272 research outputs found
-Minkowski Spacetimes and DSR Algebras: Fresh Look and Old Problems
Some classes of Deformed Special Relativity (DSR) theories are reconsidered
within the Hopf algebraic formulation. For this purpose we shall explore a
minimal framework of deformed Weyl-Heisenberg algebras provided by a smash
product construction of DSR algebra. It is proved that this DSR algebra, which
uniquely unifies -Minkowski spacetime coordinates with Poincar\'e
generators, can be obtained by nonlinear change of generators from undeformed
one. Its various realizations in terms of the standard (undeformed)
Weyl-Heisenberg algebra opens the way for quantum mechanical interpretation of
DSR theories in terms of relativistic (St\"uckelberg version) Quantum
Mechanics. On this basis we review some recent results concerning twist
realization of -Minkowski spacetime described as a quantum covariant
algebra determining a deformation quantization of the corresponding linear
Poisson structure. Formal and conceptual issues concerning quantum
-Poincar\'e and -Minkowski algebras as well as DSR theories are
discussed. Particularly, the so-called "-analog" version of DSR algebra is
introduced. Is deformed special relativity quantization of doubly special
relativity remains an open question. Finally, possible physical applications of
DSR algebra to description of some aspects of Planck scale physics are shortly
recalled
Differential structure on kappa-Minkowski space, and kappa-Poincare algebra
We construct realizations of the generators of the -Minkowski space
and -Poincar\'{e} algebra as formal power series in the -adic
extension of the Weyl algebra. The Hopf algebra structure of the
-Poincar\'{e} algebra related to different realizations is given. We
construct realizations of the exterior derivative and one-forms, and define a
differential calculus on -Minkowski space which is compatible with the
action of the Lorentz algebra. In contrast to the conventional bicovariant
calculus, the space of one-forms has the same dimension as the
-Minkowski space.Comment: 20 pages. Accepted for publication in International Journal of Modern
Physics
Dynamics with Infinitely Many Derivatives: The Initial Value Problem
Differential equations of infinite order are an increasingly important class
of equations in theoretical physics. Such equations are ubiquitous in string
field theory and have recently attracted considerable interest also from
cosmologists. Though these equations have been studied in the classical
mathematical literature, it appears that the physics community is largely
unaware of the relevant formalism. Of particular importance is the fate of the
initial value problem. Under what circumstances do infinite order differential
equations possess a well-defined initial value problem and how many initial
data are required? In this paper we study the initial value problem for
infinite order differential equations in the mathematical framework of the
formal operator calculus, with analytic initial data. This formalism allows us
to handle simultaneously a wide array of different nonlocal equations within a
single framework and also admits a transparent physical interpretation. We show
that differential equations of infinite order do not generically admit
infinitely many initial data. Rather, each pole of the propagator contributes
two initial data to the final solution. Though it is possible to find
differential equations of infinite order which admit well-defined initial value
problem with only two initial data, neither the dynamical equations of p-adic
string theory nor string field theory seem to belong to this class. However,
both theories can be rendered ghost-free by suitable definition of the action
of the formal pseudo-differential operator. This prescription restricts the
theory to frequencies within some contour in the complex plane and hence may be
thought of as a sort of ultra-violet cut-off.Comment: 40 pages, no figures. Added comments concerning fractional operators
and the implications of restricting the contour of integration. Typos
correcte
Holographic dual of the five-point conformal block
We present the holographic object which computes the five-point global
conformal block in arbitrary dimensions for external and exchanged scalar
operators. This object is interpreted as a weighted sum over infinitely many
five-point geodesic bulk diagrams. These five-point geodesic bulk diagrams
provide a generalization of their previously studied four-point counterparts.
We prove our claim by showing that the aforementioned sum over geodesic bulk
diagrams is the appropriate eigenfunction of the conformal Casimir operator
with the right boundary conditions. This result rests on crucial inspiration
from a much simpler -adic version of the problem set up on the Bruhat-Tits
tree.Comment: 20 pages + references, several figures. v2: Minor typos fixed,
matches published versio
Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions
This paper gives -dimensional analogues of the Apollonian circle packings
in parts I and II. We work in the space \sM_{\dd}^n of all -dimensional
oriented Descartes configurations parametrized in a coordinate system,
ACC-coordinates, as those real matrices \bW with \bW^T
\bQ_{D,n} \bW = \bQ_{W,n} where is the -dimensional Descartes quadratic
form, , and \bQ_{D,n} and
\bQ_{W,n} are their corresponding symmetric matrices. There are natural
actions on the parameter space \sM_{\dd}^n. We introduce -dimensional
analogues of the Apollonian group, the dual Apollonian group and the
super-Apollonian group. These are finitely generated groups with the following
integrality properties: the dual Apollonian group consists of integral matrices
in all dimensions, while the other two consist of rational matrices, with
denominators having prime divisors drawn from a finite set depending on the
dimension. We show that the the Apollonian group and the dual Apollonian group
are finitely presented, and are Coxeter groups. We define an Apollonian cluster
ensemble to be any orbit under the Apollonian group, with similar notions for
the other two groups. We determine in which dimensions one can find rational
Apollonian cluster ensembles (all curvatures rational) and strongly rational
Apollonian sphere ensembles (all ACC-coordinates rational).Comment: 37 pages. The third in a series on Apollonian circle packings
beginning with math.MG/0010298. Revised and extended. Added: Apollonian
groups and Apollonian Cluster Ensembles (Section 4),and Presentation for
n-dimensional Apollonian Group (Section 5). Slight revision on March 10, 200
Generalized Poincare algebras, Hopf algebras and kappa-Minkowski spacetime
We propose a generalized description for the kappa-Poincare-Hopf algebra as a
symmetry quantum group of underlying kappa-Minkowski spacetime. We investigate
all the possible implementations of (deformed) Lorentz algebras which are
compatible with the given choice of kappa-Minkowski algebra realization. For
the given realization of kappa-Minkowski spacetime there is a unique
kappa-Poincare-Hopf algebra with undeformed Lorentz algebra. We have
constructed a three-parameter family of deformed Lorentz generators with
kappa-Poincare algebras which are related to kappa-Poincare-Hopf algebra with
undeformed Lorentz algebra. Known bases of kappa-Poincare-Hopf algebra are
obtained as special cases. Also deformation of igl(4) Hopf algebra compatible
with the kappa-Minkowski spacetime is presented. Some physical applications are
briefly discussed.Comment: 15 pages; journal version; Physics Letters B (2012
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