24,747 research outputs found
Are Rindler Quanta Real? Inequivalent particle concepts in quantum field theory
Philosophical reflection on quantum field theory has tended to focus on how
it revises our conception of what a particle is. However, there has been
relatively little discussion of the threat to the "reality" of particles posed
by the possibility of inequivalent quantizations of a classical field theory,
i.e., inequivalent representations of the algebra of observables of the field
in terms of operators on a Hilbert space. The threat is that each
representation embodies its own distinctive conception of what a particle is,
and how a "particle" will respond to a suitably operated detector. Our main
goal is to clarify the subtle relationship between inequivalent representations
of a field theory and their associated particle concepts. We also have a
particular interest in the Minkowski versus Rindler quantizations of a free
Boson field, because they respectively entail two radically different
descriptions of the particle content of the field in the very same region of
spacetime. We shall defend the idea that these representations provide
complementary descriptions of the same state of the field against the claim
that they embody completely incommensurable theories of the field.Comment: 62 pages, LaTe
Alternative Algebraic Structures from Bi-Hamiltonian Quantum Systems
We discuss the alternative algebraic structures on the manifold of quantum
states arising from alternative Hermitian structures associated with quantum
bi-Hamiltonian systems. We also consider the consequences at the level of the
Heisenberg picture in terms of deformations of the associative product on the
space of observables.Comment: Accepted for publication in Int. J. Geom. Meth. Mod. Phy
Schwinger's Picture of Quantum Mechanics I: Groupoids
A new picture of Quantum Mechanics based on the theory of groupoids is
presented. This picture provides the mathematical background for Schwinger's
algebra of selective measurements and helps to understand its scope and
eventual applications. In this first paper, the kinematical background is
described using elementary notions from category theory, in particular the
notion of 2-groupoids as well as their representations. Some basic results are
presented, and the relation with the standard Dirac-Schr\"odinger and
Born-Jordan-Heisenberg pictures are succinctly discussed.Comment: 32 pages. Comments are welcome
Microlocal sheaves and quiver varieties
We relate Nakajima Quiver Varieties (or, rather, their multiplicative
version) with moduli spaces of perverse sheaves. More precisely, we consider a
generalization of the concept of perverse sheaves: microlocal sheaves on a
nodal curve X. They are defined as perverse sheaves on normalization of X with
a Fourier transform condition near each node and form an abelian category M(X).
One has a similar triangulated category DM(X) of microlocal complexes. For a
compact X we show that DM(X) is Calabi-Yau of dimension 2. In the case when all
components of X are rational, M(X) is equivalent to the category of
representations of the multiplicative pre-projective algebra associated to the
intersection graph of X. Quiver varieties in the proper sense are obtained as
moduli spaces of microlocal sheaves with a framing of vanishing cycles at
singular points. The case when components of X have higher genus, leads to
interesting generalizations of preprojective algebras and quiver varieties. We
analyze them from the point of view of pseudo-Hamiltonian reduction and
group-valued moment maps.Comment: 49 page
Perverse sheaves on Grassmannians
We give a complete quiver description of the category of perverse sheaves on
Hermitian symmetric spaces in types A and D, constructible with respect to the
Schubert stratification. The calculation is microlocal, and uses the action of
the Borel group to study the geometry of the conormal variety.Comment: AMS-LaTeX, 35 pages, 11 figure
The Geometry of Integrable and Superintegrable Systems
The group of automorphisms of the geometry of an integrable system is
considered. The geometrical structure used to obtain it is provided by a normal
form representation of integrable systems that do not depend on any additional
geometrical structure like symplectic, Poisson, etc. Such geometrical structure
provides a generalized toroidal bundle on the carrier space of the system.
Non--canonical diffeomorphisms of such structure generate alternative
Hamiltonian structures for complete integrable Hamiltonian systems. The
energy-period theorem provides the first non--trivial obstruction for the
equivalence of integrable systems
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