scholarly journals On the Hopf algebra structure of perturbative quantum field theories

1998 ◽  
Vol 2 (2) ◽  
pp. 303-334 ◽  
Author(s):  
Dirk Kreimer
Keyword(s):  
Hopf Algebra ◽  
Field Theories ◽  
Algebra Structure ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Hopf Algebra Structure ◽  
Perturbative Quantum

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.

TWO-DIMENSIONAL TOPOLOGICAL QUANTUM FIELD THEORIES AND FROBENIUS ALGEBRAS

1996 ◽  
Vol 05 (05) ◽  
pp. 569-587 ◽  
Author(s):  
LOWELL ABRAMS
Keyword(s):  
Field Theories ◽  
Algebra Structure ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Two Dimensional ◽  
Direct Sums ◽  
Frobenius Algebras ◽  
Topological Quantum ◽  
Simple Demonstration ◽  
Topological Quantum Field Theories

We characterize Frobenius algebras A as algebras having a comultiplication which is a map of A-modules. This characterization allows a simple demonstration of the compatibility of Frobenius algebra structure with direct sums. We then classify the indecomposable Frobenius algebras as being either “annihilator algebras” — algebras whose socle is a principal ideal — or field extensions. The relationship between two-dimensional topological quantum field theories and Frobenius algebras is then formulated as an equivalence of categories. The proof hinges on our new characterization of Frobenius algebras. These results together provide a classification of the indecomposable two-dimensional topological quantum field theories.

Counterterms in the context of the universal Hopf algebra of renormalization

2014 ◽  
Vol 29 (08) ◽  
pp. 1450045 ◽  
Author(s):  
Ali Shojaei-Fard
Keyword(s):  
Hopf Algebra ◽  
Rooted Trees ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
New Interpretation

The manuscript discovers a new interpretation of counterterms of renormalizable Quantum Field Theories in terms of formal expansions of decorated rooted trees.

scholarly journals Renormalization in Combinatorially Non-Local Field Theories: The Hopf Algebra of 2-Graphs

2021 ◽  
Vol 24 (2) ◽  
Author(s):  
Johannes Thürigen
Keyword(s):  
Hopf Algebra ◽  
Local Field ◽  
Broad Class ◽  
Feynman Diagrams ◽  
Field Theories ◽  
Quantum Field ◽  
Random Geometry ◽  
Perturbative Renormalization ◽  
Non Local ◽  
Perturbative Quantum

AbstractRenormalization in perturbative quantum field theory is based on a Hopf algebra of Feynman diagrams. A precondition for this is locality. Therefore one might suspect that non-local field theories such as matrix or tensor field theories cannot benefit from a similar algebraic understanding. Here I show that, on the contrary, perturbative renormalization of a broad class of such field theories is based in the same way on a Hopf algebra. Their interaction vertices have the structure of graphs. This gives the necessary concept of locality and leads to Feynman diagrams defined as “2-graphs” which generate the Hopf algebra. These results set the stage for a systematic study of perturbative renormalization as well as non-perturbative aspects, e.g. Dyson-Schwinger equations, for a number of combinatorially non-local field theories with possible applications to random geometry and quantum gravity.

scholarly journals q-DEFORMED CONFORMAL CORRELATION FUNCTIONS

2005 ◽  
Vol 20 (07) ◽  
pp. 1471-1479 ◽  
Author(s):  
L. MESREF
Keyword(s):  
Hopf Algebra ◽  
Correlation Functions ◽  
General Structure ◽  
Field Theories ◽  
Algebra Structure ◽  
Quantum Deformation ◽  
Hopf Algebra Structure

In this paper, we compute the general structure of two- and three-point functions in field theories that are assumed to possess an invariance under a quantum deformation of SO (4, 2). The computation is elaborated in order to fit the Hopf algebra structure.

scholarly journals A bound on thermal relativistic correlators at large spacelike momenta

2020 ◽  
Vol 8 (4) ◽  
Author(s):  
Souvik Banerjee ◽  
Kyriakos Papadodimas ◽  
Suvrat Raju ◽  
Prasant Samantray ◽  
Pushkal Shrivastava
Keyword(s):  
Field Theory ◽  
Relativistic Quantum ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Relativistic Quantum Field Theory ◽  
Point Function ◽  
Open Problems ◽  
Perturbative Quantum ◽  
Free Fields

We consider thermal Wightman correlators in a relativistic quantum field theory in the limit where the spatial momenta of the insertions become large while their frequencies stay fixed. We show that, in this limit, the size of these correlators is bounded by e^{-\beta R}e−βR, where RR is the radius of the smallest sphere that contains the polygon formed by the momenta. We show that perturbative quantum field theories can saturate this bound through suitably high-order loop diagrams. We also consider holographic theories in dd-spacetime dimensions, where we show that the leading two-point function of generalized free-fields saturates the bound in d = 2d=2 and is below the bound for d > 2d>2. We briefly discuss interactions in holographic theories and conclude with a discussion of several open problems.

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