Algebra Of Quasilocal Observables Research Articles

Unitarily Inequivalent Representations in Algebraic Quantum Theory

It has been maintained that the physical content of a model of a system is completely contained in the C∗-algebra of quasi-local observables \({\cal A}\) that is associated with the system. The reason given for this is that the unitarily inequivalent representations of \({\cal A}\) are physically equivalent. But, this view is dubious for at least two reasons. First, it is not clear why the physical content does not extend to the elements of the von Neumann algebras that are generated by representations of \({\cal A}\). It is shown here that although the unitarily inequivalent representations of \({\cal A}\) are physically equivalent, the extended representations are not. Second, this view detracts from special global features of physical systems such as temperature and chemical potential by effectively relegating them to the status of fixed parameters. It is desirable to characterize such observables theoretically as elements of the algebra that is associated with a system rather than as parameters, and thereby give a uniform treatment to all observables. This can be accomplished by going to larger algebras. One such algebra is the universal enveloping von Neumann algebra, which is generated by the universal representation of \({\cal A}\); another is the direct integral of factor representations that are associated with the set of values of the global features. Placing interpretive significance on the von Neumann algebras mentioned earlier sheds light on the significance of unitarily inequivalent representations of \({\cal A}\), and it serves to show the limitations of the notion of physical equivalence.

The limiting dynamics of an imperfect Bose gas

For an imperfect Bose gas consisting of massive particles and interacting through a long-range pair potential in a mean-field approximation, both in equilibrium and far from equilibrium, the dynamics are derived in the thermodynamic limit. This is done in the framework of the operator algebraic quantum theory. The limiting dynamics is formulated in the Schrödinger picture as a group of affine transformations of certain folia of physical states. Similar to the mean-field models for spin lattices, the algebra of quasilocal observables is not invariant under the Heisenberg dynamics in the GNS representation of general time invariant states. It is known that above critical density the infinite system exhibits condensation, and the condensate is characterized by the classical parameters density and phase. For these parameters a classical nonlinear equation of motion is derived.

Vacuum states and superselection rules in quantum field theory

Let (A(O),54,α) be a theory of local observables and let A denote the C*-algebra of quasilocal observables. Assume that every representation of A which satisfies the spectrum condition is type I. Then if (π,H) is an irreducible representation of A which satisfies the spectrum condition there exists a unique vacuum state asociated with (π,H) by large spacelike translations.

On the existence of the Heisenberg equations on the C* algebra of quasilocal observables of bose and fermi systems with a finite-range potential

On the existence of the Heisenberg equations on the C* algebra of quasilocal observables of bose and fermi systems with a finite-range potential

Approach to Equilibrium in a Simple Model

The time evolution of a class of generalized quantum Ising models (with various long-range interactions, including Dyson's 1/rα) has been studied from the C*-algebraic point of view. We establish that: (1) All 〈A〉t are weakly almost periodic in time; (2) there exists a unique averaging procedure over time; (3) the time evolution in the thermodynamical limit can be locally implemented by effective Hamiltonians in the algebra of quasilocal observables; (4) there exists a specific connection between the spectral properties of the time evolution of the initial state and the approach to equilibrium; (5) there are examples in which the time evolution is not G-Abelian.

Interaction picture in the algebraic quantum theory

It is shown that in the purely algebraic frame for quantum theory there is a possibility to define the Heisenberg, Schrodinger and interaction picture on the algebra of quasi-local observables. In the different pictures the equations of motion are derived. Some remarks are made concerning the quantum mechanics where the « algebraic frame » can be given explicitly and also checked directly by the usual « Hilbert space approach ».

The representations of symmetry groups on the algebra of quasi-local observables

Operator representations of a symmetry groupG on the algebra introduced by Haag and Kastler are investigated. Considering the representation of the Lorentz group the «stability condition» can be formulated with the help of the infinitesimal operators on the algebra itself. The connection between the representations ofG on the algebra and on the Hilbert space of itsG-invariant *-representations is discussed. A necessary and sufficient condition for theG-invariance of a *-representationR is obtained. IfR contains aG-invariant state, this condition is fulfilled and the spectrum of the infinitesimal operators on the Hilbert space ofR is included in the spectrum of the corresponding infinitesimal operators on the algebra.