From local perturbation theory to Hopf- and Lie-algebras of Feynman graphs

2001 ◽  
pp. 105-114
Author(s):  
A. Connes ◽  
D. Kreimer
Keyword(s):  
Perturbation Theory ◽  
Lie Algebras ◽  
Local Perturbation ◽  
Feynman Graphs

scholarly journals Tensor Renormalization Group for interacting quantum fields

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 586
Author(s):  
Manuel Campos ◽  
German Sierra ◽  
Esperanza Lopez
Keyword(s):  
Free Energy ◽  
Perturbation Theory ◽  
Renormalization Group ◽  
Real Space ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Excellent Performance ◽  
Tensor Network ◽  
Feynman Graphs

We present a new tensor network algorithm for calculating the partition function of interacting quantum field theories in 2 dimensions. It is based on the Tensor Renormalization Group (TRG) protocol, adapted to operate entirely at the level of fields. This strategy was applied in Ref.[1] to the much simpler case of a free boson, obtaining an excellent performance. Here we include an arbitrary self-interaction and treat it in the context of perturbation theory. A real space analogue of the Wilsonian effective action and its expansion in Feynman graphs is proposed. Using a λϕ4 theory for benchmark, we evaluate the order λ correction to the free energy. The results show a fast convergence with the bond dimension, implying that our algorithm captures well the effect of interaction on entanglement.

scholarly journals Gauge invariant perturbation theory via homotopy transfer

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Christoph Chiaffrino ◽  
Olaf Hohm ◽  
Allison F. Pinto
Keyword(s):  
Perturbation Theory ◽  
Lie Algebras ◽  
Closed String ◽  
Cosmological Perturbation ◽  
Gauge Invariant ◽  
Yang Mills ◽  
Cosmological Perturbation Theory ◽  
Invariant Perturbation Theory ◽  
General Gauge ◽  
Algorithmic Procedure

Abstract We show that the perturbative expansion of general gauge theories can be expressed in terms of gauge invariant variables to all orders in perturbations. In this we generalize techniques developed in gauge invariant cosmological perturbation theory, using Bardeen variables, by interpreting the passing over to gauge invariant fields as a homotopy transfer of the strongly homotopy Lie algebras encoding the gauge theory. This is illustrated for Yang-Mills theory, gravity on flat and cosmological backgrounds and for the massless sector of closed string theory. The perturbation lemma yields an algorithmic procedure to determine the higher corrections of the gauge invariant variables and the action in terms of these.

Perturbation Theory and Lie Algebras

1962 ◽  
Vol 3 (3) ◽  
pp. 451-468 ◽  
Author(s):  
F. J. Murray
Keyword(s):  
Perturbation Theory ◽  
Lie Algebras

Weights of Feynman diagrams, link polynomials and Vassiliev knot invariants

1995 ◽  
Vol 04 (01) ◽  
pp. 163-188 ◽  
Author(s):  
Sergey Piunikhin
Keyword(s):  
Perturbation Theory ◽  
Irreducible Representation ◽  
Hopf Algebra ◽  
Lie Algebras ◽  
Feynman Diagrams ◽  
Primitive Elements ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Vassiliev Invariant ◽  
Chern Simons

We prove that the construction of Vassiliev invariants by expanding the link polynomials Pg,V(q, q−1) at the point q=1 is equivalent to the construction of Vassiliev invariants from Chern-Simons perturbation theory. In both constructions a simple Lie algebra g and an irreducible representation V of g should be specified. We give an example of a Vassiliev invariant of order six which cannot be obtained by these constructions if we restrict ourselves to simple Lie algebras and do not allow semisimple ones. The explicit description of primitive elements in the Kontsevich Hopf algebra is given.

scholarly journals Loop amplitudes monodromy relations and color-kinematics duality

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Eduardo Casali ◽  
Sebastian Mizera ◽  
Piotr Tourkine
Keyword(s):  
Field Theory ◽  
Perturbation Theory ◽  
Gauge Theory ◽  
Tree Level ◽  
First Principle ◽  
Loop Momentum ◽  
New Developments ◽  
Feynman Graphs ◽  
Definition Of ◽  
Double Copy

Abstract Color-kinematics duality is a remarkable conjectured property of gauge theory which, together with double copy, is at the heart of a wealth of new developments in scattering amplitudes. So far, its validity has been verified in most cases only empirically, with limited ab initio understanding beyond tree-level. In this paper we provide initial steps in a first-principle understanding of color-kinematics duality and double-copy at loop level, through a detailed analysis of the field-theory limit of the monodromy relations of string theory at one loop. In this limit, we dissect the type of Feynman graphs generated and the relations they obey. We find that graphs with contact-terms are unavoidable and are generated in the field theory limit of “bulk” contours which do not have a standard physical interpretation in string perturbation theory. We show how they are related to ambiguities in the definition of the loop momentum and that their role is precisely to cancel those ambiguities.

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