Hopf Algebra Of Renormalization Research Articles

Graph polynomials associated with Dyson-Schwinger equations

Quantum motions are encoded by a particular family of recursive Hochschild equations in the renormalization Hopf algebra which represent Dyson-Schwinger equations, combinatorially. Feynman graphons, which topologically complete the space of Feynman diagrams of a gauge field theory, are considered to formulate some random graph representations for solutions of quantum motions. This framework leads us to explain the structures of Tutte and Kirchhoff-Symanzik polynomials associated with solutions of Dyson-Schwinger equations. These new graph polynomials are applied to formulate a new parametric representation for large Feynman diagrams and their corresponding Feynman rules.

FROM DYSON–SCHWINGER EQUATIONS TO QUANTUM ENTANGLEMENT

We apply combinatorial Dyson–Schwinger equations and their Feynman graphon representations to study quantum entanglement in a gauge field theory $${\varPhi }$$ in terms of cut-distance regions of Feynman diagrams in the topological renormalization Hopf algebra $$H^{\text {cut}}_{\text {FG}}({\varPhi })$$ and lattices of intermediate structures. Feynman diagrams in $$H_{\text {FG}}({\varPhi })$$ are applied to describe states in $${\varPhi }$$ where we build the Fisher information metric on finite dimensional linear subspaces of states in terms of homomorphism densities of Feynman graphons which are continuous functionals on the topological space $$\mathcal {S}^{{\varPhi },M \subseteq [0,\infty )}_{\text {graphon}}([0,1])$$ . We associate Hopf subalgebras of $$H_{\text {FG}}({\varPhi })$$ generated by quantum motions to separated regions of space-time to address some new correlations. These correlations are encoded by assigning a statistical manifold to the space of 1PI Green’s functions of $${\varPhi }$$ . These correlations are applied to build lattices of Hopf subalgebras, Lie subgroups, and Tannakian subcategories, derived from towers of combinatorial Dyson–Schwinger equations, which contribute to separated but correlated cut-distance topological regions. This lattice setting is applied to formulate a new tower of renormalization groups which encodes quantum entanglement of space-time separated particles under different energy scales.

Gauge Symmetries and Renormalization

We study the perturbative renormalization of quantum gauge theories in the Hopf algebra setup of Connes and Kreimer. It was shown by van Suijlekom (Commun Math Phys 276:773–798, 2007) that the quantum counterparts of gauge symmetries—the so-called Ward–Takahashi and Slavnov–Taylor identities—correspond to Hopf ideals in the respective renormalization Hopf algebra. We generalize this correspondence to super- and non-renormalizable Quantum Field Theories, extend it to theories with multiple coupling constants and add a discussion on transversality. In particular, this allows us to apply our results to (effective) Quantum General Relativity, possibly coupled to matter from the Standard Model, as was suggested by Kreimer (Ann Phys 323:49–60, 2008). To this end, we introduce different gradings on the renormalization Hopf algebra and study combinatorial properties of the superficial degree of divergence. Then we generalize known coproduct and antipode identities to the super- and non-renormalizable cases and to theories with multiple vertex residues. Building upon our main result, we provide criteria for the compatibility of these Hopf ideals with the corresponding renormalized Feynman rules. A direct consequence of our findings is the well-definedness of the Corolla polynomial for Quantum Yang–Mills theory without reference to a particular renormalization scheme.

Halting problem in Feynman graphon processes derived from the renormalization Hopf algebra

Thanks to the theory of graphons and random graphs, Feynman graphons are new analytic tools for the study of infinities in (strongly coupled) gauge field theories. We formulate the Halting problem in Feynman graphon processes to build a new theory of computation in dealing with solutions of combinatorial Dyson–Schwinger equations in the context of the Turing machines and Manin’s renormalization Hopf algebra.

Non-perturbative graph languages, halting problem and complexity

Abstract We explain the foundations of a new class of formal languages for the construction of large Feynman diagrams which contribute to solutions of all combinatorial Dyson–Schwinger equations in a given strongly coupled gauge field theory. Then we build a new Hopf algebraic structure on non-perturbative production rules which leads us to formulate the halting problem for the corresponding replacing–gluing graph grammars in our formal graph languages on the basis of Manin’s renormalization Hopf algebra. In addition, we apply topology of graphons to associate a complexity parameter to this new class of graph grammars. At the final step, we address some applications of our new formal language platform to Quantum Field Theory. The first application concerns the constructive role of non-perturbative graph languages in dealing with quantum gauge symmetries in the context of the Hopf ideals generated by Slavnov–Taylor or Ward–Takahashi identities. The second application concerns the importance of the complexities of non-perturbative replacing–gluing graph grammars in formulating a new generalization of the circuit complexity on the space of Dyson–Schwinger equations. We provide a geometric interpretation of non-perturbative circuit complexities. The third application concerns the impact of non-perturbative replacing–gluing graph grammars in providing some new tools for the computation of the Kolmogorov complexity of Dyson–Schwinger equations.

The dynamics of non-perturbative phases via Banach bundles

Strongly coupled Dyson–Schwinger equations generate infinite power series of running coupling constants together with Feynman diagrams with increasing loop orders as coefficients. Theory of graphons for sparse graphs can address a new useful approach for the study of graph limits of sequences of partial sums corresponding to these infinite power series in the context of Feynman graphons and the cut-distance topology. Graphon models enable us to associate some new analytic graphs to non-perturbative solutions of Dyson–Schwinger equations. Homomorphism densities of Feynman graphons provide a new way of analyzing non-perturbative phase transitions. We explain the structures of topological renormalization quotient Hopf algebras of Feynman graphons which encode gauge symmetries Hopf ideals in the context of the weakly isomorphic equivalence classes corresponding to the Slavnov–Taylor / Ward–Takahashi elements. We apply Feynman graphon representations of combinatorial Dyson–Schwinger equations underlying the Connes–Kreimer renormalization Hopf algebra to construct a new class of Banach bundles for the study of the dynamics of non-perturbative phases in strongly coupled gauge field theories.

Graphons and renormalization of large Feynman diagrams

The article builds a new enrichment of the Connes-Kreimer renormalization Hopf algebra of Feynman diagrams in the language of graph functions.

A new perspective on intermediate algorithms via the Riemann–Hilbert correspondence

The article deals with the Halting problem in the theory of computation with respect to the renormalization Hopf algebra. As the consequence, we will address a new treatment in the study of intermediate algorithms in the context of the Riemann–Hilbert correspondence.

Counterterms in the context of the universal Hopf algebra of renormalization

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

MOTIVIC DYSON–SCHWINGER EQUATIONS

We consider Dyson–Schwinger Equations (DSEs) in the context of Connes–Kreimer renormalization Hopf algebra of Feynman diagrams and Connes–Marcolli universal Tannakian formalism. This study leads us to formulate a family of Picard–Fuchs equations and a category of Feynman motivic sheaves with respect to each combina torial DSE.

FROM DYSON–SCHWINGER EQUATIONS TO THE RIEMANN–HILBERT CORRESPONDENCE

In this paper, with the study of combinatorial Dyson–Schwinger equations at the level of the universal Hopf algebra of renormalization and with the extension of the universality of this specific Hopf algebra and also category of flat equisingular vector bundles to the level of these equations, we are going to consider the notion of a geometric description from nonperturbative theory.

Combinatorial Hopf algebras from renormalization

In this paper we describe the right-sided combinatorial Hopf structure of three Hopf algebras appearing in the context of renormalization in quantum field theory: the non-commutative version of the Fa\`a di Bruno Hopf algebra, the non-commutative version of the charge renormalization Hopf algebra on planar binary trees for quantum electrodynamics, and the non-commutative version of the Pinter renormalization Hopf algebra on any bosonic field. We also describe two general ways to define the associative product in such Hopf algebras, the first one by recursion, and the second one by grafting and shuffling some decorated rooted trees.

Fixed Point Equations Related to Motion Integrals in Renormalization Hopf Algebra

Fixed Point Equations Related to Motion Integrals in Renormalization Hopf Algebra

A noncommutative Bohnenblust–Spitzer identity for Rota–Baxter algebras solves Bogoliubov’s recursion

The Bogoliubov recursion is a particular procedure appearing in the process of renormalization in perturbative quantum field theory. It provides convergent expressions for otherwise divergent integrals. We develop here a theory of functional identities for noncommutative Rota-Baxter algebras which is shown to encode, among others, this process in the context of Connes-Kreimer's Hopf algebra of renormalization. Our results generalize the seminal Cartier-Rota theory of classical Spitzer-type identities for commutative Rota-Baxter algebras. In the classical, commutative, case, these identities can be understood as deriving from the theory of symmetric functions. Here, we show that an analogous property holds for noncommutative Rota-Baxter algebras. That is, we show that functional identities in the noncommutative setting can be derived from the theory of noncommutative symmetric functions. Lie idempotents, and particularly the Dynkin idempotent play a crucial role in the process. Their action on the pro-unipotent groups such as those of perturbative renormalization is described in detail along the way.

The Structure of Renormalization Hopf Algebras for Gauge Theories I: Representing Feynman Graphs on BV-Algebras

We study the structure of renormalization Hopf algebras of gauge theories. We identify certain Hopf subalgebras in them, whose character groups are semidirect products of invertible formal power series with formal diffeomorphisms. This can be understood physically as wave function renormalization and renormalization of the coupling constants, respectively. After taking into account the Slavnov–Taylor identities for the couplings as generators of a Hopf ideal, we find Hopf subalgebras in the corresponding quotient as well. In the second part of the paper, we explain the origin of these Hopf ideals by considering a coaction of the renormalization Hopf algebras on the Batalin-Vilkovisky (BV) algebras generated by the fields and couplings constants. The so-called classical master equation satisfied by the action in the BV-algebra implies the existence of the above Hopf ideals in the renormalization Hopf algebra. Finally, we exemplify our construction by applying it to Yang–Mills gauge theory.

The necklace Lie coalgebra and renormalization algebras

We give a natural monomorphism from the necklace Lie coalgebra, defined for any quiver, to Connes and Kreimer’s Lie coalgebra of trees, and extend this to a map from a certain quivertheoretic Hopf algebra to Connes and Kreimer’s renormalization Hopf algebra, as well as to pre-Lie versions. These results are direct analogues of Turaev’s results in 2004, by replacing algebras of loops on surfaces with algebras of paths on quivers. We also factor the morphism through an algebra of chord diagrams and explain the geometric version. We then explain how all of the Hopf algebras are uniquely determined by the pre-Lie structures, and discuss noncommutative versions of the Hopf algebras.

Renormalization as a functor on bialgebras

The Hopf algebra of renormalization in quantum field theory is described at a general level. The products of fields at a point are assumed to form a bialgebra B and renormalization endows T ( T ( B ) + ) , the double tensor algebra of B , with the structure of a noncommutative bialgebra. When the bialgebra B is commutative, renormalization turns S ( S ( B ) + ) , the double symmetric algebra of B , into a commutative bialgebra. The usual Hopf algebra of renormalization is recovered when the elements of S 1 ( B ) are not renormalized, i.e., when Feynman diagrams containing one single vertex are not renormalized. When B is the Hopf algebra of a commutative group, a homomorphism is established between the bialgebra S ( S ( B ) + ) and the Faà di Bruno bialgebra of composition of series. The relation with the Connes–Moscovici Hopf algebra is given. Finally, the bialgebra S ( S ( B ) + ) is shown to give the same results as the standard renormalization procedure for the scalar field.

On matrix differential equations in the Hopf algebra of renormalization

We establish Sakakibara's differential equations in a matrix setting for the counter term (respectively renormalized character) in Connes-Kreimer's Birkhoff decomposition in any connected graded Hopf algebra, thus including Feynman rules in perturbative renormalization as a key example.

Non-commutative Hopf algebra of formal diffeomorphisms

This paper deals with two Hopf algebras which are the non-commutative analogues of two different groups of formal power series. The first group is the set of invertible series with the group law being multiplication of series, while the second is the set of formal diffeomorphisms with the group law being a composition of series. The motivation to introduce these Hopf algebras comes from the study of formal series with non-commutative coefficients. Invertible series with non-commutative coefficients still form a group, and we interpret the corresponding new non-commutative Hopf algebra as an alternative to the natural Hopf algebra given by the co-ordinate ring of the group, which has the advantage of being functorial in the algebra of coefficients. For the formal diffeomorphisms with non-commutative coefficients, this interpretation fails, because in this case the composition is not associative anymore. However, we show that for the dual non-commutative algebra there exists a natural co-associative co-product defining a non-commutative Hopf algebra. Moreover, we give an explicit formula for the antipode, which represents a non-commutative version of the Lagrange inversion formula, and we show that its coefficients are related to planar binary trees. Then we extend these results to the semi-direct co-product of the previous Hopf algebras, and to series in several variables. Finally, we show how the non-commutative Hopf algebras of formal series are related to some renormalization Hopf algebras, which are combinatorial Hopf algebras motivated by the renormalization procedure in quantum field theory, and to the renormalization functor given by the double-tensor algebra on a bi-algebra.

The Hopf algebra of renormalization, normal coordinates and Kontsevich deformation quantization

Using normal coordinates in a Poincar\'e-Birkhoff-Witt basis for the Hopf algebra of renormalization in perturbative quantum field theory, we investigate the relation between the twisted antipode axiom in that formalism, the Birkhoff algebraic decomposition and the universal formula of Kontsevich for quantum deformation.