Non-perturbative graph languages, halting problem and complexity

Author:

Shojaei-Fard Ali1ORCID

Affiliation:

1. 1461863596 Marzdaran Blvd. , Tehran , Iran

Abstract

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.

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,General Mathematics

Cited by 4 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Subsystems via quantum motions;Analysis and Mathematical Physics;2024-05-16

2. A statistical mechanical model for non-perturbative regimes;Nuclear Physics B;2023-06

3. Graph polynomials associated with Dyson-Schwinger equations;Mathematica Moravica;2023

4. FROM DYSON–SCHWINGER EQUATIONS TO QUANTUM ENTANGLEMENT;Journal of Mathematical Sciences;2022-10

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