1,984 research outputs found
Clifford algebra is the natural framework for root systems and Coxeter groups. Group theory: Coxeter, conformal and modular groups
In this paper, we make the case that Clifford algebra is the natural
framework for root systems and reflection groups, as well as related groups
such as the conformal and modular groups: The metric that exists on these
spaces can always be used to construct the corresponding Clifford algebra. Via
the Cartan-Dieudonn\'e theorem all the transformations of interest can be
written as products of reflections and thus via `sandwiching' with Clifford
algebra multivectors. These multivector groups can be used to perform concrete
calculations in different groups, e.g. the various types of polyhedral groups,
and we treat the example of the tetrahedral group in detail. As an aside,
this gives a constructive result that induces from every 3D root system a root
system in dimension four, which hinges on the facts that the group of spinors
provides a double cover of the rotations, the space of 3D spinors has a 4D
euclidean inner product, and with respect to this inner product the group of
spinors can be shown to be closed under reflections. In particular the 4D root
systems/Coxeter groups induced in this way are precisely the exceptional ones,
with the 3D spinorial point of view also explaining their unusual automorphism
groups. This construction simplifies Arnold's trinities and puts the McKay
correspondence into a wider framework. We finally discuss extending the
conformal geometric algebra approach to the 2D conformal and modular groups,
which could have interesting novel applications in conformal field theory,
string theory and modular form theory.Comment: 14 pages, 1 figure, 5 table
The Birth of out of the Spinors of the Icosahedron
is prominent in mathematics and theoretical physics, and is generally
viewed as an exceptional symmetry in an eight-dimensional space very different
from the space we inhabit; for instance the Lie group features heavily in
ten-dimensional superstring theory. Contrary to that point of view, here we
show that the root system can in fact be constructed from the icosahedron
alone and can thus be viewed purely in terms of three-dimensional geometry. The
roots of arise in the 8D Clifford algebra of 3D space as a double
cover of the elements of the icosahedral group, generated by the root
system . As a by-product, by restricting to even products of root vectors
(spinors) in the 4D even subalgebra of the Clifford algebra, one can show that
each 3D root system induces a root system in 4D, which turn out to also be
exactly the exceptional 4D root systems. The spinorial point of view explains
their existence as well as their unusual automorphism groups. This spinorial
approach thus in fact allows one to construct all exceptional root systems
within the geometry of three dimensions, which opens up a novel interpretation
of these phenomena in terms of spinorial geometry.Comment: 14 pages, 2 figures, 1 tabl
Stochastic time-evolution, information geometry and the Cramer-Rao Bound
We investigate the connection between the time-evolution of averages of
stochastic quantities and the Fisher information and its induced statistical
length. As a consequence of the Cramer-Rao bound, we find that the rate of
change of the average of any observable is bounded from above by its variance
times the temporal Fisher information. As a consequence of this bound, we
obtain a speed limit on the evolution of stochastic observables: Changing the
average of an observable requires a minimum amount of time given by the change
in the average squared, divided by the fluctuations of the observable times the
thermodynamic cost of the transformation. In particular for relaxation
dynamics, which do not depend on time explicitly, we show that the Fisher
information is a monotonically decreasing function of time and that this
minimal required time is determined by the initial preparation of the system.
We further show that the monotonicity of the Fisher information can be used to
detect hidden variables in the system and demonstrate our findings for simple
examples of continuous and discrete random processes.Comment: 25 pages, 4 figure
Using systems theory to conceptualize the implementation of undergraduate online education in a university setting
As participants in the process of exploring how to formalize and develop undergraduate online education at the University of Connecticut, the authors share their experiences relative to the challenges of identifying and addressing the diverse factors involved in such an endeavor. Recognizing the importance of multi-level organizational change in building, integrating, and sustaining an online learning environment, they utilize systems theory as a unifying framework to better analyze the nature and impact of the changes required to create an environment to support online education within a university
- …