Implementation of comparative probability by normal states. Infinite dimensional case

Let be an infinite dimensional Hilbert space and () the set of all (orthogonal) projections on . A comparative probability on () is a linear preorder on () such thatOP1,1O and such that ifP┴R,Q┴R, thenPQP+RQ+R for allP, Q, R in (). We give a sufficient and necessary condition for to be implemented in a canonical way by a normal state onB(), the bounded linear operators on .     

An asymptotically flat ℋ space

The null geodesic equation is solved for the space of Sparling and Tod. Bondi coordinates are found and it is verified that the space construction is idempotent, i.e., the -space of this space is itself, symbolically ² = . Properties of the solution are used to motivate a definition of asymptotic flatness.S.R.C. Postdoctoral Research Fellow.     

On the characterization of the stationary state of a class of dynamical semigroups

We consider a representation of the entropy production for a completely positive, trace-preserving dynamical semigroup satisfying detailed balance with respect to its faithful stationary state denned on aW*-algebra(): it is expressed as a positive Hermitian form on(), which is analogous to the quantum correlation functions used in the Kubo theory. By considering this Hermitian form as a variation function of a vector in(), an exact characterization of the stationary states of semigroups in a certain class is obtained. On this basis, the problem of characterizing the stationary states discussed by Spohn and Lebowitz for manyreservoir open systems is solved without the restriction to situations near thermal equilibrium.     

Generating fields from data onH − ⋃H +andH − ⋃I +

There are series solutions for characteristic boundary value problems for fields on black hole backgrounds that converge when the data are given on =^{– +}, or on =^{– +}, but may not converge when the data are given on ^{– –}, or on ^{+ +}. We specialize to oscillatory data of frequency and calculate approximate reflection and transmission coefficientsR() andT(), using a field generated by data on =^{– +}, and again, using a field generated by data on ^{– –}. The first calculation gives qualitatively good results at all frequencies at each order of approximation, and quantitatively better results at higher orders of approximation; the second calculation, using the series which may not converge, gives bad results except at very high frequencies. Thus for the physically unnatural case of a field that vanishes on ^– and goes toe ^iv on ⁺ we have a series that is convergent, and uniformly so with respect to frequency, while for the natural case of a field that vanishes on ^– and goes toe ^iv on ^– we are limited to high frequencies. It is argued that a frequency-dependent renormalization of a series of the first type provides an approximation scheme that is convergent, and uniformly so with respect to frequency, for the physically important problems of the second type. The difficulties posed by the -dependent renormalization for the study of incident pulses are discussed.     

Maximal violation of Bell's inequalities is generic in quantum field theory

Under weak technical assumptions on a net of local von Neumann algebras {A(O)} in a Hilbert space , which are fulfilled by any net associated to a quantum field satisfying the standard axioms, it is shown that for every vector state in there exist observables localized in complementary wedge-shaped regions in Minkowski space-time that maximally violate Bell's inequalities in the state . If, in addition, the algebras corresponding to wedge-shaped regions are injective (which is known to be true in many examples), then the maximal violation occurs in any state on () given by a density matrix.     

Schütte's tautology and the Kochen-Specker theorem

I present a new 33-ray proof of the Kochen and Specker no-go hidden variable theorem in ₃, based on a classical tautology that corresponds to a contingent quantum proposition in ₃ proposed by Kurt Schütte in an unpublished letter to Specker in 1965. 1 discuss the relation of this proof to a 31-ray proof by Conway and Kochen, and to a 33-ray proof by Peres.     

Statistics of complex levels of random matrices for decaying systems

We present analytical and numerical results for the level density of a certain class of random non-Hermitian matrices =H+i. The conservative partH belongs to the Gaussian orthogonal ensemble while the damping piece is quadratic in Gaussian random numbers and may describe the decay of resonances through various channels. In the limit of a large matrix dimension the level density assumes a surprisingly simple dependence on the relative strength of the damping and the number of channels. Moreover, we identify situations with cubic repulsion between the complex eigenvalues of , to within a logarithmic correction.     

On solving Einstein's field equations in the Newman-Penrose formalism

Using the Newman-Penrose formalism and Penrose's conformai rescaling a method is presented for finding systematically solutions of (or, at least, reduced equations for) the general field equations. These solutions are necessarily (locally) asymptotically flat and are represented in a coordinate system based on a geodesic, twist-free, expanding null congruence. All redundant equations are disposed of and the freely specifiable data are clearly exhibited. Although the few equations that remain to be solved are, in general, intractable, well-known theorems guarantee the existence and uniqueness of solutions. The method applies to spaces and spaces as well as to real space-times.     

Construction of Covariant Differential Calculi on Quantum Homogeneous Spaces

A method of constructing covariant differential calculi on a quantum homogeneous space is devised. The function algebra of the quantum homogeneous space is assumed to be a left coideal of a coquasitriangular Hopf algebra and to contain the coefficients of any matrix over which is the two-sided inverse of one with entries in . The method is based on partial derivatives. For the quantum sphere of Podle and the quantizations of symmetric spaces due to Noumi, Dijkhuizen and Sugitani, the construction produces the subcalculi of the standard bicovariant calculus on the quantum group.     

Product states for local algebras

Let ()() and () () be the von Neumann algebras associated with the space-tiem regions and respectively in the vacuum representation of the free neutral massive scalar field. For suitably chosen spacelike separated regions and it is proved that there exists a normal product state of (), Some consequences for the algebraic structure of the local rings are pointed out.     

On quantum holonomy for mixed states

There is a natural connection and parallel transport on the Hilbert tensor product (or, equivalently, the space of Hilbert-Schmidt operators), the elements of which represent density matrices in up to unitary operators. We postulate a time evolution equation, which leads to this connection after extracting a proper dynamical unitary phase. As an example, we compute the holonomy of a loop of temperature states for the spin in a rotating magnetic field.Supported by Fonds zur Förderung der wissenschaftlichen Forschung in Österreich, Project No. P5588.     

A general class of (essentially) iso-spectral perturbations

We consider (essentially) iso-spectral perturbations of operators of the formH _A=A ^* A withA being a densely defined closed linear operator from a Hilbert space to another Hilbert space . We perturbH _A by perturbingA asA+B withB being a linear operator from to . Two classes ofB are defined so as to obtain (essentially) iso-spectral perturbations ofH _A. The abstract results are applied to Schrödinger operators. Our approach gives also a mathematical unification for the so-called factorization method in quantum mechanics.     

Fuzzy quantum logic II. The logics of unsharp quantum mechanics

A survey of the main results of the Italian group about the logics of unsharp quantum mechanics is presented. In particular partial ordered structures playing with respect to effect operators (linear bounded operatorsF on a Hilbert space such that, 0¦F²) the role played by orthomodular posets with respect to orthogonal projections (corresponding to sharp effects) are analyzed. These structures are generally characterized by the splitting of standard orthocomplementation on projectors into two nonusual orthocomplementations (afuzzy-like and anintuitionistic-like) giving rise to different kinds of Brouwer-Zadeh (BZ) posets: de Morgan BZ posets, BZ^* posets, and BZ³ posets. Physically relevant generalizations of ortho-pair semantics (paraconsistent, regular paraconsistent, and minimal quantum logics) are introduced and their relevance with respect to the logic of unsharp quantum mechanics are discussed.     

Types of von Neumann algebras associated with extremal invariant states

A globalized version of the following is proved. Let be a factor acting on a Hilbert space ,G a group of unitary operators on inducing automorphisms of ,x a vector separating and cyclic for which is up to a scalar multiple the unique vector invariant under the unitaries inG. Then either is of type III or _x is a trace of . The theorem is then applied to study the representations due to invariant factors state of asymptotically abelianC*-algebras, and to show that in quantum field theory certain regions in the Minkowski space give type III factors.     

Non-uniqueness of the heat-current operator and the hopping thermoelectric and thermomagnetic phenomena

The Hardy's corrections to the Maradudin's lattice part of the total heat current operator are shown to yield no corrections to the hopping thermoelectric coefficients of amorphous solids and glasses in the lowest (minus-first) order of the electron phonon coupling constant. But even after including the magnetic field (0, 0,), the lowest non-zero (zeroth order) contributions to the Peltier and Seebeck tensors are shown to be diagonal, yielding thus no contribution to the Nernst and Ettingshaussen coefficients. Therefore, the Hardy corrections, which might yield the next (first) order contributions to these coefficients, appear to be crucial for treating the thermomagnetic phenomena in amorphous and glassy materials.     

On the relative entropy

ForA any subset of () (the bounded operators on a Hilbert space) containing the unit, and and restrictions of states on () toA, ent _A (|)—the entropy of relative to given the information inA—is defined and given an axiomatic characterisation. It is compared with ent _A ^A (|)—the relative entropy introduced by Umegaki and generalised by various authors—which is defined only forA an algebra. It is proved that ent and ent ^S agree on pairs of normal states on an injective von Neumann algebra. It is also proved that ent always has all the most important properties known for ent ^S : monotonicity, concavity,w* upper semicontinuity, etc.     

On the Finite-Dimensional Quantum Group M3⊕(M2|1(Λ2))0

We describe a few properties of the nonsemisimple associative algebra =M₃ (M_2|1 (²))₀, where ² is the Grassmann algebra with two generators. We show that is not only a finite-dimensional algebra but also a (noncommutative) Hopf algebra, hence a finite-dimensional quantum group. By selecting a system of explicit generators, we show how it is related with the quantum enveloping of SL_q(2) when the parameter q is a cubic root of unity. We describe its indecomposable projective representations as well as the irreducible ones. We also comment about the relation between this object and the theory of modular representation of the group SL(2, F₃), i.e. the binary tetrahedral group. Finally, we briefly discuss its relation with the Lorentz group and, as already suggested by A.~Connes, make a few comments about the possible use of this algebra in a modification of the Standard Model of particle physics (the unitary group of the semisimple algebra associated with is U(3) × U(2) × U(1)).     

Geometro-stochastic quantization of a theory for extended elementary objects

The geometro-stochastic quantization of a gauge theory based on the (4,1)-de Sitter group is presented. The theory contains an intrinsic elementary length parameter R of geometric origin taken to be of a size typical for hadron physics. Use is made of a soldered Hilbert bundle over curved spacetime carrying a phase space representation of SO(4, 1) with the Lorentz subgroup related to a vierbein formulation of gravitation. The typical fiber of is a resolution kernel Hilbert space constructed in terms of generalized coherent states related to the principal series of unitary irreducible representations of SO(4, 1), namely de Sitter horospherical waves for spinless particles characterized by the parameter . The framework is, finally, extended to a quantum field-theoretical formalism by using bundles with Fock space fibers constructed from .Supported in part by NSERC Research Grant No. A5206.     

On the generators of quantum dynamical semigroups

The notion of a quantum dynamical semigroup is defined using the concept of a completely positive map. An explicit form of a bounded generator of such a semigroup onB() is derived. This is a quantum analogue of the Lévy-Khinchin formula. As a result the general form of a large class of Markovian quantum-mechanical master equations is obtained.     

Algebras of bounded operators on nonclassical orthomodular spaces

A Hermitian space is called orthomodular if the Projection Theorem holds: every orthogonally closed subspace is an orthogonal summand. Besides the familiar real or complex Hilbert spaces there are non-classical infinite dimensional examples constructed over certain non-Archimedeanly valued, complete fields. We study bounded linear operators on such spaces. In particular we construct an operator algebraA of von Neumann type that contains no orthogonal projections at all. For operators inA we establish a representation theorem from which we deduce thatA is commutative. We then focus on a subalgebra which turns out to be an integral domain with unique maximal ideal. Both analytic and topological characterizations of are given.