×

Renormalization of Feynman amplitudes on manifolds by spectral zeta regularization and blow-ups. (English) Zbl 1460.81061

Summary: Our goal in this paper is to present a generalization of the spectral zeta regularization for general Feynman amplitudes on Riemannian manifolds. Our method uses complex powers of elliptic operators but involves several complex parameters in the spirit of analytic renormalization by Speer, to build mathematical foundations for the renormalization of perturbative interacting quantum field theories. Our main result shows that spectrally regularized Feynman amplitudes admit analytic continuation as meromorphic germs with linear poles in the sense of the works of Guo-Paycha and the second author. We also give an explicit determination of the affine hyperplanes supporting the poles. Our proof relies on suitable resolution of singularities of products of heat kernels to make them smooth.
As an application of the analytic continuation result, we use a universal projection from meromorphic germs with linear poles on holomorphic germs to construct renormalization maps which subtract singularities of Feynman amplitudes of Euclidean fields. Our renormalization maps are shown to satisfy consistency conditions previously introduced in the work of Nikolov-Todorov-Stora in the case of flat space-times.

MSC:

81T20 Quantum field theory on curved space or space-time backgrounds
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
81T18 Feynman diagrams
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
35B44 Blow-up in context of PDEs
35K08 Heat kernel
81T08 Constructive quantum field theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ackermann, T.: A note on the Wodzicki residue. J. Geom. Phys.20, 404-406 (1996) Zbl 0864.58057 MR 1419429 · Zbl 0864.58057
[2] Albert, B. I.: Heat kernel renormalization on manifolds with boundary.arXiv:1609.02220 (2016)
[3] Albert, B. I.: Effective field theory on manifolds with boundary. Ph.D. thesis, Univ. of Pennsylvania (2017)MR 3731923
[4] Ammann, B., B¨ar, C.: The Einstein-Hilbert action as a spectral action. In: Noncommutative Geometry and the Standard Model of Elementary Particle Physics, Springer, 75-108 (2002) Zbl 1255.81218 MR 1998531 · Zbl 1255.81218
[5] Atiyah, M. F.: Resolution of singularities and division of distributions. Comm. Pure Appl. Math.23, 145-150 (1970)Zbl 0188.19405 MR 0256156 · Zbl 0188.19405
[6] Atiyah, M. F., Bott, R., Patodi, V. K.: On the heat equation and the index theorem. Invent. Math.19, 279-330 (1973)Zbl 0257.58008 MR 0650828 · Zbl 0257.58008
[7] B¨ar, C., Ginoux, N., Pf¨affle, F.: Wave Equations on Lorentzian Manifolds and Quantization. Eur. Math. Soc. (2007)Zbl 1118.58016 MR 2298021 · Zbl 1118.58016
[8] B¨ar, C., Moroianu, S.: Heat kernel asymptotics for roots of generalized Laplacians. Int. J. Math.14, 397-412 (2003)Zbl 1079.58020 MR 1984660 · Zbl 1079.58020
[9] B¨ar, C., Strohmaier, A.: An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary. Amer. J. Math.141, 1421-1455 (2019)Zbl 1429.83004 MR 4011805 · Zbl 1429.83004
[10] Bergbauer, C., Brunetti, R., Kreimer, D.: Renormalization and resolution of singularities. arXiv:0908.0633(2009) · Zbl 1175.81154
[11] Bergbauer, C., Kreimer, D.: The Hopf algebra of rooted trees in Epstein-Glaser renormalization. Ann. Henri Poincar´e6, 343-367 (2005)Zbl 1065.81096 MR 2136195 · Zbl 1065.81096
[12] Berghoff, M.: Wonderful renormalization. PhD thesis, Humboldt-Univ. Berlin (2015) · Zbl 1376.81052
[13] Berghoff, M.: Wonderful compactifications in quantum field theory. Comm. Number Theory Phys.9, 477-547 (2015)Zbl 1376.81052 MR 3399925 · Zbl 1376.81052
[14] Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators. Springer, Berlin (2004)Zbl 1037.58015 MR 2273508 · Zbl 1037.58015
[15] Berline, N., Vergne, M.: Local asymptotic Euler-Maclaurin expansion for Riemann sums over a semi-rational polyhedron. In: Configuration Spaces, Springer, 67-105 (2016) Zbl 1387.52026 MR 3615728 · Zbl 1387.52026
[16] Brunetti, R., Fredenhagen, K.: Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds. Comm. Math. Phys.208, 623-661 (2000) Zbl 1040.81067 MR 1736329 · Zbl 1040.81067
[17] Bytsenko, A. A., Cognola, G., Elizalde, E., Moretti, V., Zerbini, S.: Analytic Aspects of Quantum Fields. World Sci. (2003)Zbl 1138.81035 MR 2022075 · Zbl 1138.81035
[18] Cardona, A., Ducourtioux, C., Paycha, S.: From tracial anomalies to anomalies in Quantum Field Theory. Comm. Math. Phys.242, 31-65 (2003)Zbl 1037.81089 MR 2018268 · Zbl 1037.81089
[19] Ceyhan, O., Marcolli, M.: Feynman integrals and motives of configuration spaces. Comm. Math. Phys.313, 35-70 (2012)Zbl 1256.81051 MR 2928218 · Zbl 1256.81051
[20] Clavier, P., Guo, L., Paycha, S., Zhang, B.: An algebraic formulation of the locality principle in renormalisation. Eur. J. Math.5, 356-394 (2019)Zbl 07073688 MR 3946584 · Zbl 1467.81074
[21] Connes, A., Kreimer, D.: Hopf algebras, renormalization and noncommutative geometry. Comm. Math. Phys.199, 203-42 (1998)Zbl 0932.16038 MR 1660199 · Zbl 0932.16038
[22] Connes, A., Kreimer, D.: 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, 249-273 (2000)Zbl 1032.81026 MR 1748177 · Zbl 1032.81026
[23] Connes, A., Kreimer, D.: Renormalization in quantum field theory and the Riemann-Hilbert problem II: theβ-function, diffeomorphisms and the renormalization group. Comm. Math. Phys.216, 215-241 (2001)Zbl 1042.81059 MR 1810779 · Zbl 1042.81059
[24] Costello, K.: Renormalization and Effective Field Theory. Math. Surveys Monogr. 170, Amer. Math. Soc. (2011)Zbl 1221.81004 MR 2778558 · Zbl 1221.81004
[25] Dang, N. V.: Complex powers of analytic functions and meromorphic renormalization in QFT. arXiv:1503.00995(2015)
[26] Derezi´nski, J., Siemssen, D.: Feynman propagators on static spacetimes. Rev. Math. Phys.30, art. 1850006, 23 pp. (2018)Zbl 1394.35407 MR 3770965 · Zbl 1394.35407
[27] Derezi´nski, J., Siemssen, D.: An evolution equation approach to the Klein-Gordon operator on curved spacetime. Pure Appl. Anal.1, 215-261 (2019)Zbl 1423.58016 MR 3949374 · Zbl 1423.58016
[28] Duistermaat, J. J.: Fourier Integral Operators. Birkh¨auser, Boston (1996)Zbl 0841.35137 MR 1362544 · Zbl 0841.35137
[29] D¨utsch, M., Fredenhagen, K., Keller, K. J., Rejzner, K.: Dimensional regularization in position space and a forest formula for Epstein-Glaser renormalization. J. Math. Phys.55, art. 122303, 37 pp. (2014)Zbl 1309.81173 MR 3390545 · Zbl 1309.81173
[30] Epstein, H., Glaser, V.: The role of locality in perturbation theory. Ann. Inst. H. Poincar´e Sect. A19, 211-295 (1973)Zbl 1216.81075 MR 0342091 · Zbl 1216.81075
[31] Fulling, S. A.: Vacuum energy as spectral geometry. Symmetry Integrability Geom. Methods Appl.3, art. 094, 23 pp. (2007)Zbl 1142.35052 MR 2366928 · Zbl 1142.35052
[32] G´erard, C., Oulghazi, O., Wrochna, M.: Hadamard states for the Klein-Gordon equation on Lorentzian manifolds of bounded geometry. Comm. Math. Phys.352, 519-583 (2017) Zbl 1364.35362 MR 3627405 · Zbl 1364.35362
[33] G´erard, C., Wrochna, M.: The massive Feynman propagator on asymptotically Minkowski spacetimes. Amer. J. Math.141, 1501-1546 (2019)Zbl 07155013 MR 4030522 · Zbl 1450.83003
[34] G´erard, C., Wrochna, M.: Feynman propagators and Hadamard states from scattering data for the Klein-Gordon equation on asymptotically Minkowski spacetimes.arXiv:1603.07465 (2016)
[35] G´erard, C., Wrochna, M.: Analytic Hadamard states, Calder´on projectors and Wick rotation near analytic Cauchy surfaces. Comm. Math. Phys.366, 29-65 (2019)Zbl 1414.58002 MR 3919442 · Zbl 1414.58002
[36] Gilkey, P.: Invariance Theory, the Heat Equation and the Atiyah-Singer Index Theorem. CRC Press (1995)Zbl 0856.58001 MR 1396308 · Zbl 0856.58001
[37] Grothendieck, A.: Sur certains espaces de fonctions holomorphes. I. J. Reine Angew. Math. 192, 35-64 (1953)Zbl 0051.08704 MR 0058865 · Zbl 0051.08704
[38] Gunning, R., Rossi, H.: Analytic Functions of Several Complex Variables. Prentice-Hall, Englewood Cliffs, NJ (1965)Zbl 0141.08601 MR 0180696 · Zbl 0141.08601
[39] Guo, L., Paycha, S., Zhang, B.: A conical approach to Laurent expansions for multivariate meromorphic germs with linear poles. Pacific J. Math.307, 159-196 (2020)Zbl 07229725 MR 4131805 · Zbl 1447.32008
[40] Guo, L., Paycha, S., Zhang, B.: Counting an infinite number of points: a testing ground for renormalization methods. In: Geometric, Algebraic, and Topological Methods for Quantum Field Theory (Villa de Leyva, 2013), World Sci., 309-352 (2017)Zbl 1369.81072 MR 3617360 · Zbl 1369.81072
[41] Gurau, R., Rivasseau, V., Sfondrini, A.: Renormalization: an advanced overview. arXiv:1401.5003(2014)
[42] Hairer, M.: An analyst’s take on the BPHZ theorem. In: Computation and Combinatorics in Dynamics, Stochastics and Control, Abel Sympos. 13, Springer, 429-476 (2018) Zbl 1408.60049 MR 3967393 · Zbl 1408.60049
[43] Hawking, S. W.: Zeta function regularization of path integrals in curved spacetime. Comm. Math. Phys.55, 133-148 (1977)Zbl 0407.58024 MR 0524257 · Zbl 0407.58024
[44] Herscovich, E.: Renormalization in quantum field theory (after R. Borcherds). Ast´erisque 412 (2019)Zbl 1425.81005 MR 3978423 · Zbl 1425.81005
[45] Hollands, S., Wald, R. M.: Local Wick polynomials and time ordered products of quantum fields in curved spacetime. Comm. Math. Phys.223, 289-326 (2001)Zbl 0989.81081 MR 1864435 · Zbl 0989.81081
[46] Hollands, S., Wald, R. M.: Existence of local covariant time ordered products of quantum fields in curved spacetime. Comm. Math. Phys.231, 309-45 (2002)Zbl 1015.81043 MR 1946335 · Zbl 1015.81043
[47] Kalau, W., Walze, M.: Gravity, non-commutative geometry and the Wodzicki residue. J. Geom. Phys.16, 327-344 (1995)Zbl 0826.58008 MR 1336738 · Zbl 0826.58008
[48] Kastler, D.: The Dirac operator and gravitation. Comm. Math. Phys.166, 633-643 (1995) Zbl 0823.58046 MR 1312438 · Zbl 0823.58046
[49] Kontsevich, M., Vishik, S.: Geometry of determinants of elliptic operators. In: Functional Analysis on the Eve of the 21st Century, Vol. 1, Progr. Math. 131, Birkh¨auser, 173-197 (1995) Zbl 0920.58061 MR 1373003 · Zbl 0920.58061
[50] Krajewski, T., Rivasseau, V., Tanas˘a, A., Wang, Z.: Topological graph polynomials and quantum field theory. Part I: Heat kernel theories. J. Noncommut. Geom.4, 29-82 (2010) Zbl 1186.81095 MR 2575389 · Zbl 1186.81095
[51] Kruskal, J. B.: On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. Amer. Math. Soc.7, 48-50 (1956)Zbl 0070.18404 MR 0078686 · Zbl 0070.18404
[52] Lefschetz, S.: Applications of Algebraic Topology. Springer (1975)Zbl 0328.55001 MR 0494126 · Zbl 0328.55001
[53] Lesch, M.: On the noncommutative residue for pseudodifferential operators with logpolyhomogeneous symbols. Ann. Global Anal. Geom.17, 151-187 (1999)Zbl 0920.58047 MR 1675408 · Zbl 0920.58047
[54] Melrose, R. B.: The Atiyah-Patodi-Singer Index Theorem. A K Peters, Wellesley, MA (1993) Zbl 0796.58050 MR 1348401 · Zbl 0796.58050
[55] Melrose, R. B., Nistor, V.: Homology of pseudodifferential operators I. Manifolds with boundary.arXiv:funct-an/9606005(1996)
[56] Mnev, P.: Lecture notes on torsions.arXiv:1406.3705(2014)
[57] Nicolaescu, L. I.: Random Morse functions and spectral geometry.arXiv:1209.0639(2012)
[58] Nikolov, N., Nedanovski, D.: Analytic renormalization and residues of Feynman diagrams, C. R. Acad. Bulgare Sci.70, 1219-1226 (2017)
[59] Nikolov, N. M., Stora, R., Todorov, I.: Renormalization of massless Feynman amplitudes in configuration space. Rev. Math. Phys.26, no. 4, art. 1430002, 65 pp. (2014)Zbl 1303.81126 MR 3208883 · Zbl 1303.81126
[60] Paycha, S.: Regularised Integrals, Sums and Traces: An Analytic Point of View. Univ. Lecture Ser. 59, Amer. Math. Soc. (2012)Zbl 1272.11103 MR 2987296 · Zbl 1272.11103
[61] Paycha, S., Scott, S.: Chern-Weil forms associated with superconnections. In: Analysis, Geometry and Topology of Elliptic Operators, World Sci., 79-104 (2006)Zbl 1272.11103 MR 2246766 · Zbl 1119.58013
[62] Paycha, S., Scott, S.: A Laurent expansion for regularized integrals of holomorphic symbols. Geom. Funct. Anal.17, 491-536 (2007)Zbl 1125.58009 MR 2322493 · Zbl 1125.58009
[63] Popineau, G., Stora, R.: A pedagogical remark on the main theorem of perturbative renormalization theory. Nuclear Phys. B912, 70-78 (2016)Zbl 1349.81142 MR 3567574 · Zbl 1349.81142
[64] Pottel, S.: A BPHZ theorem in configuration space.arXiv:1706.06762(2017) · Zbl 1380.81195
[65] Pottel, S.: Configuration space BPHZ renormalization on analytic spacetimes. Nuclear Phys. B927, 274-293 (2018)Zbl 1380.81195 MR 3763330 · Zbl 1380.81195
[66] Ray, D. B., Singer, I. M.:R-torsion and the Laplacian on Riemannian manifolds. Adv. Math. 7, 145-210 (1971)Zbl 0239.58014 MR 0295381 · Zbl 0239.58014
[67] Riesz, M.: L’int´egrale de Riemann-Liouville et le probl‘eme de Cauchy. Acta Math.81, 1-222 (1949)Zbl 0033.27601 MR 0030102 · Zbl 0033.27601
[68] Rivasseau, V.: From Perturbative to Constructive Renormalization. Princeton Univ. Press, Princeton (1991)MR 1174294
[69] Roe, J.: Elliptic Operators, Topology, and Asymptotic Methods. Longman (1998) Zbl 0919.58060 MR 1670907 · Zbl 0919.58060
[70] Schwarz, A. S.: The partition function of degenerate quadratic functional and Ray-Singer invariants. Lett. Math. Phys.2, 247-252 (1978)Zbl 0383.70017 MR 0676337 · Zbl 0383.70017
[71] Scott, S.: Traces and Determinants of Pseudodifferential Operators. Oxford Univ. Press, Oxford (2010)Zbl 1216.35192 MR 2683288 · Zbl 1216.35192
[72] Seeley, R. T.: Complex powers of an elliptic operator. In: Singular Integrals (Chicago, IL, 1966), Proc. Sympos. Pure Math. 10, Amer. Math. Soc., 288-307 (1967)Zbl 0159.15504 MR 0237943 · Zbl 0159.15504
[73] Speer, E. R.: Analytic renormalization. J. Math. Phys.9, 1404-1410 (1968)
[74] Speer, E. R.: Dimensional and analytic renormalization. In: Renormalization Theory, Reidel, 25-93 (1976)MR 0522287
[75] Speer, E. R.: Lectures on analytic renormalization. Technical report 73-067, Univ. of Maryland (1972)
[76] Taylor, M.: Partial Differential Equations II: Qualitative Studies of Linear Equations. Appl. Math. Sci. 116, Springer (2011)Zbl 1206.35003 MR 2743652 · Zbl 1206.35003
[77] Vasy, A., Wrochna, M.: Quantum fields from global propagators on asymptotically Minkowski and extended de Sitter spacetimes. Ann. Henri Poincar´e19, 1529-1586 (2018) Zbl 06871241 MR 3784921 · Zbl 1457.81037
[78] Wodzicki, M.: Noncommutative residue. Chapter I. Fundamentals. In: K-Theory, Arithmetic and Geometry, Springer, 320-399 (1987)Zbl 0649.58033 MR 0923140
[79] Wodzicki, M.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.