Abstract
We set up a general framework for Calderón projectors (and their generalization to non-compact manifolds), associated with complex Laplacians obtained by Wick rotation of a Lorentzian metric. In the analytic case, we use this to show that the Laplacian’s Green’s functions have analytic continuations whose boundary values are two-point functions of analytic Hadamard states. The result does not require the metric to be stationary. As an aside, we describe how thermal states are obtained as a special case of this construction if the coefficients are time independent.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ammann, B., Grosse, N., Nistor, V.: Well-posedness of the Laplacian on manifolds with boundary and bounded geometry. Math. Nachr. 292(6), 1213–1237 (2019)
Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics II, 2nd edn. Springer, Berlin (2002)
Candelas, P., Raine, D.J.: Feynman propagator in curved space-time. Phys. Rev. D 15(6), 1494–1500 (1977)
Chruściel, P.T., Delay, E.: Non-singular, vacuum, stationary space-times with a negative cosmological constant. Ann. Henri Poincaré 8, 219–239 (2007)
Dang, N.V.: Renormalization of determinant lines in quantum field theory, preprint arXiv:1901.10542 (2019)
Dappiaggi, C., Drago, N., Rinaldi, P.: The algebra of Wick polynomials of a scalar field on a Riemannian manifold, preprint arXiv:1901.10542 (2019)
Dereziński, J., Siemssen, D.: Feynman propagators on static spacetimes. Rev. Math. Phys. 30(03), 1850006 (2018)
Dang, N.V., Zhang, B.: Renormalization of Feynman amplitudes on manifolds by spectral zeta regularization and blow-ups, preprint arXiv:1903.01258 (2019)
Fulling, S.A., Ruijsenaars, S.N.: Temperature, periodicity and horizons. Phys. Rep. 152, 135–176 (1987)
Gérard, C.: On the Hartle–Hawking–Israel states for spacetimes with static bifurcate Killing horizons, preprint arXiv:1608.06739 (2016)
Gérard, C.: The Hartle–Hawking–Israel state on stationary black hole spacetimes, preprint arXiv:1806.07645 (2018)
Gérard, C.: Microlocal Analysis of Quantum Fields on Curved Spacetimes, preprint arXiv:1901.10175 (2019)
Gell-Redman, J., Haber, N., Vasy, A.: The Feynman propagator on perturbations of Minkowski space. Commun. Math. Phys. 342(1), 333–384 (2016)
Grubb, G.: Distributions and Operators. Graduate Texts in Mathematics. Springer, Berlin (2009)
Glimm, J., Jaffe, A.: Quantum Physics: A Functional Integral Point of View. Springer, Berlin (1987)
Grosse, N., Nistor, V.: Neumann and mixed problems on manifolds with boundary and bounded geometry, preprint arXiv:1703.07228 (2017)
Gérard, C., Wrochna, M.: Analytic Hadamard states, Calderón projectors and Wick rotation near analytic Cauchy surfaces. Commun. Math. Phys. 366(1), 29–65 (2019)
Hack, T.-P., Moretti, V.: 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 Math. Theor. 45, 374019 (2012)
Hörmander, L.: The Analysis of Linear Partial Differential Operators, vol. 3. Springer, Berlin (1994)
Komatsu, H.: Microlocal analysis in Gevrey classes and in complex domains. In: Cattabriga, L., Rodino, L. (eds.) Microlocal Analysis and Applications. C.I.M.E. Lectures Montecatini Terme. Springer, Berlin (1989)
Moretti, V.: One-loop stress-tensor renormalization in curved background: the relation between \(\zeta \)-function and point-splitting approaches, and an improved point-splitting procedure. J. Math. Phys. 40(8), 3843–3875 (1999)
Moretti, V.: Proof of the symmetry of the off-diagonal Hadamard/Seeley–deWitt’s coefficients in \(C^\infty \) Lorentzian manifolds by a ‘local Wick rotation’. Commun. Math. Phys. 212, 165 (2000)
Osterwalder, K., Schrader, R.: Axioms for Euclidean Green’s functions. Commun. Math. Phys. 31, 83–112 (1973)
Sanders, K.: Thermal equilibrium states of a linear scalar quantum field in stationary spacetimes. Int. J. Mod. Phys. A 28, 1330010 (2013)
Sanders, K.: On the construction of Hartle–Hawking–Israel state across a static bifurcate Killing horizon. Lett. Math. Phys. 105(4), 575–640 (2015)
Schapira, P.: Wick rotation for \(D\)-modules. Math. Phys. Anal. Geom. 20, 21 (2017)
Sewell, G.L.: Relativity of temperature and the Hawking effect. Phys. Lett. A 79, 23–24 (1980)
Sahlmann, H., Verch, R.: Passivity and microlocal spectrum condition. Commun. Math. Phys. 214, 705–731 (2000)
Strohmaier, A., Verch, R., Wollenberg, M.: Microlocal analysis of quantum fields on curved space-times: analytic wave front sets and Reeh–Schlieder theorems. J. Math. Phys. 43, 5514–5530 (2002)
Wald, R.M.: On the Euclidean approach to quantum field theory in curved spacetime. Commun. Math. Phys. 70(3), 221–242 (1979)
Witten, E.: Invited article on entanglement properties of quantum field theory. Rev. Mod. Phys. 90, 045003 (2018)
Acknowledgements
The author is very grateful to Christian Gérard for inspiring discussions and helpful suggestions, and would also like to thank the anonymous referees for their help in improving the paper. Support from the Grant ANR-16-CE40-0012-01 is gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Wrochna, M. Wick rotation of the time variables for two-point functions on analytic backgrounds. Lett Math Phys 110, 585–609 (2020). https://doi.org/10.1007/s11005-019-01230-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-019-01230-7