Non-perturbative graph languages, halting problem and complexity

References

[1] M. Bachmann, H. Kleinert and A. Pelster, Recursive graphical construction of Feynman diagrams in quantum electrodynamics, Phys. Rev. D (3) 61 (2000), no. 8, Article ID 085017. 10.1103/PhysRevD.61.085017Search in Google Scholar

[2] C. Borgs, J. T. Chayes, H. Cohn and N. Holden, Sparse exchangeable graphs and their limits via graphon processes, J. Mach. Learn. Res. 18 (2017), Paper No. 210. Search in Google Scholar

[3] C. Brouder, A. Frabetti and F. Menous, Combinatorial Hopf algebras from renormalization, J. Algebraic Combin. 32 (2010), no. 4, 557–578. 10.1007/s10801-010-0227-7Search in Google Scholar

[4] D. Calaque and T. Strobl, Mathematical Aspects of Quantum Field Theory, Math. Phys. Stud., Springer, Cham, 2015. 10.1007/978-3-319-09949-1Search in Google Scholar

[5] E. Dagotto, A new phase of QED in strong coupling: A guide for the perplexed, Vacuum Structure in Intense Fields, NATO ASI Ser. 255, Plenum Press, New York (1991), 195–221. 10.1007/978-1-4757-0441-9_12Search in Google Scholar

[6] M. D. Davis and E. J. Weyuker, Computability, Complexity, and Languages, Comput. Sci. Appl. Math., Academic Press, New York, 1983. 10.1016/B978-0-12-206380-0.50020-1Search in Google Scholar

[7] C. Delaney and M. Marcolli, Dyson–Schwinger equations in the theory of computation, Feynman Amplitudes, Periods and Motives, Contemp. Math. 648, American Mathematical Society, Providence (2015), 79–107. 10.1090/conm/648/12999Search in Google Scholar

[8] M. Dütsch and K. Fredenhagen, The master Ward identity and generalized Schwinger–Dyson equation in classical field theory, Comm. Math. Phys. 243 (2003), no. 2, 275–314. 10.1007/s00220-003-0968-4Search in Google Scholar

[9] H. Ehrig, Tutorial introduction to the algebraic approach of graph grammars, Graph-Grammars and Their Application to Computer Science, Lecture Notes in Comput. Sci. 291, Springer, Berlin (1987), 3–14. 10.1007/3-540-18771-5_40Search in Google Scholar

[10] H. Ehrig, A. Habel and H. J. Kreowski, Introduction to graph grammars with applications to semantic networks, Comput. Math. Appl. 23 (1992), no. 6–9, 557–572. 10.1016/0898-1221(92)90124-ZSearch in Google Scholar

[11] H. Ehrig, H.-J. Kreowski and G. Rozenberg, Graph Grammars and Their Application to Computer Science, Lecture Notes in Comput. Sci. 532, Springer, Berlin, 1990. 10.1007/BFb0017372Search in Google Scholar

[12] H. Fahmy and D. Blostein, A survey of graph grammars: Theory and applications, 11th IAPR International Conference on Pattern Recognition. Vol. II. Conference B: Pattern Recognition Methodology and Systems, IEEE Press, Piscataway (1992), 294–298. 10.1109/ICPR.1992.201776Search in Google Scholar

[13] S. Janson, Graphons, Cut Norm and Distance, Couplings and Rearrangements, NYJM Monogr. 4, State University of New York, New York, 2013. Search in Google Scholar

[14] R. A. Jefferson and R. C. Myers, Circuit complexity in quantum field theory, J. High Energy Phys. 2017 (2017), no. 10, Paper No. 107. 10.1007/JHEP10(2017)107Search in Google Scholar

[15] H. Kleinert, A. Pelster, B. Kastening and M. Bachmann, Recursive graphical construction of Feynman diagrams and their multiplicities in ϕ 4 and in ϕ 2 A theory, Phys. Rev. E 62 (2000), 1537–1559. 10.1103/PhysRevE.62.1537Search in Google Scholar

[16] K. Kondo, Transverse Ward–Takahashi identity, anomaly and Schwinger–Dyson equation, Internat. J. Modern Phys. A 12 (1997), 5651–5686. 10.1142/S0217751X97002978Search in Google Scholar

[17] D. Kreimer, On the Hopf algebra structure of perturbative quantum field theories, Adv. Theor. Math. Phys. 2 (1998), no. 2, 303–334. 10.4310/ATMP.1998.v2.n2.a4Search in Google Scholar

[18] D. Kreimer, Unique factorization in perturbative QFT, Nuclear Phys. B 116 (2003), 392–396. 10.1016/S0920-5632(03)80206-2Search in Google Scholar

[19] D. Kreimer, Structures in Feynman graphs: Hopf algebras and symmetries, Graphs and Patterns in Mathematics and Theoretical Physics, Proc. Sympos. Pure Math. 73, American Mathematical Society, Providence (2005), 43–78. 10.1090/pspum/073/2131011Search in Google Scholar

[20] D. Kreimer, Anatomy of a gauge theory, Ann. Phys. 321 (2006), no. 12, 2757–2781. 10.1016/j.aop.2006.01.004Search in Google Scholar

[21] L. Lovász, Very large graphs, Current Developments in Mathematics 2008, International Press, Somerville (2009), 67–128. 10.4310/CDM.2008.v2008.n1.a2Search in Google Scholar

[22] Y. I. Manin, Infinities in quantum field theory and in classical computing: Renormalization program, Programs, Proofs, Processes, Lecture Notes in Comput. Sci. 6158, Springer, Berlin (2010), 307–316. 10.1007/978-3-642-13962-8_34Search in Google Scholar

[23] Y. I. Manin, Renormalisation and computation II: Time cut-off and the halting problem, Math. Structures Comput. Sci. 22 (2012), no. 5, 729–751. 10.1017/S0960129511000508Search in Google Scholar

[24] Y. I. Manin, Renormalization and computation I: Motivation and background, OPERADS 2009, Sémin. Congr. 26, Société Mathématique de France, Paris (2013), 181–222. Search in Google Scholar

[25] M. Marcolli and A. Port, Graph grammars, insertion Lie algebras, and quantum field theory, Math. Comput. Sci. 9 (2015), no. 4, 391–408. 10.1007/s11786-015-0236-ySearch in Google Scholar

[26] H. Okabe, Formal expressions of infinite graphs and their families, Inform. and Control 44 (1980), no. 2, 164–186. 10.1016/S0019-9958(80)90074-1Search in Google Scholar

[27] A. Shojaei-Fard, A new perspective on intermediate algorithms via the Riemann–Hilbert correspondence, Quantum Stud. Math. Found. 4 (2017), no. 2, 127–148. 10.1007/s40509-016-0088-4Search in Google Scholar

[28] A. Shojaei-Fard, A measure theoretic perspective on the space of Feynman diagrams, Bol. Soc. Mat. Mex. (3) 24 (2018), no. 2, 507–533. 10.1007/s40590-017-0166-6Search in Google Scholar

[29] A. Shojaei-Fard, Graphons and renormalization of large Feynman diagrams, Opuscula Math. 38 (2018), no. 3, 427–455. 10.7494/OpMath.2018.38.3.427Search in Google Scholar

[30] A. Shojaei-Fard, Formal aspects of non-perturbative quantum field theory via an operator theoretic setting, Int. J. Geom. Methods Mod. Phys. 16 (2019), no. 12, Article ID 1950192. 10.1142/S0219887819501925Search in Google Scholar

[31] A. Shojaei-Fard, Non-perturbative β-functions via Feynman graphons, Modern Phys. Lett. A 34 (2019), no. 14, Article ID 1950109. 10.1142/S0217732319501098Search in Google Scholar

[32] A. Shojaei-Fard, The analytic evolution of Dyson–Schwinger equations via homomorphism densities, Math. Phys. Anal. Geom. 24 (2021), no. 2, Paper No. 18. 10.1007/s11040-021-09389-zSearch in Google Scholar

[33] A. Shojaei-Fard, The complexities of nonperturbative computations, Russ. J. Math. Phys. 28 (2021), no. 3, 358–376. 10.1134/S1061920821030092Search in Google Scholar

[34] A. Shojaei-Fard, The dynamics of non-perturbative phases via Banach bundles, Nuclear Phys. B 969 (2021), Paper No. 115478. 10.1016/j.nuclphysb.2021.115478Search in Google Scholar

[35] I. Tsutsui, Origin of anomalies in the path-integral formalism, Phys. Rev. D (3) 40 (1989), no. 10, 3543–3546. 10.1103/PhysRevD.40.3543Search in Google Scholar

[36] W. D. van Suijlekom, The Hopf algebra of Feynman graphs in quantum electrodynamics, Lett. Math. Phys. 77 (2006), no. 3, 265–281. 10.1007/s11005-006-0092-4Search in Google Scholar

[37] W. D. van Suijlekom, Renormalization of gauge fields: A Hopf algebra approach, Comm. Math. Phys. 276 (2007), no. 3, 773–798. 10.1007/s00220-007-0353-9Search in Google Scholar

[38] S. Weinzierl, Introduction to Feynman integrals, Geometric and Topological Methods for Quantum Field Theory, Cambridge University, Cambridge (2013), 144–187. 10.1017/CBO9781139208642.005Search in Google Scholar

[39] S. Weinzierl, Hopf algebras and Dyson–Schwinger equations, Front. Phys. 11 (2016), Article ID 111206. 10.1007/s11467-016-0562-9Search in Google Scholar