Abstract
Inspired by the notion that physical systems can contain only a finite amount of information or complexity, we introduce a framework that allows for quantifying the amount of logical information needed to specify a function or set. We then apply this methodology to a variety of physical systems and derive the complexity of parameter-dependent physical observables and coupling functions appearing in effective Lagrangians. In order to implement these ideas, it is essential to consider physical theories that can be defined in an o-minimal structure. O-minimality, a concept from mathematical logic, encapsulates a tameness principle. It was recently argued that this property is inherent to many known quantum field theories and is linked to the UV completion of the theory. To assign a complexity to each statement in these theories one has to further constrain the allowed o-minimal structures. To exemplify this, we show that many physical systems can be formulated using Pfaffian o-minimal structures, which have a well-established notion of complexity. More generally, we propose adopting sharply o-minimal structures, recently introduced by Binyamini and Novikov, as an overarching framework to measure complexity in quantum theories.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
L.P.D. Dries, Tame Topology and O-minimal Structures, Cambridge University Press (1998) [https://doi.org/10.1017/cbo9780511525919].
T.W. Grimm, Taming the landscape of effective theories, JHEP 11 (2022) 003 [arXiv:2112.08383] [INSPIRE].
M.R. Douglas, T.W. Grimm and L. Schlechter, The Tameness of Quantum Field Theory, Part I Amplitudes, arXiv:2210.10057 [INSPIRE].
M.R. Douglas, T.W. Grimm and L. Schlechter, The Tameness of Quantum Field Theory, Part II Structures and CFTs, arXiv:2302.04275 [INSPIRE].
G. Binyamini and D. Novikov, Tameness in geometry and arithmetic: beyond o-minimality, in International Congress of Mathematicians, EMS Press (2023), p. 1440–1461 [https://doi.org/10.4171/icm2022/117].
G. Binyamini, D. Novikov and B. Zack, Sharply o-minimal structures and sharp cellular decomposition, arXiv:2209.10972.
A.A. Kytmanov, A.M. Kytmanov and E.K. Myshkina, Residue Integrals and Waring’s Formulas for a Class of Systems of Transcendental Equations in ℂn, arXiv:1709.00791.
A.J. Wilkie, On the theory of the real exponential field, Illinois J. Math. 33 (1989) 384 .
A. Gabrièlov and N. Vorobjov, Complexity of stratifications of semi-Pfaffian sets, Discrete Comput. Geom. 14 (1995) 71.
A. Gabrièlov and N. Vorobjov, Complexity of cylindrical decompositions of sub-Pfaffian sets, J. Pure Appl. Algebra 164 (2001) 179.
A. Gabrièlov and N. Vorobjov, Complexity of computations with Pfaffian and Noetherian functions, in Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, Springer Netherlands (2004), p. 211–250 [https://doi.org/10.1007/978-94-007-1025-2_5].
F. Gasparotto, A. Rapakoulias and S. Weinzierl, Nonperturbative computation of lattice correlation functions by differential equations, Phys. Rev. D 107 (2023) 014502 [arXiv:2210.16052] [INSPIRE].
S. Chapman and G. Policastro, Quantum computational complexity from quantum information to black holes and back, Eur. Phys. J. C 82 (2022) 128 [arXiv:2110.14672] [INSPIRE].
N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [hep-th/9407087] [INSPIRE].
L. van den Dries and C. Miller, On the real exponential field with restricted analytic functions, Israel J. Math. 85 (1994) 19.
B. Bakker, B. Klingler and J. Tsimerman, Tame topology of arithmetic quotients and algebraicity of Hodge loci, arXiv:1810.04801.
A.J. Wilkie, A theorem of the complement and some new o-minimal structures, Selecta Math. (N.S.) 5 (1999) 397.
D. Dorigoni, An Introduction to Resurgence, Trans-Series and Alien Calculus, Annals Phys. 409 (2019) 167914 [arXiv:1411.3585] [INSPIRE].
Crutchfield, II and W. Y., A Method for Borel Summing Instanton Singularities. I. Introduction, Phys. Rev. D 19 (1979) 2370 [INSPIRE].
G. Teschl, Ordinary differential equations and dynamical systems, American Mathematical Society, Providence, RI (2012) [https://doi.org/10.1090/gsm/140].
R. Jefferson and R.C. Myers, Circuit complexity in quantum field theory, JHEP 10 (2017) 107 [arXiv:1707.08570] [INSPIRE].
N. Chagnet, S. Chapman, J. de Boer and C. Zukowski, Complexity for Conformal Field Theories in General Dimensions, Phys. Rev. Lett. 128 (2022) 051601 [arXiv:2103.06920] [INSPIRE].
L. Alvarez-Gaume and S.F. Hassan, Introduction to S duality in N = 2 supersymmetric gauge theories: A pedagogical review of the work of Seiberg and Witten, Fortsch. Phys. 45 (1997) 159 [hep-th/9701069] [INSPIRE].
G. Binyamini and N. Vorobjov, Effective cylindrical cell decompositions for restricted sub-Pfaffian sets, arXiv:2004.06411.
G. Binyamini, D. Novikov and S. Yakovenko, On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem, arXiv:0808.2952 [https://doi.org/10.1007/s00222-010-0244-0].
S. Weinzierl, Feynman Integrals, arXiv:2201.03593 [https://doi.org/10.1007/978-3-030-99558-4] [INSPIRE].
B. Bakker and J. Tsimerman, Functional Transcendence of Periods and the Geometric André-Grothendieck Period Conjecture, arXiv:2208.05182.
A. Klemm et al., Selfdual strings and N = 2 supersymmetric field theory, Nucl. Phys. B 477 (1996) 746 [hep-th/9604034] [INSPIRE].
Thomas W. Grimm, David Prieto and Mick van Vliet Tameness, Complexity, and the Swampland, in progress.
L. Susskind, Computational Complexity and Black Hole Horizons, Fortsch. Phys. 64 (2016) 24 [arXiv:1403.5695] [INSPIRE].
A.R. Brown et al., Holographic Complexity Equals Bulk Action?, Phys. Rev. Lett. 116 (2016) 191301 [arXiv:1509.07876] [INSPIRE].
B. Bakker and J. Tsimerman, The Ax-Schanuel conjecture for variations of Hodge structures, arXiv:1712.05088.
M. Kontsevich and D. Zagier, Periods, in Mathematics Unlimited — 2001 and Beyond, B. Engquist and W. Schmid (eds) Springer Berlin Heidelberg (2001), p. 771–808 [https://doi.org/10.1007/978-3-642-56478-9_39].
T.W. Grimm, S. Lanza and C. Li, Tameness, Strings, and the Distance Conjecture, JHEP 09 (2022) 149 [arXiv:2206.00697] [INSPIRE].
Acknowledgments
We profited immensely from discussions with Lou van den Dries and Gal Binyamini and would like to thank them for sharing their insights and understanding with us. Furthermore, would like to thank Benjamin Bakker, Michael Douglas, Gerard ’t Hooft, Damian van de Heisteeg, Ro Jefferson, and Cumrun Vafa for useful discussions and comments. This research is supported, in part, by the Dutch Research Council (NWO) via a Vici grant.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2310.01484
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Grimm, T.W., Schlechter, L. & van Vliet, M. Complexity in tame quantum theories. J. High Energ. Phys. 2024, 1 (2024). https://doi.org/10.1007/JHEP05(2024)001
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2024)001