
Special Issue on Deformations of SpaceTime and its Symmetries
The Guest Editors for this special issue are
Gaetano Fiore (Università di Napoli "Federico II", and INFN  Sez. di
Napoli, Italy)
T.R. Govindarajan (Chennai Mathematical Institute, India)
Jerzy KowalskiGlikman (University of Wroclaw, Poland)
Petr Kulish (St. Petersburg Department of Steklov Institute of Mathematics, Russia)
Jerzy Lukierski (University of Wroclaw, Poland)
Preface
There is a general agreement that the main challenge
fundamental physics is facing nowadays is to find a new paradigm which
reconciles the world of quantum with the world of gravity. It is
also expected that within this new paradigm some new mathematics
will have to be introduced
in order
to describe the
expected discreteness and/or noncommutativity of spacetime, and
deformations of its symmetries, at the Planck scale.
Deformations of the algebra of classical spacetime functions were
firstly considered by introducing
Groenewold–Moyal starproduct
technique. This simple deformation
was however not sufficient when the spacetime geometry became curved,
as required by the Einstein construction in general relativity.
Further, one needs to introduce curved momentum space
if one wishes
to incorporate Born's reciprocity principle.
The
first technical realization of a deformed relativistic spacetime
algebra was proposed in 1940’s by Snyder with the hope to resolve the
divergences problem of relativistic quantum field theories. These expectations
were not fulfilled, but after a couple of decades the efforts of linking geometry,
gravity and quantum theory by using noncommutative geometry was undertaken
by many mathematicians and theoretical physicists. They
developed the theory of quantum groups, noncommutative spaces and noncommutative
algebraic geometry
and
provided various models of deformed
relativistic symmetries and quantum spacetimes.
Further, in late 1990’s the approaches using noncommutative (super)spacetimes did find important
support in the framework of quantized (super)string theory.
It is interesting to recall that the inadequacies of Riemannian
geometry at infinitesimal scales were mentioned by Riemann himself
who stated “… it seems that empirical notions on which the
metrical determinations of space are founded, the notion of a solid
body and a ray of light cease to be valid for the infinitely small.
We are therefore quite at liberty to suppose that the metric
relations of space in the infinitely small do not conform to
hypotheses of geometry; …” in his famous 1854 habilitation
lecture. The physical arguments about the inapplicability of
classical (pseudo)Riemannian geometry to the description of quantum
world were presented only when
general relativity and quantum mechanics had been invented.
In 1930’s M. Bronstein, using Heisenberg
uncertainty principle and the features of Einstein general
relativity firstly argued that gravitational dynamics does not allow
measurement of arbitrarily small spacetime distances, concluding
that the notions of classical Riemannian geometry should be duly modified.
In this Special Issue of SIGMA many problems of the deformation theory
of algebras and deformed symmetries
have
been investigated. In eighteen
contributions to the Issue there were covered several broad themes, namely
 models of noncommutative spacetime and its symmetries;
 novel physical features of quantum theories on Moyal spacetimes;
 κdeformed algebras and quantum symmetries;
 QFTs/gauge theories on κMinkowski and κAdS
spaces;
 group field theory and its symmetries;
 deformations of special relativity (DSR) and curved momentum space
formalism;
 matrix models and fuzzy geometries;
 noncommutativity, thermodynamics and phase transitions.
We believe that the presented results may help in the search for
the
new paradigm mentioned
at the beginning.
We would like to thank warmly all the contributors to this Special
Issue and the referees for their
careful and insightful
reviews.
The Guest Editors
Papers in this Issue:

κDeformations and Extended κMinkowski Spacetimes

Andrzej Borowiec and Anna Pachoł
SIGMA 10 (2014), 107, 24 pages [ abs
pdf ]

κDeformed Phase Space, Hopf Algebroid and Twisting

Tajron Jurić, Domagoj Kovačević and Stjepan Meljanac
SIGMA 10 (2014), 106, 18 pages [ abs
pdf ]

Effects of a Maximal Energy Scale in Thermodynamics for Photon Gas and Construction of Path Integral

Sudipta Das, Souvik Pramanik and Subir Ghosh
SIGMA 10 (2014), 104, 26 pages [ abs
pdf ]

Wong's Equations and Charged Relativistic Particles in NonCommutative Space

Herbert Balasin, Daniel N. Blaschke, François Gieres and Manfred Schweda
SIGMA 10 (2014), 099, 21 pages [ abs
pdf ]

Matrix Bases for Star Products: a Review

Fedele Lizzi and Patrizia Vitale
SIGMA 10 (2014), 086, 36 pages [ abs
pdf ]

Group Momentum Space and Hopf Algebra Symmetries of Point Particles Coupled to 2+1 Gravity

Michele Arzano, Danilo Latini and Matteo Lotito
SIGMA 10 (2014), 079, 23 pages [ abs
pdf ]

Energy Spectrum and Phase Transition of Superfluid Fermi Gas of Atoms on Noncommutative Space

YanGang Miao and Hui Wang
SIGMA 10 (2014), 075, 19 pages [ abs
pdf ]

The SoccerBall Problem

Sabine Hossenfelder
SIGMA 10 (2014), 074, 8 pages [ abs
pdf ]

Asymptotic Analysis of the PonzanoRegge Model with NonCommutative Metric Boundary Data

Daniele Oriti and Matti Raasakka
SIGMA 10 (2014), 067, 32 pages [ abs
pdf ]

Gauge Theory on Twisted κMinkowski: Old Problems and Possible Solutions

Marija Dimitrijević, Larisa Jonke and Anna Pachoł
SIGMA 10 (2014), 063, 22 pages [ abs
pdf ]

Deformations of the Canonical Commutation Relations and Metric Structures

Francesco D'Andrea, Fedele Lizzi and Pierre Martinetti
SIGMA 10 (2014), 062, 14 pages [ abs
pdf ]

TwoPoint Functions on Deformed Spacetime

Josip Trampetić and Jiangyang You
SIGMA 10 (2014), 054, 20 pages [ abs
pdf ]

Towards NonCommutative Deformations of Relativistic Wave Equations in 2+1 Dimensions

Bernd J. Schroers and Matthias Wilhelm
SIGMA 10 (2014), 053, 23 pages [ abs
pdf ]

Twisted (2+1) κAdS Algebra, Drinfel'd Doubles and NonCommutative Spacetimes

Ángel Ballesteros, Francisco J. Herranz, Catherine Meusburger and Pedro Naranjo
SIGMA 10 (2014), 052, 26 pages [ abs
pdf ]

Symmetries of the Free Schrödinger Equation in the NonCommutative Plane

Carles Batlle, Joaquim Gomis and Kiyoshi Kamimura
SIGMA 10 (2014), 011, 15 pages [ abs
pdf ]

Dirac Operators on Noncommutative Curved Spacetimes

Alexander Schenkel and Christoph F. Uhlemann
SIGMA 9 (2013), 080, 19 pages [ abs
pdf ]

An Index for Intersecting Branes in Matrix Models

Harold Steinacker and Jochen Zahn
SIGMA 9 (2013), 067, 7 pages [ abs
pdf ]

Generalized Fuzzy Torus and its Modular Properties

Paul Schreivogl and Harold Steinacker
SIGMA 9 (2013), 060, 23 pages [ abs
pdf ]

