Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

Special Issue on Deformations of Space-Time 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 Kowalski-Glikman (University of Wroclaw, Poland)
Petr Kulish (St. Petersburg Department of Steklov Institute of Mathematics, Russia)
Jerzy Lukierski (University of Wroclaw, Poland)


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 non-commutativity of space-time, and deformations of its symmetries, at the Planck scale.

Deformations of the algebra of classical space-time functions were firstly considered by introducing Groenewold–Moyal star-product technique. This simple deformation was however not sufficient when the space-time 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 space-time 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 space-times. Further, in late 1990’s the approaches using noncommutative (super)space-times 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 space-time 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 space-time and its symmetries;
  • novel physical features of quantum theories on Moyal spacetimes;
  • κ-deformed algebras and quantum symmetries;
  • QFTs/gauge theories on κ-Minkowski and κ-AdS spaces;
  • group field theory and its symmetries;
  • deformations of special relativity (DSR) and curved momentum space formalism;
  • matrix models and fuzzy geometries;
  • non-commutativity, 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:

κ-Deformations and Extended κ-Minkowski Spacetimes
Andrzej Borowiec and Anna Pachoł
SIGMA 10 (2014), 107, 24 pages   [ abs   pdf ]
κ-Deformed 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 Non-Commutative 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
Yan-Gang Miao and Hui Wang
SIGMA 10 (2014), 075, 19 pages   [ abs   pdf ]
The Soccer-Ball Problem
Sabine Hossenfelder
SIGMA 10 (2014), 074,  8 pages   [ abs   pdf ]
Asymptotic Analysis of the Ponzano-Regge Model with Non-Commutative Metric Boundary Data
Daniele Oriti and Matti Raasakka
SIGMA 10 (2014), 067, 32 pages   [ abs   pdf ]
Gauge Theory on Twisted κ-Minkowski: 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 ]
Two-Point Functions on Deformed Spacetime
Josip Trampetić and Jiangyang You
SIGMA 10 (2014), 054, 20 pages   [ abs   pdf ]
Towards Non-Commutative 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) κ-AdS Algebra, Drinfel'd Doubles and Non-Commutative 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 Non-Commutative 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 ]

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