Ricci flow, quantum mechanics and gravity

It has been argued that, underlying any given quantum-mechanical model, there exists at least one deterministic system that reproduces, after prequantisation, the given quantum dynamics. For a quantum mechanics with a complex d-dimensional Hilbert space, the Lie group SU(d) represents classical canonical transformations on the projective space CP^{d-1} of quantum states. Let R stand for the Ricci flow of the manifold SU(d-1) down to one point, and let P denote the projection from the Hopf bundle onto its base CP^{d-1}. Then the underlying deterministic model we propose here is the Lie group SU(d), acted on by the operation PR. Finally we comment on some possible consequences that our model may have on a quantum theory of gravity.Comment: 8 page

Remarks on the representation theory of the Moyal plane

We present an explicit construction of a unitary representation of the commutator algebra satisfied by position and momentum operators on the Moyal plane.Comment: 10 pages, minor changes, refs. adde

Central potentials on spaces of constant curvature: The Kepler problem on the two-dimensional sphere S2S^2 and the hyperbolic plane H2H^2

The Kepler problem is a dynamical system that is well defined not only on the Euclidean plane but also on the sphere and on the Hyperbolic plane. First, the theory of central potentials on spaces of constant curvature is studied. All the mathematical expressions are presented using the curvature \k as a parameter, in such a way that they reduce to the appropriate property for the system on the sphere $S^2$, or on the hyperbolic plane $H^2$, when particularized for \k>0, or \k<0, respectively; in addition, the Euclidean case arises as the particular case \k=0. In the second part we study the main properties of the Kepler problem on spaces with curvature, we solve the equations and we obtain the explicit expressions of the orbits by using two different methods: first by direct integration and second by obtaining the \k-dependent version of the Binet's equation. The final part of the article, that has a more geometric character, is devoted to the study of the theory of conics on spaces of constant curvature.Comment: 37 pages, 7 figure

On the noncommutative eikonal

We study the eikonal approximation to quantum mechanics on the Moyal plane. Instead of using a star product, the analysis is carried out in terms of operator-valued wavefunctions depending on noncommuting, operator-valued coordinates.Comment: 18 page

Trigonometry of 'complex Hermitian' type homogeneous symmetric spaces

This paper contains a thorough study of the trigonometry of the homogeneous symmetric spaces in the Cayley-Klein-Dickson family of spaces of 'complex Hermitian' type and rank-one. The complex Hermitian elliptic CP^N and hyperbolic CH^N spaces, their analogues with indefinite Hermitian metric and some non-compact symmetric spaces associated to SL(N+1,R) are the generic members in this family. The method encapsulates trigonometry for this whole family of spaces into a single "basic trigonometric group equation", and has 'universality' and '(self)-duality' as its distinctive traits. All previously known results on the trigonometry of CP^N and CH^N follow as particular cases of our general equations. The physical Quantum Space of States of any quantum system belongs, as the complex Hermitian space member, to this parametrised family; hence its trigonometry appears as a rather particular case of the equations we obtain.Comment: 46 pages, LaTe

Classical and quantum integrability in 3D systems

In this contribution, we discuss three situations in which complete integrability of a three dimensional classical system and its quantum version can be achieved under some conditions. The former is a system with axial symmetry. In the second, we discuss a three dimensional system without spatial symmetry which admits separation of variables if we use ellipsoidal coordinates. In both cases, and as a condition for integrability, certain conditions arise in the integrals of motion. Finally, we study integrability in the three dimensional sphere and a particular case associated with the Kepler problem in $S^3$.Comment: plenary talk on the Conference QTS-5, July 2007, Valladolid, Spai

Through the magnifying glass: ALMA acute viewing of the intricate nebular architecture of OH231.8+4.2

We present continuum and molecular line emission ALMA observations of OH 231.8+4.2, a well studied bipolar nebula around an asymptotic giant branch (AGB) star. The high angular resolution (~0.2-0.3 arcsec) and sensitivity of our ALMA maps provide the most detailed and accurate description of the overall nebular structure and kinematics of this object to date. We have identified a number of outflow components previously unknown. Species studied in this work include 12CO, 13CO, CS, SO, SO2, OCS, SiO, SiS, H3O+, Na37Cl, and CH3OH. The molecules Na37Cl and CH3OH are first detections in OH 231.8+4.2, with CH3OH being also a first detection in an AGB star. Our ALMA maps bring to light the totally unexpected position of the mass-losing AGB star (QX Pup) relative to the large-scale outflow. QX Pup is enshrouded within a compact (<60 AU) parcel of dust and gas (clump S) in expansion (V~5-7 km/s) that is displaced by 0.6arcsec to the south of the dense equatorial region (or waist) where the bipolar lobes join. Our SiO maps disclose a compact bipolar outflow that emerges from QX Pup's vicinity. This outflow is oriented similarly to the large-scale nebula but the expansion velocities are about ten times lower (~35 km/s). We deduce short kinematical ages for the SiO outflow, ranging from ~50-80 yr, in regions within ~150 AU, to ~400-500 yr at the lobe tips (~3500 AU). Adjacent to the SiO outflow, we identify a small-scale hourglass-shaped structure (mini-hourglass) that is probably made of compressed ambient material formed as the SiO outflow penetrates the dense, central regions of the nebula. The lobes and the equatorial waist of the mini-hourglass are both radially expanding with a constant velocity gradient. The mini-waist is characterized by extremely low velocities, down to ~1 km/s at ~150 AU, which tentatively suggest the presence of a stable structure. (abridged