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Hadamard states for the Klein-Gordon equation on Lorentzian manifolds of bounded geometry. (English) Zbl 1364.35362

Summary: We consider the Klein-Gordon equation on a class of Lorentzian manifolds with Cauchy surface of bounded geometry, which is shown to include examples such as exterior Kerr, Kerr-de Sitter spacetime and the maximal globally hyperbolic extension of the Kerr outer region. In this setup, we give an approximate diagonalization and a microlocal decomposition of the Cauchy evolution using a time-dependent version of the pseudodifferential calculus on Riemannian manifolds of bounded geometry. We apply this result to construct all pure regular Hadamard states (and associated Feynman inverses), where regular refers to the state’s two-point function having Cauchy data given by pseudodifferential operators. This allows us to conclude that there is a one-parameter family of elliptic pseudodifferential operators that encodes both the choice of (pure, regular) Hadamard state and the underlying spacetime metric.

MSC:

35Q75 PDEs in connection with relativity and gravitational theory
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83F05 Relativistic cosmology
35S05 Pseudodifferential operators as generalizations of partial differential operators
53B20 Local Riemannian geometry
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[1] Ammann B., Lauter R., Nistor V., Vasy A.: Complex powers and non-compact manifolds. Commun. Partial Differ Equ 29, 671-705 (2004) · Zbl 1071.58022 · doi:10.1081/PDE-120037329
[2] Avetisyan Z.: A unified mode decomposition method for physical fields in homogeneous cosmology. Rev. Math. Phys. 26, 1430001 (2014) · Zbl 1290.83001 · doi:10.1142/S0129055X14300015
[3] Brouder C., Dang N.V., Hélein F.: Boundedness and continuity of the fundamental operations on distributions having a specified wave front set, (with a counter example by Semyon Alesker). Stud. Math. 232, 201-226 (2016) · Zbl 1366.46027
[4] Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics, vol 1, 2nd edn. Springer, Berlin (1987) · Zbl 0905.46046
[5] Brum, M., Fredenhagen, K.: ‘Vacuum-like’ Hadamard states for quantum fields on curved spacetimes. Class. Quantum Gravity 31(2), 025024 (2014) · Zbl 1292.83027
[6] Brunetti R., Fredenhagen K.: Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds. Commun. Math. Phys. 208, 623-661 (2000) · Zbl 1040.81067 · doi:10.1007/s002200050004
[7] Bär, C., Ginoux, N., Pfäffle, F.: Wave equation on Lorentzian manifolds and quantization. ESI Lectures in Mathematics and Physics, EMS (2007) · Zbl 1118.58016
[8] Brum, M., Jorás, S.E.: Hadamard state in Schwarzschild-de Sitter spacetime. Class. Quantum Grav. 32(1) (2014) · Zbl 1309.83061
[9] Bernal A.M., Sánchez M.: On smooth Cauchy hypersurfaces and Geroch’s splitting theorem. Commun. Math. Phys. 243, 461-470 (2013) · Zbl 1085.53060 · doi:10.1007/s00220-003-0982-6
[10] Brum M., Them K.: States of low energy in homogeneous and inhomogeneous expanding spacetimes. Class. Quantum Gravity 30(23), 235035 (2013) · Zbl 1284.83024 · doi:10.1088/0264-9381/30/23/235035
[11] Chazarain J.: Opérateurs hyperboliques à caractéristiques de multiplicité constante. Ann. Inst. Fourier 24, 173-202 (1974) · Zbl 0274.35045 · doi:10.5802/aif.497
[12] Choquet-Bruhat, Y.: Hyperbolic partial differential equations on a manifold. In: Battelle Rencontres, Lectures Math. Phys., Benjamin, New York, 1968, pp. 84-106 (1967) · Zbl 0169.43202
[13] Cheeger J., Gromov M.: Bounds on the von Neumann dimension of L2-cohomology and the Gauss-Bonnet theorem for open manifolds. J. Differ. Geom. 21, 1-34 (1985) · Zbl 0614.53034 · doi:10.4310/jdg/1214439461
[14] Cheeger J., Gromov M., Taylor M.: Finite propagation speed, kernel estimates for functions of the Laplace operator and the geometry of complete Riemannian manifolds. J. Differ. Geom. 17, 15-54 (1982) · Zbl 0493.53035 · doi:10.4310/jdg/1214436699
[15] Dang, N.V.: Complex powers of analytic functions and meromorphic renormalization in QFT, preprint. arXiv:1503.00995 (2015) · Zbl 0861.53074
[16] Dang, N.V.: Renormalization of quantum field theory on curved spacetimes, a causal approach. Ph.D. thesis, Paris Diderot University (2013)
[17] Dang, N.V.: The extension of distributions on manifolds, a microlocal approach. Ann. Henri Poincaré Online First (2015). doi:10.1007/s00023-015-0419-8 · Zbl 1038.81052
[18] Dappiaggi C., Drago, N.: Constructing Hadamard states via an extended Møller operator. Lett. Math. Phys. 106, 11 (2016). Read More: http://www.worldscientific.com/doi/abs/10.1142/S0129055X14300015 · Zbl 1362.81073
[19] Dappiaggi, C., Moretti, V., Pinamonti, N.: Distinguished quantum states in a class of cosmological spacetimes and their Hadamard property. J. Math. Phys. 50, 062304 (2009) · Zbl 1216.81112
[20] Dappiaggi C., Moretti V., Pinamonti N.: Rigorous construction and Hadamard property of the Unruh state in Schwarzschild spacetime. Adv. Theor. Math. Phys. 15, 355 (2011) · Zbl 1257.83008 · doi:10.4310/ATMP.2011.v15.n2.a4
[21] Dereziński, J., Gérard, C.: Mathematics of Quantization and Quantum Fields. Cambridge Monographs in Mathematical Physics, Cambridge University Press, Cambridge (2013) · Zbl 1271.81004
[22] Dimock J.: Algebras of local observables on a manifold. Commun. Math. Phys. 77, 219-228 (1980) · Zbl 0455.58030 · doi:10.1007/BF01269921
[23] Duistermaat J.J., Hörmander L.: Fourier integral operators, II. Acta Math. 128, 183-269 (1972) · Zbl 0232.47055 · doi:10.1007/BF02392165
[24] Eldering, J.: Persistence of non compact normally hyperbolic invariant manifolds in bounded geometry. PhD Thesis Utrecht University (2012) · Zbl 1257.53054
[25] Fulling S.A., Narcowich F.J., Wald R.M.: Singularity structure of the two-point function in quantum field theory in curved spacetime II. Ann. Phys. 136, 243-272 (1981) · Zbl 0495.35054 · doi:10.1016/0003-4916(81)90098-1
[26] Fewster C.J., Verch R.: The necessity of the Hadamard condition. Class. Quant. Gravity 30, 235027 (2013) · Zbl 1284.83057 · doi:10.1088/0264-9381/30/23/235027
[27] Fewster, C.J., Verch, R.: Algebraic quantum field theory in curved spacetimes. In: Advances in Algebraic Quantum Field Theory, Springer (2015) · Zbl 1334.81079
[28] Gell-Redman J., Haber N., Vasy A.: The Feynman propagator on perturbations of Minkowski space. Commun. Math. Phys. 342(1), 333-384 (2016) · Zbl 1335.81085 · doi:10.1007/s00220-015-2520-8
[29] Gérard C., Wrochna M.: Construction of Hadamard states by pseudo-differential calculus. Commun. Math. Phys. 325(2), 713-755 (2014) · Zbl 1298.81214 · doi:10.1007/s00220-013-1824-9
[30] Gérard C., Wrochna M.: Hadamard states for the linearized Yang-Mills equation on curved spacetime. Commun. Math. Phys. 337, 253-320 (2015) · Zbl 1314.83025 · doi:10.1007/s00220-015-2305-0
[31] Gérard C., Wrochna M.: Construction of Hadamard states by characteristic Cauchy problem. Anal. PDE 9(1), 111-149 (2016) · Zbl 1334.83020 · doi:10.2140/apde.2016.9.111
[32] Häfner D.: Sur la théorie de la diffusion pour l’équation de Klein-Gordon dans la métrique de Kerr. Diss. Math. 421, 1-102 (2003) · Zbl 1075.35093
[33] Hollands S.: The Hadamard condition for Dirac fields and adiabatic states on Robertson-Walker spacetimes. Commun. Math. Phys. 216, 635-661 (2001) · Zbl 0976.58023 · doi:10.1007/s002200000350
[34] Hörmander, L.: The analysis of linear partial differential operators I. In: Distribution Theory and Fourier Analysis, Springer, Berlin (1985) · Zbl 0601.35001
[35] Hollands, S., Wald, R.M.: Quantum fields in curved spacetime. In: General Relativity and Gravitation: A Centennial Perspective, Cambridge University Press, Cambridge (2015) · Zbl 1357.81144
[36] Kordyukov Y.: Lp-theory of elliptic differential operators on manifolds of bounded geometry. Acta Appl. Math. 23, 223-260 (1991) · Zbl 0743.58030
[37] Khavkine, I., Moretti, V.: Algebraic QFT in curved spacetime and quasifree Hadamard states: an introduction. In: Advances in Algebraic Quantum Field Theory, Springer (2015) · Zbl 1334.81081
[38] Kay B.S., Wald R.M.: Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon. Phys. Rep. 207, 49 (1991) · Zbl 0861.53074 · doi:10.1016/0370-1573(91)90015-E
[39] Junker W.: Hadamard states, adiabatic vacua and the construction of physical states for scalar quantum fields on curved spacetime. Rev. Math. Phys. 8, 1091-1159 (1996) · Zbl 0869.53053 · doi:10.1142/S0129055X9600041X
[40] Junker W.: Erratum to “Hadamard states, adiabatic vacua and the construction of physical states.”. Rev. Math. Phys. 207, 511-517 (2002) · doi:10.1142/S0129055X02001326
[41] Junker W., Schrohe E.: Adiabatic vacuum states on general space-time manifolds: definition, construction, and physical properties. Ann. Henri Poincaré 3, 1113-1181 (2002) · Zbl 1038.81052 · doi:10.1007/s000230200001
[42] Leray, J.: Hyperbolic Differential Equations. Unpublished Lecture Notes, Princeton (1953)
[43] Moretti V.: Quantum out-states holographically induced by asymptotic flatness: invariance under spacetime symmetries, energy positivity and Hadamard property. Commun. Math. Phys. 279, 31-75 (2008) · Zbl 1145.83016 · doi:10.1007/s00220-008-0415-7
[44] Olbermann H.: States of low energy on Robertson-Walker spacetimes. Class. Quantum Gravity 24, 5011 (2007) · Zbl 1206.83171 · doi:10.1088/0264-9381/24/20/007
[45] O’ Neill B.: Semi-Riemannian Geometry. Academic Press, Cambridge (1983) · Zbl 0531.53051
[46] O’ Neill B.: The Geometry of Kerr Black Holes. A K Peters, Natick (1995) · Zbl 0828.53078
[47] Oulghazi, O.: Ph.D. Thesis in Preparation · Zbl 1335.81116
[48] Radzikowski M.: Micro-local approach to the Hadamard condition in quantum field theory on curved spacetime. Commun. Math. Phys. 179, 529-553 (1996) · Zbl 0858.53055 · doi:10.1007/BF02100096
[49] Reed M., Simon B.: Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness. Academic Press, Cambridge (1975) · Zbl 0308.47002
[50] Roe J.: An index theorem on open manifolds I. J. Differ. Geom. 27, 87-113 (1988) · Zbl 0657.58041 · doi:10.4310/jdg/1214441652
[51] Sahlmann H., Verch R.: Passivity and microlocal spectrum condition. Commun. Math. Phys. 214, 705-731 (2000) · Zbl 1010.81046 · doi:10.1007/s002200000297
[52] Sanders K.: On the construction of Hartle-Hawking-Israel states across a static bifurcate Killing horizon. Lett. Math. Phys. 105(4), 575-640 (2015) · Zbl 1325.81129 · doi:10.1007/s11005-015-0745-2
[53] Seeley, R.: Complex powers of an elliptic operator. In: Singular Integrals, Proc. Symp. Pure Math., pp. 288-307. AMS, Providence, RI (1967) · Zbl 0159.15504
[54] Shubin M.A.: Pseudo-differential Operators and Spectral Theory. Springer, Berlin (2001) · Zbl 0980.35180 · doi:10.1007/978-3-642-56579-3
[55] Shubin M.A.: Spectral theory of elliptic operators on non-compact manifolds. Astérisque 207, 37-108 (1992) · Zbl 0793.58039
[56] Stottmeister A., Thiemann T.: The microlocal spectrum condition, initial value formulations and background independence. J. Math. Phys. 57, 022303 (2016) · Zbl 1335.81116 · doi:10.1063/1.4940052
[57] Taylor M.: Pseudo-differential Operators and Nonlinear PDE. Birkhäuser, Cambridge (1991) · Zbl 0746.35062 · doi:10.1007/978-1-4612-0431-2
[58] Vasy, A.: On the positivity of propagator differences. Ann. Henri Poincaré. doi:10.1007/s00023-016-0527-0 (2016) · Zbl 1360.81173
[59] Vasy, A., Wrochna, M.: Quantum fields from global propagators on asymptotically Minkowski and extended de Sitter spacetimes, preprint. arXiv:1512.08052 (2015) · Zbl 1457.81037
[60] Wald R.M.: General Relativity. University of Chicago Press, Chicago (1985) · Zbl 0549.53001
[61] Wrochna, M.: Singularities of two-point functions in quantum field theory. Ph.D. Thesis, University of Göttingen (2013) · Zbl 1290.83001
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