmidAAidmmAA@ >mA (0.4)

Next suppose we consider ‘reversing the arrows’, and take an algebra A equipped with a linear homorphisms Δ:AAA, satisfying, for a,bA :

Δ(ab) =Δ(a)Δ(b) (0.5)
(Δid)Δ =(idΔ)Δ.

We call Δ a comultiplication, which is said to be coasociative in so far that the following diagram commutes

AAAΔidAAidΔΔAA@ <ΔA (0.6)

There is also a counterpart to η, the counity map ε:A satisfying

(idε)Δ=(εid)Δ=id. (0.7)

A bialgebraPlanetmathPlanetmath (A,m,Δ,η,ε) is a linear space A with maps m,Δ,η,ε satisfying the above properties.

Now to recover anything resembling a group structure, we must append such a bialgebra with an antihomomorphism S:AA, satisfying S(ab)=S(b)S(a), for a,bA . This map is defined implicitly via the property :

m(Sid)Δ=m(idS)Δ=ηε. (0.8)

We call S the antipode map.

A Hopf algebra is then a bialgebra (A,m,η,Δ,ε) equipped with an antipode map S .

CommutativePlanetmathPlanetmathPlanetmath and non-commutative Hopf algebras form the backbone of quantum ‘groups’ and are essential to the generalizationsPlanetmathPlanetmath of symmetry. Indeed, in most respects a quantum ‘group’ is closely related to its dual Hopf algebra; in the case of a finite, commutative quantum groupPlanetmathPlanetmathPlanetmathPlanetmath its dual Hopf algebra is obtained via Fourier transformationPlanetmathPlanetmath of the group elements. When Hopf algebras are actually associated with their dual, proper groups of matrices, there is considerable scope for their representations on both finite and infinite dimensional Hilbert spaces.

0.3.3 Groupoids

Recall that a groupoid 𝖦 is, loosely speaking, a small category with inversesMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath over its set of objects X=Ob(𝖦) . One often writes 𝖦xy for the set of morphismsMathworldPlanetmathPlanetmath in 𝖦 from x to y . A topological groupoid consists of a space 𝖦, a distinguished subspacePlanetmathPlanetmath 𝖦(0)=Ob(𝖦)𝖦, called the space of objects of 𝖦, together with maps

r,s: (0.9)