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 Non-unitarity of Minkowskian non-local quantum field theories

2021 ◽  
Vol 81 (8) ◽  
Author(s):  
Fabio Briscese ◽  
Leonardo Modesto
Keyword(s):  
Scalar Field ◽  
Imaginary Part ◽  
Local Field ◽  
Field Theories ◽  
Unitarity Condition ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Not Given ◽  
Local Quantum ◽  
Non Local

AbstractWe show that Minkowskian non-local quantum field theories are not unitary. We consider a simple one loop diagram for a scalar non-local field and show that the imaginary part of the corresponding complex amplitude is not given by Cutkosky rules, indeed this diagram violates the unitarity condition. We compare this result with the case of an Euclidean non-local scalar field, that has been shown to satisfy the Cutkosky rules, and we clearly identify the reason of the breaking of unitarity of the Minkowskian theory.

scholarly journals Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme

2021 ◽  
Author(s):  
Johannes Thürigen ◽  
◽  
◽  
◽  
◽  
...  
Keyword(s):  
Field Theory ◽  
Hopf Algebra ◽  
Local Field ◽  
Algebraic Method ◽  
Field Theories ◽  
Noncommutative Field Theory ◽  
Matrix Formulation ◽  
Perturbative Calculations ◽  
Local Field Theory ◽  
Non Local

Various combinatorially non-local field theories are known to be renormalizable. Still, explicit calculations of amplitudes are very rare and restricted to matrix field theory. In this contribution I want to demonstrate how the BPHZ momentum scheme in terms of the Connes-Kreimer Hopf algebra applies to any combinatorially non-local field theory which is renormalizable. This algebraic method improves the understanding of known results in noncommutative field theory in its matrix formulation. Furthermore, I use it to provide new explicit perturbative calculations of amplitudes in tensorial field theories of rank r>2.

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 RENORMALIZATION WITHOUT INFINITIES

2005 ◽  
Vol 20 (06) ◽  
pp. 1336-1345 ◽  
Author(s):  
GERARD 'T HOOFT
Keyword(s):  
Renormalization Group ◽  
Conceptual Understanding ◽  
Feynman Diagrams ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Renormalization Group Equations

Most renormalizable quantum field theories can be rephrased in terms of Feynman diagrams that only contain dressed irreducible 2-, 3-, and 4-point vertices. These irreducible vertices in turn can be solved from equations that also only contain dressed irreducible vertices. The diagrams and equations that one ends up with do not contain any ultraviolet divergences. The original bare Lagrangian of the theory only enters in terms of freely adjustable integration constants. It is explained how the procedure proposed here is related to the renormalization group equations. The procedure requires the identification of unambiguous "paths" in a Feynman diagrams, and it is shown how to define such paths in most of the quantum field theories that are in use today. We do not claim to have a more convenient calculational scheme here, but rather a scheme that allows for a better conceptual understanding of ultraviolet infinities.

On differential conservation laws in non-local field theories

1953 ◽  
Vol 10 (2) ◽  
pp. 182-185 ◽  
Author(s):  
J. Rzewuski
Keyword(s):  
Conservation Laws ◽  
Local Field ◽  
Field Theories ◽  
Non Local

On the hamiltonian structure of non-local field theories

1953 ◽  
Vol 10 (5) ◽  
pp. 648-667 ◽  
Author(s):  
W. Pauli
Keyword(s):  
Local Field ◽  
Hamiltonian Structure ◽  
Field Theories ◽  
Non Local

Thermodynamics and cosmological constant of non-local field theories from p-adic strings

2010 ◽  
Vol 2010 (10) ◽  
Author(s):  
Tirthabir Biswas ◽  
Jose A. R. Cembranos ◽  
Joseph I. Kapusta
Keyword(s):  
Cosmological Constant ◽  
Local Field ◽  
Field Theories ◽  
Non Local

Quantum Field Theory

2020 ◽  
pp. 237-288
Author(s):  
Giuseppe Mussardo
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Perturbation Theory ◽  
Canonical Quantization ◽  
Feynman Diagrams ◽  
Field Theories ◽  
Coordinate Space ◽  
Matrix Formalism ◽  
Quantum Field ◽  
Transfer Matrix Formalism

Chapter 7 covers the main reasons for adopting the methods of quantum field theory (QFT) to study the critical phenomena. It presents both the canonical quantization and the path integral formulation of the field theories as well as the analysis of the perturbation theory. The chapter also covers transfer matrix formalism and the Euclidean aspects of QFT, the field theory of the Ising model, Feynman diagrams, correlation functions in coordinate space, the Minkowski space and the Legendre transformation and vertex functions. Everything in this chapter will be needed sooner or later, since it highlights most of the relevant aspects of quantum field theory.

FIRST NON-VANISHING SELF-LINKING OF KNOTS (I) COMBINATORIC AND DIAGRAMMATIC STUDY

2011 ◽  
Vol 20 (12) ◽  
pp. 1637-1648 ◽  
Author(s):  
CHUN-CHUNG HSIEH
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Knot Theory ◽  
Feynman Diagrams ◽  
Quantum Field ◽  
Borromean Rings ◽  
Diagrammatic Representation ◽  
Perturbative Quantum ◽  
Perturbative Quantum Field Theory ◽  
Chern Simons

In this paper, following the scheme of [Borromean rings and linkings, J. Geom. Phys.60 (2010) 823–831; Combinatoric and diagrammatic study in knot theory, J. Knot Theory Ramifications16 (2007) 1235–1253; Massey–Milnor linking = Chern–Simons–Witten graphs, J. Knot Theory Ramifications17 (2008) 877–903], we study the first non-vanishing self-linkings of knots, aiming at the study of combinatorial formulae and diagrammatic representation. The upshot of perturbative quantum field theory is to compute the Feynman diagrams explicitly, though it is impossible in general. Along this line in this paper we could not only compute some Feynman diagrams, but also give the explicit and combinatorial formulae.

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