36 research outputs found
On the geometry of the energy operator in quantum mechanics
We analyze the different ways to define the energy operator in geometric
theories of quantum mechanics. In some formulations the operator contains the
scalar curvature as a multiplicative term. We show that such term can be
canceled or added with an arbitrary constant factor, both in the mainstream
Geometric Quantization and in the Covariant Quantum Mechanics, developed by
Jadczyk and Modugno with several contributions from many authors.Comment: 18 pages; paper in honour of the 70th birthday of Luigi Mangiarotti
and Marco Modugn
The geometry of real reducible polarizations in quantum mechanics
The formulation of Geometric Quantization contains several axioms and
assumptions. We show that for real polarizations we can generalize the standard
geometric quantization procedure by introducing an arbitrary connection on the
polarization bundle. The existence of reducible quantum structures leads to
considering the class of Liouville symplectic manifolds. Our main application
of this modified geometric quantization scheme is to Quantum Mechanics on
Riemannian manifolds. With this method we obtain an energy operator without the
scalar curvature term that appears in the standard formulation, thus agreeing
with the usual expression found in the Physics literature.Comment: 29 page
Fourier-Mukai transforms for coherent systems on elliptic curves
We determine all the Fourier-Mukai transforms for coherent systems consisting
of a vector bundle over an elliptic curve and a subspace of its global
sections, showing that these transforms are indexed by the positive integers.
We prove that the natural stability condition for coherent systems, which
depends on a parameter, is preserved by these transforms for small and large
values of the parameter. By means of the Fourier-Mukai transforms we prove that
certain moduli spaces of coherent systems corresponding to small and large
values of the parameter are isomorphic. Using these results we draw some
conclusions about the possible birational type of the moduli spaces. We prove
that for a given degree of the vector bundle and a given dimension of the
subspace of its global sections there are at most different possible
birational types for the moduli spaces.Comment: LaTeX2e, 21 pages, some proofs simplified, typos corrected. Final
version to appear in Journal of the London Mathematical Societ
The meaning of the Sky of Salamanca
After presenting the main characteristics of the Sky of Salamanca, we analyse
whether what is represented in it is motivated by astrological or astronomical
considerations. We will see that the astrological explanation based on the
system of planetary houses described in Claudius Ptolemy's Tetrabiblos does not
correspond to the planetary configuration we see in the salmantine vault.
Finally, we will conclude that the astronomical interpretation of the Sky of
Salamanca is justified by the representation of the Universe known in the 15th
century according to the Almagest, by the placement of the stars in the
constellations according to that work and by the dating of the painted
planetary configuration, which circumstantial evidence places in August 1475.Comment: 11 pages, 8 figure
Moduli Spaces of Semistable Sheaves on Singular Genus One Curves
We find some equivalences of the derived category of coherent sheaves on a
Gorenstein genus one curve that preserve the (semi)-stability of pure
dimensional sheaves. Using them we establish new identifications between
certain Simpson moduli spaces of semistable sheaves on the curve. For rank
zero, the moduli spaces are symmetric powers of the curve whilst for a fixed
positive rank there are only a finite number of non-isomorphic spaces. We prove
similar results for the relative semistable moduli spaces on an arbitrary genus
one fibration with no conditions either on the base or on the total space. For
a cycle of projective lines, we show that the unique degree 0 stable
sheaves are the line bundles having degree 0 on every irreducible component and
the sheaves supported on one irreducible component. We also
prove that the connected component of the moduli space that contains vector
bundles of rank is isomorphic to the -th symmetric product of the
rational curve with one node.Comment: 26 pages, 4 figures. Added the structure of the biggest component of
the moduli space of sheaves of degree 0 on a cycle of projective lines. Final
version; to appear en IMRS (International Mathematics Research Notices 2009
The Grothendieck and Picard groups of finite rank torsion free {sl}(2)-modules
[EN] The classification problem for simple sl(2)-modules leads in a natural way to the study of
the category of finite rank torsion free sl(2)-modules and its subcategory of rational sl(2)-modules. We prove that the rationalization functor induces an identification between theisomorphism classes of simple modules of these categories. This raises the question of what is the precise relationship between other invariants associated with them. We give a complete solution to this problem for the Grothendieck and Picard groups, obtaining along the way several new results regarding these categories that are interesting in their own right.Publicación en abierto financiada por el Consorcio de Bibliotecas Universitarias de Castilla y León (BUCLE), con cargo al Programa Operativo 2014ES16RFOP009 FEDER 2014-2020 DE CASTILLA Y LEÓN, Actuación:20007-CL - Apoyo Consorcio BUCLE