The Measurement Process in Local Quantum Theory and the EPR Paradox

We describe in a qualitative way a possible picture of the Measurement Process in Quantum Mechanics, which takes into account: 1. the finite and non zero time duration T of the interaction between the observed system and the microscopic part of the measurement apparatus; 2. the finite space size R of that apparatus; 3. the fact that the macroscopic part of the measurement apparatus, having the role of amplifying the effect of that interaction to a macroscopic scale, is composed by a very large but finite number N of particles. The conventional picture of the measurement, as an instantaneous action turning a pure state into a mixture, arises only in the limit in which N and R tend to infinity, and T tends to 0. We sketch here a proposed scheme, which still ought to be made mathematically precise in order to analyse its implications and to test it in specific models, where we argue that in Quantum Field Theory this picture should apply to the unique time evolution expressing the dynamics of a given theory, and should comply with the Principle of Locality. We comment on the Einstein Podolski Rosen thought experiment (partly modifying the discussion on this point in an earlier version of this note), reformulated here only in terms of local observables (rather than global ones, as one particle or polarisation observables). The local picture of the measurement process helps to make it clear that there is no conflict with the Principle of Locality.Comment: 18 page

CFT fusion rules, DHR gauge groups, and CAR algebras

It is demonstrated that several series of conformal field theories, while satisfying braid group statistics, can still be described in the conventional setting of the DHR theory, i.e. their superselection structure can be understood in terms of a compact DHR gauge group. Besides theories with only simple sectors, these include (the untwisted part of) c=1 orbifold theories and level two so(N) WZW theories. We also analyze the relation between these models and theories of complex free fermions.Comment: 22 pages, LaTeX2

Space-time noncommutativity and (1+1) Higgs Model

We compare the classical scattering of kinks in (1+1) Higgs model with its analogous noncommutative counterpart. While at a classical level we are able to solve the scattering at all orders finding a smooth solution, at a noncommutative level we present only perturbative results, suggesting the existence of a smooth solution also in this case.Comment: 18 pages, 2 figure

Space-Time Symmetries of Noncommutative Spaces

We define a noncommutative Lorentz symmetry for canonical noncommutative spaces. The noncommutative vector fields and the derivatives transform under a deformed Lorentz transformation. We show that the star product is invariant under noncommutative Lorentz transformations. We then apply our idea to the case of actions obtained by expanding the star product and the fields taken in the enveloping algebra via the Seiberg-Witten maps and verify that these actions are invariant under these new noncommutative Lorentz transformations. We finally consider general coordinate transformations and show that the metric is undeformed.Comment: 7 pages, v2: typos corrected, to appear in Phys. Rev.

Quantum Field Theory on Quantum Spacetime

Condensed account of the Lectures delivered at the Meeting on {\it Noncommutative Geometry in Field and String Theory}, Corfu, September 18 - 20, 2005.Comment: 10 page

Generalized Particle Statistics in Two-Dimensions: Examples from the Theory of Free Massive Dirac Field

In the framework of algebraic quantum field theory we analyze the anomalous statistics exhibited by a class of automorphisms of the observable algebra of the two-dimensional free massive Dirac field, constructed by fermionic gauge group methods. The violation of Haag duality, the topological peculiarity of a two-dimensional space-time and the fact that unitary implementers do not lie in the global field algebra account for strange behaviour of statistics, which is no longer an intrinsic property of sectors. Since automorphisms are not inner, we exploit asymptotic abelianness of intertwiners in order to construct a braiding for a suitable $C^*$-tensor subcategory of End($\mathscr{A}$). We define two inequivalent classes of path connected bi-asymptopias, selecting only those sets of nets which yield a true generalized statistics operator.Comment: 24 page

Some Remarks on Group Bundles and C*-dynamical systems

We introduce the notion of fibred action of a group bundle on a C(X)-algebra. By using such a notion, a characterization in terms of induced C*-bundles is given for C*-dynamical systems such that the relative commutant of the fixed-point algebra is minimal (i.e., it is generated by the centre of the given C*-algebra and the centre of the fixed-point algebra). A class of examples in the setting of the Cuntz algebra is given, and connections with superselection structures with nontrivial centre are discussed.Comment: 22 pages; to appear on Comm. Math. Phy

An Algebraic Duality Theory for Multiplicative Unitaries

Multiplicative Unitaries are described in terms of a pair of commuting shifts of relative depth two. They can be generated from ambidextrous Hilbert spaces in a tensor C*-category. The algebraic analogue of the Takesaki-Tatsuuma Duality Theorem characterizes abstractly C*-algebras acted on by unital endomorphisms that are intrinsically related to the regular representation of a multiplicative unitary. The relevant C*-algebras turn out to be simple and indeed separable if the corresponding multiplicative unitaries act on a separable Hilbert space. A categorical analogue provides internal characterizations of minimal representation categories of a multiplicative unitary. Endomorphisms of the Cuntz algebra related algebraically to the grading are discussed as is the notion of braided symmetry in a tensor C*-category.Comment: one reference adde

Noncommutative Electrodynamics with covariant coordinates

We study Noncommutative Electrodynamics using the concept of covariant coordinates. We propose a scheme for interpreting the formalism and construct two basic examples, a constant field and a plane wave. Superposing these two, we find a modification of the dispersion relation. Our results differ from those obtained via the Seiberg-Witten map.Comment: 5 pages, published versio

On the extension of stringlike localised sectors in 2+1 dimensions

In the framework of algebraic quantum field theory, we study the category \Delta_BF^A of stringlike localised representations of a net of observables O \mapsto A(O) in three dimensions. It is shown that compactly localised (DHR) representations give rise to a non-trivial centre of \Delta_BF^A with respect to the braiding. This implies that \Delta_BF^A cannot be modular when non-trival DHR sectors exist. Modular tensor categories, however, are important for topological quantum computing. For this reason, we discuss a method to remove this obstruction to modularity. Indeed, the obstruction can be removed by passing from the observable net A(O) to the Doplicher-Roberts field net F(O). It is then shown that sectors of A can be extended to sectors of the field net that commute with the action of the corresponding symmetry group. Moreover, all such sectors are extensions of sectors of A. Finally, the category \Delta_BF^F of sectors of F is studied by investigating the relation with the categorical crossed product of \Delta_BF^A by the subcategory of DHR representations. Under appropriate conditions, this completely determines the category \Delta_BF^F.Comment: 36 pages, 1 eps figure; v2: appendix added, minor corrections and clarification