Abstract
We formulate a bi-Heyting algebra structure on the space of Feynman diagrams of a quantum field theory. This particular bi-Heyting algebra, which is originated from Feynman graphon representations of 1PI Green’s functions together with renormalization and core coproducts, provides some new topological treatments for the study of the asymptotics of 1PI Green’s functions and their corrections.
Acknowledgements
The author would like to thank the referees because of constructive comments.
References
[1] N. M. Atakishiev, Asymptotic behavior of the Green’s function in some models of nonrenormalizable field theory, Theoret. and Math. Phys. 1 (1969), 46–50. 10.1007/BF01028570Search in Google Scholar
[2] C. Bergbauer and D. Kreimer, The Hopf algebra of rooted trees in Epstein–Glaser renormalization, Ann. Henri Poincaré 6 (2005), no. 2, 343–367. 10.1007/s00023-005-0210-3Search in Google Scholar
[3] C. Bergbauer and D. Kreimer, New algebraic aspects of perturbative and non-perturbative quantum field theory, New Trends in Mathematical Physcis, Springer, Dordrecht (2009), 45–58. 10.1007/978-90-481-2810-5_4Search in Google Scholar
[4] 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
[5] D. J. Broadhurst and D. Kreimer, Renormalization automated by Hopf algebra, J. Symbolic Comput. 27 (1999), no. 6, 581–600. 10.1006/jsco.1999.0283Search in Google Scholar
[6] C. Brouder, The structure of Green functions in quantum field theory with a general state, Quantum Field Theory, Birkhäuser, Basel (2009), 163–175. 10.1007/978-3-7643-8736-5_10Search in Google Scholar
[7] S. A. Celani and R. Jansana, Easkia duality and its extensions, Leo Esakia on Duality in Modal and Intuitionistic Logics, Outst. Contrib. Log. 4, Springer, Dordrecht (2014), 63–98. 10.1007/978-94-017-8860-1_4Search in Google Scholar
[8] A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann–Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys. 210 (2000), no. 1, 249–273. 10.1007/s002200050779Search in Google Scholar
[9] A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann–Hilbert problem. II. The 𝛽-function, diffeomorphisms and the renormalization group, Comm. Math. Phys. 216 (2001), no. 1, 215–241. 10.1007/PL00005547Search in Google Scholar
[10] 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
[11] A. Döring and C. Isham, “What is a thing?”: Topos theory in the foundations of physics, New Structures for Physics, Lecture Notes in Phys. 813, Springer, Heidelberg (2011), 753–937. 10.1007/978-3-642-12821-9_13Search in Google Scholar
[12] K. Ebrahimi-Fard and D. Kreimer, The Hopf algebra approach to Feynman diagram calculations, J. Phys. A 38 (2005), no. 50, R385–R407. 10.1088/0305-4470/38/50/R01Search in Google Scholar
[13] L. Esakia, Heyting Algebras, Trends Log. Stud. Log. Libr. 50, Springer, Cham, 2019. 10.1007/978-3-030-12096-2Search in Google Scholar
[14] H. Figueroa and J. M. Gracia-Bondía, Combinatorial Hopf algebras in quantum field theory. I, Rev. Math. Phys. 17 (2005), no. 8, 881–976. 10.1142/S0129055X05002467Search in Google Scholar
[15] A. Frabetti, Renormalization Hopf algebras and combinatorial groups, Geometric and Topological Methods for Quantum Field Theory, Cambridge University, Cambridge (2010), 159–219. 10.1017/CBO9780511712135.005Search in Google Scholar
[16] S. Ghilardi, Free Heyting algebras as bi-Heyting algebras, C. R. Math. Rep. Acad. Sci. Canada 14 (1992), no. 6, 240–244. Search in Google Scholar
[17] Z. Haba, Lectures on Quantum Field Theory and Functional Integration, Springer, Cham, 2023. 10.1007/978-3-031-30712-6Search in Google Scholar
[18] F. Herzog, Zimmermann’s forest formula, infrared divergences and the QCD beta function, Nuclear Phys. B 926 (2018), 370–380. 10.1016/j.nuclphysb.2017.11.011Search in Google Scholar
[19] A. Heyting, Die formalen Regeln der intuitionistischen Logik. I, Sitzungsberichte Akad. Berlin 1930 (1930), 42–56. Search in Google Scholar
[20] A. Heyting, Die formalen Regeln der intuitionistischen Logik. II, Sitzungsberichte Akad. Berlin 1930 (1930), 57–71. Search in Google Scholar
[21] A. Heyting, Die formalen Regeln der intuitionistischen Logik. III, Sitzungsberichte Akad. Berlin 1930 (1930), 158–169. Search in Google Scholar
[22] C. J. Isham, Topos methods in the foundations of physics, Deep Beauty, Cambridge University, Cambridge (2011), 187–205. 10.1017/CBO9780511976971.005Search in Google Scholar
[23] D. Kreimer, Combinatorics of (perturbative) quantum field theory, Phys. Rep. 363 (2002), no. 4–6, 387–424. 10.1016/S0370-1573(01)00099-0Search in Google Scholar
[24] D. Kreimer, A remark on quantum gravity, Ann. Physics 323 (2008), no. 1, 49–60. 10.1016/j.aop.2007.06.005Search in Google Scholar
[25] D. Kreimer, The core Hopf algebra, Quanta of Maths, Clay Math. Proc. 11, American Mathematical Society, Providence (2010), 313–321. Search in Google Scholar
[26] D. Kreimer and R. Delbourgo, Using the Hopf algebra structure of QFT in calculations, Phys. Rev. D (3) 60 (1999), no. 10, Article ID 105025. 10.1103/PhysRevD.60.105025Search in Google Scholar
[27] D. Kreimer and W. D. van Suijlekom, Recursive relations in the core Hopf algebra, Nuclear Phys. B 820 (2009), no. 3, 682–693. 10.1016/j.nuclphysb.2009.04.025Search in Google Scholar
[28] O. Krüger, Log expansions from combinatorial Dyson–Schwinger equations, Lett. Math. Phys. 110 (2020), no. 8, 2175–2202. 10.1007/s11005-020-01288-8Search in Google Scholar
[29] O. Krüger and D. Kreimer, Filtrations in Dyson–Schwinger equations: Next-to {j} -leading log expansions systematically, Ann. Physics 360 (2015), 293–340. 10.1016/j.aop.2015.05.013Search in Google Scholar
[30] W. Kuich, N. Sauer and F. Urbanek, Heyting algebras and formal languages, J.UCS 8 (2002), no. 7, 722–736. Search in Google Scholar
[31] F. W. Lawvere, Intrinsic co-Heyting boundaries and the Leibniz rule in certain toposes, Category Theory (Como 1990), Lecture Notes in Math. 1488, Springer, Berlin (1991), 279–281. 10.1007/BFb0084226Search in Google Scholar
[32] 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
[33] 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
[34] F. Paugam, Towards the Mathematics of Quantum Field Theory, Ergeb. Math. Grenzgeb. (3)59, Springer, Cham, 2014. 10.1007/978-3-319-04564-1Search in Google Scholar
[35] G. E. Reyes and H. Zolfaghari, Bi-Heyting algebras, toposes and modalities, J. Philos. Logic 25 (1996), no. 1, 25–43. 10.1007/BF00357841Search in Google Scholar
[36] E. Rydzyńska, Probability Heyting algebras, Demonstr. Math. 19 (1986), no. 3, 573–589. 10.1515/dema-1986-0304Search in Google Scholar
[37] E. Rydzyńska, Differential calculus in probability Heyting algebras, Demonstr. Math. 25 (1992), no. 4, 887–904. 10.1515/dema-1992-0420Search in Google Scholar
[38] H. P. Sankappanavar, Heyting algebras with dual pseudocomplementation, Pacific J. Math. 117 (1985), no. 2, 405–415. 10.2140/pjm.1985.117.405Search in Google Scholar
[39] A. Shojaei-Fard, A geometric perspective on counterterms related to Dyson–Schwinger equations, Internat. J. Modern Phys. A 28 (2013), no. 32, Article ID 1350170. 10.1142/S0217751X13501704Search in Google Scholar
[40] 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
[41] A. Shojaei-Fard, The complexities of nonperturbative computations, Russ. J. Math. Phys. 28 (2021), no. 3, 358–376. 10.1134/S1061920821030092Search in Google Scholar
[42] A. Shojaei-Fard, The dynamics of non-perturbative phases via Banach bundles, Nuclear Phys. B 969 (2021), Article ID 115478. 10.1016/j.nuclphysb.2021.115478Search in Google Scholar
[43] A. Shojaei-Fard, From Dyson–Schwinger equations to quantum entanglement, J. Math. Sci. (N. Y.) 266 (2022), no. 6, 892–916. 10.1007/s10958-022-06171-6Search in Google Scholar
[44] A. Shojaei-Fard, Halting problem in Feynman graphon processes derived from the renormalization Hopf algebra, Bull. Transilv. Univ. Braşov Ser. III. Math. Comput. Sci. 2(64) (2022), no. 1, 139–157. 10.31926/but.mif.2022.2.64.1.10Search in Google Scholar
[45] A. Shojaei-Fard, Non-perturbative graph languages, halting problem and complexity, Forum Math. 34 (2022), no. 5, 1159–1185. 10.1515/forum-2021-0119Search in Google Scholar
[46] A. Shojaei-Fard, A statistical mechanical model for non-perturbative regimes, Nuclear Phys. B 991 (2023), Article ID 116220. 10.1016/j.nuclphysb.2023.116220Search in Google Scholar
[47] A. Shojaei-Fard, Non-perturbative topos and its application, Rev. Roumaine Math. Pures Appl. 68 (2023), no. 3–4, 329–345. Search in Google Scholar
[48] A. Shojaei-Fard, Subsystems via quantum motions, Anal. Math. Phys. 14 (2024), no. 3, Paper No. 61. 10.1007/s13324-024-00912-3Search in Google Scholar
[49] A. V. Smirnov and V. A. Smirnov, How to choose master integrals, Nuclear Phys. B 960 (2020), Article ID 115213. 10.1016/j.nuclphysb.2020.115213Search in Google Scholar
[50] V. A. Smirnov, Feynman Integral Calculus, Springer, Berlin, 2006. Search in Google Scholar
[51] W. D. van Suijlekom, The structure of renormalization Hopf algebras for gauge theories. I. Representing Feynman graphs on BV-algebras, Comm. Math. Phys. 290 (2009), no. 1, 291–319. 10.1007/s00220-009-0829-xSearch in Google Scholar
[52] A. Vourdas, The complete Heyting algebra of subsystems and contextuality, J. Math. Phys. 54 (2013), no. 8, Article ID 082105. 10.1063/1.4817855Search in Google Scholar
[53] S. Weinzierl, Feynman Integrals—A Comprehensive Treatment for Students and Researchers, Unitext Phys., Springer, Cham, 2022. 10.1007/978-3-030-99558-4Search in Google Scholar
[54] W. Zimmermann, Convergence of Bogoliubov’s method of renormalization in momentum space, Comm. Math. Phys. 15 (1969), 208–234. 10.1007/BF01645676Search in Google Scholar
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