Dynamical residues of Lorentzian spectral zeta functions
Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 1245-1292.

We define a dynamical residue which generalizes the Guillemin–Wodzicki residue density of pseudo-differential operators. More precisely, given a Schwartz kernel, the definition refers to Pollicott–Ruelle resonances for the dynamics of scaling towards the diagonal. We apply this formalism to complex powers of the wave operator and we prove that residues of Lorentzian spectral zeta functions are dynamical residues. The residues are shown to have local geometric content as expected from formal analogies with the Riemannian case.

Nous définissons un résidu dynamique qui généralise la densité de résidus de Guillemin-Wodzicki des opérateurs pseudo-différentiels. Plus précisément, étant donné un noyau de Schwartz, la définition fait référence aux résonances de Pollicott-Ruelle pour la dynamique de l’échelonnement vers la diagonale. Nous appliquons ce formalisme aux puissances complexes de l’opérateur des ondes et nous prouvons que les résidus des fonctions zêta spectrales lorentziennes sont des résidus dynamiques. Nous montrons que les résidus ont un contenu géométrique local, comme prévu par les analogies formelles avec le cas riemannien.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.205
Classification: 58J40, 35A17, 37C30
Keywords: Guillemin–Wodzicki residue, spectral zeta functions, wave equation, Hadamard parametrix, Pollicott–Ruelle resonances
Mot clés : Résidu de Guillemin–Wodzicki, fonctions zêta spectrales, équation des ondes, paramétrixe d’Hadamard, résonances de Pollicott–Ruelle

Nguyen Viet Dang 1; Michał Wrochna 2

1 Institut de Mathématiques de Jussieu, Sorbonne Université – Université de Paris 4 pl. Jussieu, 75252 Paris, France
2 Laboratoire Analyse Géométrie Modélisation, CY Cergy Paris Université 2 av. Adolphe Chauvin, 95302 Cergy-Pontoise, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{JEP_2022__9__1245_0,
     author = {Nguyen Viet Dang and Micha{\l} Wrochna},
     title = {Dynamical residues of {Lorentzian} spectral zeta functions},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {1245--1292},
     publisher = {\'Ecole polytechnique},
     volume = {9},
     year = {2022},
     doi = {10.5802/jep.205},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.205/}
}
TY  - JOUR
AU  - Nguyen Viet Dang
AU  - Michał Wrochna
TI  - Dynamical residues of Lorentzian spectral zeta functions
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2022
SP  - 1245
EP  - 1292
VL  - 9
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.205/
DO  - 10.5802/jep.205
LA  - en
ID  - JEP_2022__9__1245_0
ER  - 
%0 Journal Article
%A Nguyen Viet Dang
%A Michał Wrochna
%T Dynamical residues of Lorentzian spectral zeta functions
%J Journal de l’École polytechnique — Mathématiques
%D 2022
%P 1245-1292
%V 9
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.205/
%R 10.5802/jep.205
%G en
%F JEP_2022__9__1245_0
Nguyen Viet Dang; Michał Wrochna. Dynamical residues of Lorentzian spectral zeta functions. Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 1245-1292. doi : 10.5802/jep.205. https://jep.centre-mersenne.org/articles/10.5802/jep.205/

[1] J. L. Antoniano & G. A. Uhlmann - “A functional calculus for a class of pseudodifferential operators with singular symbols”, in Pseudodifferential operators and applications (Notre Dame, Ind., 1984), Proc. Sympos. Pure Math., vol. 43, American Mathematical Society, Providence, RI, 1985, p. 5-16 | DOI | MR | Zbl

[2] V. Baladi - Dynamical zeta functions and dynamical determinants for hyperbolic maps, Ergeb. Math. Grenzgeb. (3), vol. 68, Springer, Cham, 2018 | DOI

[3] C. Bär & A. Strohmaier - “An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary”, Amer. J. Math. 141 (2019) no. 5, p. 1421-1455 | DOI | MR | Zbl

[4] C. Bär & A. Strohmaier - “Local index theory for Lorentzian manifolds”, 2020 | arXiv

[5] F. Bischoff, H. Bursztyn, H. Lima & E. Meinrenken - “Deformation spaces and normal forms around transversals”, Compositio Math. 156 (2020) no. 4, p. 697-732 | DOI | MR | Zbl

[6] J. Bourgain, P. Shao, C. D. Sogge & X. Yao - “On L p -resolvent estimates and the density of eigenvalues for compact Riemannian manifolds”, Comm. Math. Phys. 333 (2015) no. 3, p. 1483-1527 | DOI | MR | Zbl

[7] R. Brunetti & K. Fredenhagen - “Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds”, Comm. Math. Phys. 208 (2000) no. 3, p. 623-661 | DOI | MR | Zbl

[8] H. Bursztyn, H. Lima & E. Meinrenken - “Splitting theorems for Poisson and related structures”, J. reine angew. Math. (2019) no. 754, p. 281-312 | DOI | MR | Zbl

[9] A. Connes - “The action functional in non-commutative geometry”, Comm. Math. Phys. 117 (1988) no. 4, p. 673-683 | DOI | Zbl

[10] A. Connes & H. Moscovici - “The local index formula in noncommutative geometry”, Geom. Funct. Anal. 5 (1995) no. 2, p. 174-243 | DOI | MR | Zbl

[11] N. V. Dang - Renormalization of quantum field theory on curved space-times, a causal approach, Ph. D. Thesis, Université Paris Diderot (Paris VII), 2013

[12] N. V. Dang - “The extension of distributions on manifolds, a microlocal approach”, Ann. Henri Poincaré 17 (2016) no. 4, p. 819-859 | DOI | MR | Zbl

[13] N. V. Dang & M. Wrochna - “Complex powers of the wave operator and the spectral action on Lorentzian scattering spaces”, 2020 | arXiv

[14] J. Dereziński & D. Siemssen - “Feynman propagators on static spacetimes”, Rev. Math. Phys. 30 (2018) no. 3, article ID 1850006, 23 pages | DOI | MR | Zbl

[15] J. Dereziński & D. Siemssen - “An evolution equation approach to linear quantum field theory”, 2019 | arXiv

[16] B. S. DeWitt - “Quantum field theory in curved spacetime”, Phys. Rep. 19 (1975) no. 6, p. 295-357 | DOI

[17] D. Dos Santos Ferreira, C. E. Kenig & M. Salo - “On L p resolvent estimates for Laplace–Beltrami operators on compact manifolds”, Forum Math. 26 (2014) no. 3, p. 815-849 | DOI | MR | Zbl

[18] J. S. Dowker & R. Critchley - “Effective Lagrangian and energy-momentum tensor in de Sitter space”, Phys. Rev. D 13 (1976) no. 12, p. 3224-3232 | DOI

[19] S. Dyatlov & M. Zworski - “Dynamical zeta functions for Anosov flows via microlocal analysis”, Ann. Sci. École Norm. Sup. (4) 49 (2016) no. 3, p. 543-577 | DOI | MR | Zbl

[20] S. A. Fulling - Aspects of quantum field theory in curved space-time, London Math. Soc. Student Texts, vol. 17, Cambridge University Press, Cambridge, 1989 | DOI

[21] J. Gell-Redman, N. Haber & A. Vasy - “The Feynman propagator on perturbations of Minkowski space”, Comm. Math. Phys. 342 (2016) no. 1, p. 333-384 | DOI | MR | Zbl

[22] C. Gérard & M. Wrochna - “The massive Feynman propagator on asymptotically Minkowski spacetimes”, Amer. J. Math. 141 (2019) no. 6, p. 1501-1546 | DOI | MR | Zbl

[23] C. Gérard & M. Wrochna - “The massive Feynman propagator on asymptotically Minkowski spacetimes II”, Internat. Math. Res. Notices (2020) no. 20, p. 6856-6870 | DOI | MR | Zbl

[24] J. M. Gracia-Bondía, H. Gutiérrez & J. C. Várilly - “Improved Epstein–Glaser renormalization in x-space versus differential renormalization”, Nuclear Phys. B 886 (2014), p. 824-869 | DOI | MR | Zbl

[25] A. Greenleaf & G. Uhlmann - “Estimates for singular Radon transforms and pseudodifferential operators with singular symbols”, J. Funct. Anal. 89 (1990) no. 1, p. 202-232 | DOI | MR | Zbl

[26] V. Guillemin - “A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues”, Adv. Math. 55 (1985) no. 2, p. 131-160 | DOI | MR | Zbl

[27] V. Guillemin - “Gauged Lagrangian distributions”, Adv. Math. 102 (1993) no. 2, p. 184-201 | DOI | MR | Zbl

[28] V. Guillemin & G. Uhlmann - “Oscillatory integrals with singular symbols”, Duke Math. J. 48 (1981) no. 1, p. 251-267 | DOI | MR | Zbl

[29] T.-P. Hack & V. Moretti - “On the stress–energy tensor of quantum fields in curved spacetimes – comparison of different regularization schemes and symmetry of the Hadamard/Seeley–DeWitt coefficients”, J. Phys. A 45 (2012) no. 37, article ID 374019 | DOI | MR | Zbl

[30] T. Hartung & S. Scott - “A generalized Kontsevich-Vishik trace for Fourier integral operators and the Laurent expansion of ζ-functions”, 2015 | arXiv

[31] S. W. Hawking - “Zeta function regularization of path integrals in curved spacetime”, Comm. Math. Phys. 55 (1977) no. 2, p. 133-148 | DOI | MR | Zbl

[32] S. Hollands & R. M. Wald - “Local Wick polynomials and time ordered products of quantum fields in curved spacetime”, Comm. Math. Phys. 223 (2001) no. 2, p. 289-326 | DOI | MR | Zbl

[33] L. Hörmander - The analysis of linear partial differential operators I. Distribution theory and Fourier analysis, Classics in Math., Springer-Verlag, Berlin, 2003

[34] L. Hörmander - The analysis of linear partial differential operators III. Pseudo-differential operators, Classics in Math., Springer, Berlin, 2007 | DOI

[35] M. S. Joshi - “An intrinsic characterisation of polyhomogeneous Lagrangian distributions”, Proc. Amer. Math. Soc. 125 (1997) no. 5, p. 1537-1543 | DOI | MR | Zbl

[36] M. S. Joshi - “Complex powers of the wave operator”, Portugal. Math. 54 (1997) no. 3, p. 345-362 | MR | Zbl

[37] M. S. Joshi - “A symbolic construction of the forward fundamental solution of the wave operator”, Comm. Partial Differential Equations 23 (1998) no. 7-8, p. 1349-1417 | DOI | MR | Zbl

[38] B. S. Kay & R. M. Wald - “Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon”, Phys. Rep. 207 (1991) no. 2, p. 49-136 | DOI | MR | Zbl

[39] M. Kontsevich & S. Vishik - “Geometry of determinants of elliptic operators”, in Functional analysis on the eve of the 21st century, Vol. 1 (New Brunswick, NJ, 1993), Progress in Math., vol. 131, Birkhäuser Boston, Boston, MA, 1995, p. 173-197 | DOI | MR | Zbl

[40] M. Lesch - “On the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols”, Ann. Global Anal. Geom. 17 (1999), p. 151-187 | DOI | MR | Zbl

[41] M. Lesch & M. J. Pflaum - “Traces on algebras of parameter dependent pseudodifferential operators and the eta–invariant”, Trans. Amer. Math. Soc. 352 (2000) no. 11, p. 4911-4936 | DOI | MR | Zbl

[42] M. Lewandowski - “Hadamard states for bosonic quantum field theory on globally hyperbolic spacetimes”, 2020 | arXiv

[43] Y. Maeda, D. Manchon & S. Paycha - “Stokes’ formulae on classical symbol valued forms and applications”, 2005 | arXiv

[44] E. Meinrenken - “Euler-like vector fields, normal forms, and isotropic embeddings”, Indag. Math. (N.S.) 32 (2021) no. 1, p. 224-245 | DOI | MR | Zbl

[45] R. B. Melrose & G. A. Uhlmann - “Lagrangian intersection and the Cauchy problem”, Comm. Pure Appl. Math. 32 (1979) no. 4, p. 483-519 | DOI | MR | Zbl

[46] Y. Meyer - Wavelets, vibrations and scalings, CRM Monograph Series, vol. 9, American Mathematical Society, Providence, RI, 1998 | DOI

[47] S. Minakshisundaram & Å. Pleijel - “Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds”, Canad. J. Math. 1 (1949) no. 3, p. 242-256 | DOI | MR | Zbl

[48] V. Moretti - “Local ζ-function techniques vs. point-splitting procedure: A few rigorous results”, Comm. Math. Phys. 201 (1999) no. 2, p. 327-363 | DOI | MR | Zbl

[49] V. Moretti - “One-loop stress-tensor renormalization in curved background: The relation between ζ-function and point-splitting approaches, and an improved point-splitting procedure”, J. Math. Phys. 40 (1999) no. 8, p. 3843-3875 | DOI | MR | Zbl

[50] S. Nakamura & K. Taira - “Essential self-adjointness of real principal type operators”, Ann. H. Lebesgue 4 (2021), p. 1035-1059 | DOI | MR | Zbl

[51] N. M. Nikolov, R. Stora & I. Todorov - “Euclidean configuration space renormalization, residues and dilation anomaly”, in Lie theory and its applications in physics, Springer Proc. Math. Stat., vol. 36, Springer, Tokyo, 2013, p. 127-147 | DOI | MR | Zbl

[52] S. Paycha - “The noncommutative residue and canonical trace in the light of Stokes’ and continuity properties”, 2007 | arXiv

[53] S. Paycha - Regularised integrals, sums and traces. An analytic point of view, University Lecture Series, vol. 59, American Mathematical Society, Providence, RI, 2012

[54] M. Pollicott - “Meromorphic extensions of generalised zeta functions”, Invent. Math. 85 (1986) no. 1, p. 147-164 | DOI | MR | Zbl

[55] M. J. Radzikowski - “Micro-local approach to the Hadamard condition in quantum field theory on curved space-time”, Comm. Math. Phys. 179 (1996) no. 3, p. 529-553 | DOI | MR | Zbl

[56] K. Rejzner - “Renormalization and periods in perturbative algebraic quantum field theory”, in Periods in quantum field theory and arithmetic, Springer Proc. Math. Stat., vol. 314, Springer, Cham, 2020, p. 345-376 | DOI | MR | Zbl

[57] D. Ruelle - “Resonances of chaotic dynamical systems”, Phys. Rev. Lett. 56 (1986) no. 5, p. 405-407 | DOI | MR

[58] R. T. Seeley - “Complex powers of an elliptic operator”, in Singular Integrals (Chicago, Ill., 1966), Proc. Symp. Pure Math., vol. 10, American Mathematical Society, Providence, RI, 1967, p. 288-307 | DOI | Zbl

[59] D. Shen & M. Wrochna - “An index theorem on asymptotically static spacetimes with compact Cauchy surface”, 2021 | arXiv

[60] M. A. Shubin - Pseudodifferential operators and spectral theory, Springer-Verlag, Berlin, 2001 | DOI

[61] C. D. Sogge - “Concerning the L p norm of spectral clusters for second-order elliptic operators on compact manifolds”, J. Funct. Anal. 77 (1988) no. 1, p. 123-138 | DOI | MR | Zbl

[62] C. D. Sogge - Hangzhou lectures on eigenfunctions of the Laplacian, Annals of Math. Studies, vol. 188, Princeton University Press, Princeton, NJ, 2014 | DOI

[63] A. Strohmaier & S. Zelditch - “A Gutzwiller trace formula for stationary space-times”, Adv. Math. 376 (2020), article ID 107434, 53 pages | DOI | MR | Zbl

[64] A. Strohmaier & S. Zelditch - “Spectral asymptotics on stationary space-times”, Rev. Math. Phys. (2020), article ID X206000 | DOI

[65] A. Strohmaier & S. Zelditch - “Semi-classical mass asymptotics on stationary spacetimes”, Indag. Math. (N.S.) 32 (2021) no. 1, p. 323-363 | DOI | MR | Zbl

[66] K. Taira - “Limiting absorption principle and equivalence of Feynman propagators on asymptotically Minkowski spacetimes”, Comm. Math. Phys. 388 (2021) no. 1, p. 625-655 | DOI | MR | Zbl

[67] A. Vasy - “On the positivity of propagator differences”, Ann. Henri Poincaré 18 (2017) no. 3, p. 983-1007 | DOI | MR | Zbl

[68] A. Vasy - “Essential self-adjointness of the wave operator and the limiting absorption principle on Lorentzian scattering spaces”, J. Spectral Theory 10 (2020) no. 2, p. 439-461 | DOI | MR | Zbl

[69] A. Vasy & M. Wrochna - “Quantum fields from global propagators on asymptotically Minkowski and extended de Sitter spacetimes”, Ann. Henri Poincaré 19 (2018) no. 5, p. 1529-1586 | DOI | MR | Zbl

[70] R. M. Wald - “On the Euclidean approach to quantum field theory in curved spacetime”, Comm. Math. Phys. 70 (1979) no. 3, p. 221-242 | DOI | MR

[71] M. Wodzicki - “Local invariants of spectral asymmetry”, Invent. Math. 75 (1984) no. 1, p. 143-177 | DOI | MR | Zbl

[72] S. Zelditch - “Pluri-potential theory on Grauert tubes of real analytic Riemannian manifolds, I”, in Spectral geometry, Proc. Sympos. Pure Math., vol. 84, American Mathematical Society, Providence, RI, 2012, p. 299-339 | DOI | MR | Zbl

[73] S. Zelditch - Eigenfunctions of the Laplacian of Riemannian manifolds, CBMS Regional Conference Series in Math., vol. 125, American Mathematical Society, Providence, RI, 2017 | DOI

Cited by Sources: