Abstract
In this paper we present an inductive renormalizability proof for massive \(\phi_4^4\) theory on Riemannian manifolds, based on the Wegner-Wilson flow equations of the Wilson renormalization group, adapted to perturbation theory. The proof goes in hand with bounds on the perturbative Schwinger functions which imply tree decay between their position arguments. An essential prerequisite is precise bounds on the short and long distance behaviour of the heat kernel on the manifold. With the aid of a regularity assumption (often taken for granted) we also show that for suitable renormalization conditions the bare action takes the minimal form, that is to say, there appear the same counterterms as in flat space, apart from a logarithmically divergent one which is proportional to the scalar curvature.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barvinsky A.O. and Vilkovisky G.A. (1985). The generalized Schwinger-De Witt technique in gauge theories and quantum gravity. Phys. Rep. 119: 1–74
Bros J., Epstein H. and Moschella U. (2002). Towards a General Theory of Quantized Fields on the Anti-de Sitter Space-Time. Commun. Math. Phys. 231: 481–528
Brunetti R., Fredenhagen K. and Verch R. (2003). The Generally Covariant Locality Principle - A New Paradigm for Local Quantum Field Theory. Commun. Math. Phys. 237: 31–68
Birrell N.D. and Davies P.C.W. (1982). Quantum Fields in Curved Space. Cambridge University Press, Cambridge
Birke L. and Fröhlich J. (2002). KMS, etc. Rev. Math. Phys. 14: 829–873
Birrell N.D. (1980). Momentum space renormalization of \(\lambda \varphi^4\) in curved space-time. J. Phys. A 13: 569–584
Bunch T.S., Panangaden P. and Parker L. (1980). On renormalization of \(\lambda \varphi^4\) field theory in curved space-time: I. J. Phys. A 13: 901–918
Brunetti R. and Fredenhagen K. (2000). Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds. Commun. Math. Phys. 208: 623–661
Bunch T.S. (1981). Local Momentum Space and Two-loop Renormalization of \(\lambda \varphi^4\) field Theory in Curved Space-Time. Gen. Rel. Grav. 13: 711–723
Bunch T.S. (1981). BPHZ Renormalization of \(\lambda \varphi^4\) field Theory in Curved Space-Time. Ann. Phys. (N.Y.) 131: 118–148
Bunch T.S. and Panangaden P. (1980). On renormalization of \(\lambda \varphi^4\)field theory in curved space-time: II. J. Phys. A 13: 919–932
Bunch T.S. and Parker L. (1979). Feynman propagator in curved space-time: A momentum-space representation. Phys. Rev. D 20: 2499–2510
Chavel I. (1993). Riemannian Geometry: A Modern Introduction. Cambridge University Press, Cambridge
Cheng S.Y., Li P. and Yau S.-T. (1981). On the upper estimate of the heat kernel of a complete Riemannian manifold. Am. J. Math. 103: 1021–1063
Davies E.B. (1989). Heat kernels and spectral theory. Cambridge University Press, Cambridge
Davies E.B. (1988). Gaussian upper bounds for the heat kernels of some second order operators on Riemannian manifolds. J. Funct. Anal. 80: 16–32
Davies E.B. (1989). Pointwise bounds on the space and time derivatives of heat kernels. J. Operator Theory 21: 367–378
Grigor’yan A. (1999). Estimates of heat kernels on Riemannian manifolds. In: Davies, E.B. and Safarov, Yu. (eds) Spectral Theory and Geometry, London Math. Soc. Lecture Notes 273, pp 140–225. Cambridge Univ. Press, Cambridge
Hollands S. and Wald R.M. (2001). Local Wick polynomials and time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 223: 289–326
Hollands S. and Wald R.M. (2002). Existence of Local Covariant Time Ordered Products of Quantum Fields in Curved Spacetime. Commun. Math. Phys. 231: 309–345
Hollands S. and Wald R.M. (2003). On the Renormalization Group in Curved Spacetime. Commun. Math. Phys. 237: 123–160
Keller G., Kopper Ch. and Salmhofer M. (1992). Perturbative renormalization and effective Lagrangians in \(\phi_4^4\). Helv. Phys. Acta 156: 32–52
Kopper, Ch.: Renormierungstheorie mit Flussgleichungen Aachen, Shaker Verlag, 1998
Kopper, Ch.: Renormalization Theory based on Flow equations. Lecture in honour of Jacques Bros. In: Rigorous Quantum Field Theory, Progress in Mathematics, Basel, Birkhäuser, 2006
Li P. and Yau S.-T. (1986). On the parabolic kernel of the Schrödinger operator. Acta Math. 156: 153–201
Lüscher M. (1982). Dimensional Regularization in the Presence of Large Background Fields. Ann. Phys. (N.Y.) 142: 359–392
Müller V.F. (2003). Perturbative Renormalization by Flow Equations. Rev. Math. Phys. 15: 491–557
Nelson B.L. and Panangaden P. (1982). Scaling behavior of interacting quantum fields in curved spacetime. Phys. Rev. D 25: 1019–1027
Polchinski J. (1984). Renormalization and Effective Lagrangians. Nucl. Phys. B 231: 269–295
Salmhofer M. (1998). Renormalization - An Introduction. Springer-Verlag, Berlin-Heidelberg-New York
Souplet P. and Zhang Q. (2006). Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds. Bull. London Math. Soc. 38(6): 14045–1053
Taylor, M.E.: Partial Differential Equations I. AMS 115. Springer-Verlag, 1996
Varopoulos N.Th. (1989). Small time Gaussian estimates of heat diffusion kernel. I. The semigroup technique. Bull. Sc. Math., 2 Série 113: 253–277
Willmore T.J. (1996). Riemannian Geometry. Oxford University Press, Oxford
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J.Z. Imbrie
Rights and permissions
About this article
Cite this article
Kopper, C., Müller, V.F. Renormalization Proof for Massive \(\phi_4^4\) Theory on Riemannian Manifolds. Commun. Math. Phys. 275, 331–372 (2007). https://doi.org/10.1007/s00220-007-0297-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-007-0297-0