Mutually Unbiased Bases and Semi-definite Programming

A complex Hilbert space of dimension six supports at least three but not more than seven mutually unbiased bases. Two computer-aided analytical methods to tighten these bounds are reviewed, based on a discretization of parameter space and on Grobner bases. A third algorithmic approach is presented: the non-existence of more than three mutually unbiased bases in composite dimensions can be decided by a global optimization method known as semidefinite programming. The method is used to confirm that the spectral matrix cannot be part of a complete set of seven mutually unbiased bases in dimension six.Comment: 11 pages

Scaling algebras for charged fields and short-distance analysis for localizable and topological charges

The method of scaling algebras, which has been introduced earlier as a means for analyzing the short-distance behaviour of quantum field theories in the setting of the model-independent, operator-algebraic approach, is extended to the case of fields carrying superselection charges. In doing so, consideration will be given to strictly localizable charges ("DHR-type" superselection charges) as well as to charges which can only be localized in regions extending to spacelike infinity ("BF-type" superselection charges). A criterion for the preservance of superselection charges in the short-distance scaling limit is proposed. Consequences of this preservance of superselection charges are studied. The conjugate charge of a preserved charge is also preserved, and for charges of DHR-type, the preservance of all charges of a quantum field theory in the scaling limit leads to equivalence of local and global intertwiners between superselection sectors.Comment: Latex 2e, 57 pages. Supersedes hep-th/030114

Reflections upon separability and distillability

We present an abstract formulation of the so-called Innsbruck-Hannover programme that investigates quantum correlations and entanglement in terms of convex sets. We present a unified description of optimal decompositions of quantum states and the optimization of witness operators that detect whether a given state belongs to a given convex set. We illustrate the abstract formulation with several examples, and discuss relations between optimal entanglement witnesses and n-copy non-distillable states with non-positive partial transpose.Comment: 12 pages, 7 figures, proceedings of the ESF QIT Conference Gdansk, July 2001, submitted to special issue of J. Mod. Op

New Concepts in Particle Physics from Solution of an Old Problem

Recent ideas on modular localization in local quantum physics are used to clarify the relation between on- and off-shell quantities in particle physics; in particular the relation between on-shell crossing symmetry and off-shell Einstein causality. Among the collateral results of this new nonperturbative approach are profound relations between crossing symmetry of particle physics and Hawking-Unruh like thermal aspects (KMS property, entropy attached to horizons) of quantum matter behind causal horizons, aspects which hitherto were exclusively related with Killing horizons in curved spacetime rather than with localization aspects in Minkowski space particle physics. The scope of this modular framework is amazingly wide and ranges from providing a conceptual basis for the d=1+1 bootstrap-formfactor program for factorizable d=1+1 models to a decomposition theory of QFT's in terms of a finite collection of unitarily equivalent chiral conformal theories placed a specified relative position within a common Hilbert space (in d=1+1 a holographic relation and in higher dimensions more like a scanning). The new framework gives a spacetime interpretation to the Zamolodchikov-Faddeev algebra and explains its thermal aspects.Comment: In this form it will appear in JPA Math Gen, 47 pages tcilate

On the consequences of twisted Poincare' symmetry upon QFT on Moyal noncommutative spaces

We explore some general consequences of a consistent formulation of relativistic quantum field theory (QFT) on the Groenewold-Moyal-Weyl noncommutative versions of Minkowski space with covariance under the twisted Poincare' group of Chaichian et al. [12], Wess [44], Koch et al. [31], Oeckl [34]. We argue that a proper enforcement of the latter requires braided commutation relations between any pair of coordinates $\hat x,\hat y$ generating two different copies of the space, or equivalently a $\star$-tensor product $f(x)\star g(y)$ (in the parlance of Aschieri et al. [3]) between any two functions depending on $x,y$. Then all differences $(x-y)^\mu$ behave like their undeformed counterparts. Imposing (minimally adapted) Wightman axioms one finds that the $n$-point functions fulfill the same general properties as on commutative space. Actually, upon computation one finds (at least for scalar fields) that the $n$-point functions remain unchanged as functions of the coordinates' differences both if fields are free and if they interact (we treat interactions via time-ordered perturbation theory). The main, surprising outcome seems a QFT physically equivalent to the undeformed counterpart (to confirm it or not one should however first clarify the relation between $n$-point functions and observables, in particular S-matrix elements). These results are mainly based on a joint work [24] with J. WessComment: Latex file, 21 page