scholarly journals An Introduction to κ-Deformed Symmetries, Phase Spaces and Field Theory

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 946
Author(s):  
Michele Arzano ◽  
Jerzy Kowalski-Glikman
Keyword(s):  
Field Theory ◽  
Phase Space ◽  
Scalar Field ◽  
Lie Groups ◽  
Relativistic Particle ◽  
Deformed Symmetries ◽  
Poincaré Algebra ◽  
Free Scalar ◽  
Poisson Lie Groups ◽  
Relativistic Phase

In this review, we give a basic introduction to the κ-deformed relativistic phase space and free quantum fields. After a review of the κ-Poincaré algebra, we illustrate the construction of the κ-deformed phase space of a classical relativistic particle using the tools of Lie bi-algebras and Poisson–Lie groups. We then discuss how to construct a free scalar field theory on the non-commutative κ-Minkowski space associated to the κ-Poincaré and illustrate how the group valued nature of momenta affects the field propagation.

scholarly journals A short introduction to κ-deformation

2017 ◽  
Vol 32 (35) ◽  
pp. 1730026 ◽  
Author(s):  
J. Kowalski-Glikman
Keyword(s):  
Field Theory ◽  
Scalar Field ◽  
Minkowski Space ◽  
Momentum Space ◽  
Short Review ◽  
Short Introduction ◽  
Scalar Field Theory ◽  
Poincaré Algebra ◽  
Free Scalar

In this short review we describe some aspects of [Formula: see text]-deformation. After discussing the algebraic and geometric approaches to [Formula: see text]-Poincaré algebra we construct the free scalar field theory, both on noncommutative [Formula: see text]-Minkowski space and on curved momentum space. Finally, we make a few remarks concerning the interacting scalar field.

POISSON–LIE GROUPS AND CLASSICAL W-ALGEBRAS

1992 ◽  
Vol 07 (05) ◽  
pp. 853-876 ◽  
Author(s):  
V. A. FATEEV ◽  
S. L. LUKYANOV
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Quantum Group ◽  
Lie Groups ◽  
Group Structure ◽  
Two Dimensional ◽  
Poisson Structures ◽  
Additional Symmetries ◽  
Conformal Field ◽  
Poisson Lie Groups

This is the first part of a paper studying the quantum group structure of two-dimensional conformal field theory with additional symmetries. We discuss the properties of the Poisson structures possessing classical W-invariance. The Darboux variables for these Poisson structures are constructed.

scholarly journals GEOMETRIC STRUCTURES OF THE CLASSICAL GENERAL RELATIVISTIC PHASE SPACE

2008 ◽  
Vol 05 (05) ◽  
pp. 699-754 ◽  
Author(s):  
JOSEF JANYŠKA ◽  
MARCO MODUGNO
Keyword(s):  
Electromagnetic Field ◽  
Phase Space ◽  
Relativistic Particle ◽  
Jet Space ◽  
Geometric Structures ◽  
Geometric Properties ◽  
Geometric Conditions ◽  
General Relativistic ◽  
Relativistic Phase

This paper is concerned with basic geometric properties of the phase space of a classical general relativistic particle, regarded as the 1st jet space of motions, i.e. as the 1st jet space of timelike 1-dimensional submanifolds of spacetime. This setting allows us to skip constraints. Our main goal is to determine the geometric conditions by which the Lorentz metric and a connection of the phase space yield contact and Jacobi structures. In particular, we specialize these conditions to the cases when the connection of the phase space is generated by the metric and an additional tensor. Indeed, the case generated by the metric and the electromagnetic field is included, as well.

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