18,711 research outputs found
Born--Oppenheimer decomposition for quantum fields on quantum spacetimes
Quantum Field Theory on Curved Spacetime (QFT on CS) is a well established
theoretical framework which intuitively should be a an extremely effective
description of the quantum nature of matter when propagating on a given
background spacetime. If one wants to take care of backreaction effects, then a
theory of quantum gravity is needed. It is now widely believed that such a
theory should be formulated in a non-perturbative and therefore background
independent fashion. Hence, it is a priori a puzzle how a background dependent
QFT on CS should emerge as a semiclassical limit out of a background
independent quantum gravity theory. In this article we point out that the
Born-Oppenheimer decomposition (BOD) of the Hilbert space is ideally suited in
order to establish such a link, provided that the Hilbert space representation
of the gravitational field algebra satisfies an important condition. If the
condition is satisfied, then the framework of QFT on CS can be, in a certain
sense, embedded into a theory of quantum gravity. The unique representation of
the holonomy-flux algebra underlying Loop Quantum Gravity (LQG) violates that
condition. While it is conceivable that the condition on the representation can
be relaxed, for convenience in this article we consider a new classical
gravitational field algebra and a Hilbert space representation of its
restriction to an algebraic graph for which the condition is satisfied. An
important question that remains and for which we have only partial answers is
how to construct eigenstates of the full gravity-matter Hamiltonian whose BOD
is confined to a small neighbourhood of a physically interesting vacuum
spacetime.Comment: 38 pages, 2 figure
Scaling Algorithms for Unbalanced Transport Problems
This article introduces a new class of fast algorithms to approximate
variational problems involving unbalanced optimal transport. While classical
optimal transport considers only normalized probability distributions, it is
important for many applications to be able to compute some sort of relaxed
transportation between arbitrary positive measures. A generic class of such
"unbalanced" optimal transport problems has been recently proposed by several
authors. In this paper, we show how to extend the, now classical, entropic
regularization scheme to these unbalanced problems. This gives rise to fast,
highly parallelizable algorithms that operate by performing only diagonal
scaling (i.e. pointwise multiplications) of the transportation couplings. They
are generalizations of the celebrated Sinkhorn algorithm. We show how these
methods can be used to solve unbalanced transport, unbalanced gradient flows,
and to compute unbalanced barycenters. We showcase applications to 2-D shape
modification, color transfer, and growth models
An introduction to Lie group integrators -- basics, new developments and applications
We give a short and elementary introduction to Lie group methods. A selection
of applications of Lie group integrators are discussed. Finally, a family of
symplectic integrators on cotangent bundles of Lie groups is presented and the
notion of discrete gradient methods is generalised to Lie groups
Vortex Scattering and Intercommuting Cosmic Strings on a Noncommutative Spacetime
We study the scattering of noncommutative vortices, based on the
noncommutative field theory developed in [Phys. Rev. D 75, 045009 (2007)], as a
way to understand the interaction of cosmic strings. In the center-of-mass
frame, the effects of noncommutativity vanish, and therefore the reconnection
of cosmic strings occurs in an identical manner to the commutative case.
However, when scattering occurs in a frame other than the center-of-mass frame,
strings still reconnect but the well known 90-degree scattering no longer need
correspond to the head on collision of the strings, due to the breakdown of
Lorentz invariance in the underlying noncommutative field theory.Comment: 18 pages, 2 figure
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