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Hamiltonian renormalization. VIII. P(Φ)2 quantum field theory

M. Rodriguez Zarate* and T. Thiemann†

  • *Contact author: melissa.rodriguez@gravity.fau.de
  • †Contact author: thomas.thiemann@gravity.fau.de
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Phys. Rev. D 112, 125021 – Published 18 December, 2025

DOI: https://doi.org/10.1103/gfvk-hpsb

Abstract

In previous works in this series we focused on Hamiltonian renormalization of free field theories in all spacetime dimensions. In this paper, we address the Hamiltonian renormalization of the self-interacting scalar field in two spacetime dimensions with polynomial potential, called P(Φ)2. We consider only the finite volume case. The P(Φ)2 theory is one of the few interacting quantum field theories that can be rigorously constructed nonperturbatively. We find that our Hamiltonian renormalization flow finds this theory indeed as a fixed point.

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See Also

Hamiltonian renormalization. IX. U(1)3 quantum gravity

A. M. Rodriguez Zarate and T. Thiemann
Phys. Rev. D 112, 126015 (2025)

Article Text

References (24)

  1. R. Haag, Local Quantum Physics (Springer Verlag, Berlin, 1984).
  2. J. Glimm and A. Jaffe, Quantum Physics (Springer Verlag, New York, 1987).
  3. K. G. Wilson, The renormalization group: Critical phenomena and the Kondo problem, Rev. Mod. Phys. 47 (1975) 773.
  4. T. Thiemann, Canonical quantum gravity, constructive QFT and renormalisation, Front. Phys. 8, 548232 (2020).
  5. T. Lang, K. Liegener, and T. Thiemann, Hamiltonian renormalisation I. Derivation from Osterwalder-Schrader reconstruction, Classical Quantum Gravity 35, 245011 (2018).
  6. T. Lang, K. Liegener, and T. Thiemann, Hamiltonian renormalisation II. Renormalisation flow of 1+1 dimensional free, scalar fields: Derivation, Classical Quantum Gravity 35, 245012 (2018).
  7. T. Lang, K. Liegener, and T. Thiemann, Hamiltonian renormalisation III. Renormalisation flow of 1+1 dimensional free, scalar fields: Properties, Classical Quantum Gravity 35, 245013 (2018).
  8. T. Lang, K. Liegener, and T. Thiemann, Hamiltonian renormalisation IV. Renormalisation flow of D+1 dimensional free scalar fields and rotation invariance, Classical Quantum Gravity 35, 245014 (2018).
  9. K. Liegener and T. Thiemann, Hamiltonian renormalisation V. Free vector bosons, Front. Astron. Space Sci. 7, 547550 (2021).
  10. T. Thiemann, Hamiltonian renormalisation VII. Free fermions and doubler free kernels., Phys. Rev. D 108, 125007 (2023).
  11. E.-A. Zwicknagel, Hamiltonian renormalization. VI. Parametrized field theory on the cylinder, Phys. Rev. D 108, 125006 (2023).
  12. B. Simon, The P(ϕ)2 Euclidean (Quantum) Field Theory (Princeton Unviersity Press, Princeton, NJ, 1974).
  13. K. Osterwalder and R. Schrader, Axioms for Euclidean Green’s functions, Commun. Math. Phys. 31, 83 (1973).
  14. J. Glimm and A. Jaffe, A λϕ4 quantum field theory without cutoffs. I, Phys. Rev. 176, 1945 (1968); The λ(ϕ4)2 quantum field theory without cutoffs: III. The physical vacuum, Acta Math. 125, 203 (1970); The λ(ϕ4)2 quantum field theory without cutoffs. II. The field operators and the approximate vacuum, Ann. Math. 91, 362 (1970); The λ(ϕ4)2 quantum field theory without cutoffs. IV. Perturbations of the Hamiltonian, J. Math. Phys. (N.Y.) 13, 1568 (1972).
  15. T. Thiemann, Renormalization, wavelets, and the Dirichlet-Shannon kernels, Phys. Rev. D 108, 125008 (2023).
  16. I. Daubechies, Ten Lectures of Wavelets (Springer Verlag, Berlin (1993); A. Cohen, I. Daubechies, and P.Vial, Wavelets on the interval and fast wavelet transforms, Appl. Comput. Harmon. Anly. 1, 54 (1993).
  17. J. Glimm, Boson fields with the :ϕ4:   Interaction in three dimensions, Commun. Math. Phys. 10, 1 (1968); J. Glimm and A. Jaffe, Positivity of the ϕ34 Hamiltonian, Fortschr. Phys. 21, 327 (1973).
  18. A. M. Rodriguez Zarate and T. Thiemann, companion paper, Hamiltonian renormalization. IX. U(1)3 quantum gravity, Phys. Rev. D 112, 126015 (2025).
  19. L. Smolin, The G(Newton) →0 limit of Euclidean quantum gravity, Classical Quantum Gravity 9, 883 (1992).
  20. Exact quantisation of U(1)3 quantum gravity via exponentiation of the hypersurface deformation algebroid, Classical Quantum Gravity 40, 245003 (2023).
  21. H. Narnhofer and W. E. Thirring, Covariant QED without indefinite metric, Rev. Math. Phys. 04, 197 (1992).
  22. T. Thiemann, Nonperturbative quantum gravity in Fock representations, Phys. Rev. D 110, 124023 (2024).
  23. C. Rovelli, Quantum Gravity (Cambridge University Press, Cambridge, England, 2004); T. Thiemann, Modern Canonical Quantum General Relativity (Cambridge University Press, Cambridge, England, 2007); J. Pullin and R. Gambini, A First Course in Loop Quantum Gravity (Oxford University Press, New York, 2011); C. Rovelli and F. Vidotto, Covariant Loop Quantum Gravity (Cambridge University Press, Cambridge, England, 2015); K. Giesel and H. Sahlmann, From classical to quantum gravity: Introduction to loop quantum gravity, Proc. Sci., QGQGS2011 (2011) 002 [arXiv:1203.2733].
  24. O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vol. 1,2 (Springer Verlag, Berlin, 1997).

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Phys. Rev. D 112, 125021– Published 18 December, 2025
  • Received 24 October 2025
  • Accepted 2 December 2025
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