Is there a Jordan geometry underlying quantum physics?

There have been several propositions for a geometric and essentially non-linear formulation of quantum mechanics. From a purely mathematical point of view, the point of view of Jordan algebra theory might give new strength to such approaches: there is a ``Jordan geometry'' belonging to the Jordan part of the algebra of observables, in the same way as Lie groups belong to the Lie part. Both the Lie geometry and the Jordan geometry are well-adapted to describe certain features of quantum theory. We concentrate here on the mathematical description of the Jordan geometry and raise some questions concerning possible relations with foundational issues of quantum theory.Comment: 30 page

Isolated deep earthquakes beneath the North Island of New Zealand

Seismicity shallows towards the south along the Tonga-Kermadec-Hikurangi margin, deep and intermediate seismicity being absent altogether in the South Island of New Zealand. Beneath the Taranaki region of the North Island the maximum depth of the main seismicity is 250 km, but very rare events occur directly below at 600 km. These could be associated with a detached slab or a vertical, aseismic continuation of the subducted Pacific Plate. Six small events that occurred in the 1990s were recorded extensively by digital instruments of the New Zealand National Network (NZNN) and temporary deployments. We relocate these events by a joint hypocentre determination (JHD) method and find their focal mechanisms using first motions and relative amplitudes of P and S arrivals. The earthquakes relocate to a remarkably uniform depth of 603 +/- 3 kmrelative error (+/- 10 km absolute error) in a line 30- km long orientated 40 NE, roughly parallel to the strike of the intermediate- depth seismicity. The only consistent component of the focal mechanisms is the tension axis: all lie close to horizontal and tend to align with the line of hypocentres. We interpret this deep seismic zone as a detached sliver of plate lying horizontally with the same orientation as the main subducted plate above. Volume change caused by a phase change controlled by the pressure at 600 km and temperature in the sliver produces a pattern of strain that places the sliver under tension along its lengt

May 12 1997 Cme Event: I. a Simplified Model of the Pre-Eruptive Magnetic Structure

A simple model of the coronal magnetic field prior to the CME eruption on May 12 1997 is developed. First, the magnetic field is constructed by superimposing a large-scale background field and a localized bipolar field to model the active region (AR) in the current-free approximation. Second, this potential configuration is quasi-statically sheared by photospheric vortex motions applied to two flux concentrations of the AR. Third, the resulting force-free field is then evolved by canceling the photospheric magnetic flux with the help of an appropriate tangential electric field applied to the central part of the AR. To understand the structure of the modeled configuration, we use the field line mapping technique by generalizing it to spherical geometry. It is demonstrated that the initial potential configuration contains a hyperbolic flux tube (HFT) which is a union of two intersecting quasi-separatrix layers. This HFT provides a partition of the closed magnetic flux between the AR and the global solar magnetic field. The vortex motions applied to the AR interlock the field lines in the coronal volume to form additionally two new HFTs pinched into thin current layers. Reconnection in these current layers helps to redistribute the magnetic flux and current within the AR in the flux-cancellation phase. In this phase, a magnetic flux rope is formed together with a bald patch separatrix surface wrapping around the rope. Other important implications of the identified structural features of the modeled configuration are also discussed.Comment: 25 pages, 11 figures, to appear in ApJ 200

Locating current sheets in the solar corona

Current sheets are essential for energy dissipation in the solar corona, in particular by enabling magnetic reconnection. Unfortunately, sufficiently thin current sheets cannot be resolved observationally and the theory of their formation is an unresolved issue as well. We consider two predictors of coronal current concentrations, both based on geometrical or even topological properties of a force free coronal magnetic field. First, there are separatrices related to magnetic nulls. Through separatrices the magnetic connectivity changes discontinuously. Coronal magnetic nulls are, however, very rare. At second, inspired by the concept of generalized magnetic reconnection without nulls, quasi-separatrix layers (QSL) were suggested. Through QSL the magnetic connectivity changes continuously, though strongly. The strength of the connectivity change can be quantified by measuring the squashing of the flux tubes which connect the magnetically conjugated photospheres. We verify the QSL and separatrix concepts by comparing the sites of magnetic nulls and enhanced squashing with the location of current concentrations in the corona. Due to the known difficulties of their direct observation we simulated the coronal current sheets by numerically calculating the response of the corona to energy input from the photosphere heating a simultaneously observed EUV Bright Point. We did not find coronal current sheets not at the separatrices but at several QSL locations. The reason is that although the geometrical properties of force free extrapolated magnetic fields can indeed, hint at possible current concentrations, a necessary condition for current sheet formation is the local energy input into the corona