scholarly journals Renormalization in Quantum Field Theory and the Riemann-Hilbert Problem I: The Hopf Algebra Structure of Graphs and the Main Theorem

2000 ◽  
Vol 210 (1) ◽  
pp. 249-273 ◽  
Author(s):  
Alain Connes ◽  
Dirk Kreimer
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Hopf Algebra ◽  
Hilbert Problem ◽  
Algebra Structure ◽  
Quantum Field ◽  
Hopf Algebra Structure ◽  
Riemann Hilbert Problem

scholarly journals ON ALGEBRAIC STRUCTURES IMPLICIT IN TOPOLOGICAL QUANTUM FIELD THEORIES

1999 ◽  
Vol 08 (02) ◽  
pp. 125-163 ◽  
Author(s):  
Louis Crane ◽  
David Yetter
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Hopf Algebra ◽  
Tensor Category ◽  
Topological Quantum Field Theory ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Topological Quantum ◽  
Natural Way

We show that any 3D topological quantum field theory satisfying physically reasonable factorizability conditions has associated to it in a natural way a Hopf algebra object in a suitable tensor category. We also show that all objects in the tensor category have the structure of left-left crossed bimodules over the Hopf algebra object. For 4D factorizable topological quantum filed theories, we provide by analogous methods a construction of a Hopf algebra category.

scholarly journals A NONASSOCIATIVE QUATERNION SCALAR FIELD THEORY

2013 ◽  
Vol 28 (35) ◽  
pp. 1350163 ◽  
Author(s):  
SERGIO GIARDINO ◽  
PAULO TEOTÔNIO-SOBRINHO
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Scalar Field ◽  
String Theory ◽  
Hopf Algebra ◽  
Scalar Product ◽  
Quantum Field ◽  
Quantum Algebras ◽  
Function Algebras ◽  
Nonlinear Quantum

A nonassociative Groenewold–Moyal (GM) plane is constructed using quaternion-valued function algebras. The symmetrized multiparticle states, the scalar product, the annihilation/creation algebra and the formulation in terms of a Hopf algebra are also developed. Nonassociative quantum algebras in terms of position and momentum operators are given as the simplest examples of a framework whose applications may involve string theory and nonlinear quantum field theory.

scholarly journals BOGOLUBOV'S RECURSION AND INTEGRABILITY OF EFFECTIVE ACTIONS

2001 ◽  
Vol 16 (09) ◽  
pp. 1531-1558 ◽  
Author(s):  
A. GERASIMOV ◽  
A. MOROZOV ◽  
K. SELIVANOV
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Hopf Algebra ◽  
Moduli Space ◽  
Feynman Diagrams ◽  
Coupling Constants ◽  
Quantum Field ◽  
Integrable Structure ◽  
Standard Formalism

The Hopf algebra of Feynman diagrams, analyzed by A. Connes and D. Kreimer, is considered from the perspective of the theory of effective actions and generalized τ-functions, which describes the action of diffeomorphism and shift groups in the moduli space of coupling constants. These considerations provide additional evidence of the hidden group (integrable) structure behind the standard formalism of quantum field theory.

scholarly journals GAUGE AND POINCARÉ INVARIANT REGULARIZATION AND HOPF SYMMETRIES

2012 ◽  
Vol 27 (17) ◽  
pp. 1250097 ◽  
Author(s):  
FEDELE LIZZI ◽  
PATRIZIA VITALE
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Gauge Symmetry ◽  
Hopf Algebra ◽  
Quantum Field ◽  
Cutoff Function ◽  
Invariant Regularization

We consider the regularization of a gauge quantum field theory following a modification of the Pochinski proof based on the introduction of a cutoff function. We work with a Poincaré invariant deformation of the ordinary pointwise product of fields introduced by Ardalan, Arfaei, Ghasemkhani and Sadooghi, and show that it yields, through a limiting procedure of the cutoff functions, to a regularized theory, preserving all symmetries at every stage. The new gauge symmetry yields a new Hopf algebra with deformed costructures, which is inequivalent to the standard one.

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