5,571 research outputs found
Modular localization and Wigner particles
We propose a framework for the free field construction of algebras of local
observables which uses as an input the Bisognano-Wichmann relations and a
representation of the Poincare' group on the one-particle Hilbert space. The
abstract real Hilbert subspace version of the Tomita-Takesaki theory enables us
to bypass some limitations of the Wigner formalism by introducing an intrinsic
spacetime localization. Our approach works also for continuous spin
representations to which we associate a net of von Neumann algebras on
spacelike cones with the Reeh-Schlieder property. The positivity of the energy
in the representation turns out to be equivalent to the isotony of the net, in
the spirit of Borchers theorem. Our procedure extends to other spacetimes
homogeneous under a group of geometric transformations as in the case of
conformal symmetries and de Sitter spacetime.Comment: 22 pages, LaTeX. Some errors have been corrected. To appear on Rev.
Math. Phy
Spectral Invariants of Operators of Dirac Type on Partitioned Manifolds
We review the concepts of the index of a Fredholm operator, the spectral flow
of a curve of self-adjoint Fredholm operators, the Maslov index of a curve of
Lagrangian subspaces in symplectic Hilbert space, and the eta invariant of
operators of Dirac type on closed manifolds and manifolds with boundary. We
emphasize various (occasionally overlooked) aspects of rigorous definitions and
explain the quite different stability properties. Moreover, we utilize the heat
equation approach in various settings and show how these topological and
spectral invariants are mutually related in the study of additivity and
nonadditivity properties on partitioned manifolds.Comment: 131 pages, 9 figure
Weak Symplectic Functional Analysis and General Spectral Flow Formula
We consider a continuous curve of self-adjoint Fredholm extensions of a curve
of closed symmetric operators with fixed minimal domain and fixed {\it
intermediate} domain . Our main example is a family of symmetric
generalized operators of Dirac type on a compact manifold with boundary with
varying well-posed boundary conditions. Here is the first Sobolev space
and the subspace of sections with support in the interior. We express the
spectral flow of the operator curve by the Maslov index of a corresponding
curve of Fredholm pairs of Lagrangian subspaces of the quotient Hilbert space
which is equipped with continuously varying {\it weak symplectic
structures} induced by the Green form.
In this paper, we specify the continuity conditions; define the Maslov index
in weak symplectic analysis; discuss the required weak inner Unique
Continuation Property; derive a General Spectral Flow Formula; and check that
the assumptions are natural and all are satisfied in geometric and
pseudo-differential context.
Applications are given to spectral flow formulae; to the splitting of
the spectral flow on partitioned manifolds; and to linear Hamiltonian systems
The Spectrum of an Adelic Markov Operator
With the help of the representation of SL(2,Z) on the rank two free module
over the integer adeles, we define the transition operator of a Markov chain.
The real component of its spectrum exhibits a gap, whereas the non-real
component forms a circle of radius 1/\sqrt{2}.Comment: 38 pages, 5 figure
A splitting formula for the spectral flow of the odd signature operator on 3-manifolds coupled to a path of SU(2) connections
We establish a splitting formula for the spectral flow of the odd signature
operator on a closed 3-manifold M coupled to a path of SU(2) connections,
provided M = S cup X, where S is the solid torus. It describes the spectral
flow on M in terms of the spectral flow on S, the spectral flow on X (with
certain Atiyah-Patodi-Singer boundary conditions), and two correction terms
which depend only on the endpoints.
Our result improves on other splitting theorems by removing assumptions on
the non-resonance level of the odd signature operator or the dimension of the
kernel of the tangential operator, and allows progress towards a conjecture by
Lisa Jeffrey in her work on Witten's 3-manifold invariants in the context of
the asymptotic expansion conjecture.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol9/paper52.abs.htm
On the Power of Manifold Samples in Exploring Configuration Spaces and the Dimensionality of Narrow Passages
We extend our study of Motion Planning via Manifold Samples (MMS), a general
algorithmic framework that combines geometric methods for the exact and
complete analysis of low-dimensional configuration spaces with sampling-based
approaches that are appropriate for higher dimensions. The framework explores
the configuration space by taking samples that are entire low-dimensional
manifolds of the configuration space capturing its connectivity much better
than isolated point samples. The contributions of this paper are as follows:
(i) We present a recursive application of MMS in a six-dimensional
configuration space, enabling the coordination of two polygonal robots
translating and rotating amidst polygonal obstacles. In the adduced experiments
for the more demanding test cases MMS clearly outperforms PRM, with over
20-fold speedup in a coordination-tight setting. (ii) A probabilistic
completeness proof for the most prevalent case, namely MMS with samples that
are affine subspaces. (iii) A closer examination of the test cases reveals that
MMS has, in comparison to standard sampling-based algorithms, a significant
advantage in scenarios containing high-dimensional narrow passages. This
provokes a novel characterization of narrow passages which attempts to capture
their dimensionality, an attribute that had been (to a large extent) unattended
in previous definitions.Comment: 20 page
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