Quantum fields in curved spacetime. (English) Zbl 1357.81144

Summary: We review the theory of quantum fields propagating in an arbitrary, classical, globally hyperbolic spacetime. Our review emphasizes the conceptual issues arising in the formulation of the theory and presents known results in a mathematically precise way. Particular attention is paid to the distributional nature of quantum fields, to their local and covariant character, and to microlocal spectrum conditions satisfied by physically reasonable states. We review the Unruh and Hawking effects for free fields, as well as the behavior of free fields in deSitter spacetime and FLRW spacetimes with an exponential phase of expansion. We review how nonlinear observables of a free field, such as the stress-energy tensor, are defined, as well as time-ordered-products. The “renormalization ambiguities” involved in the definition of time-ordered products are fully characterized. Interacting fields are then perturbatively constructed. Our main focus is on the theory of a scalar field, but a brief discussion of gauge fields is included. We conclude with a brief discussion of a possible approach towards a nonperturbative formulation of quantum field theory in curved spacetime and some remarks on the formulation of quantum gravity.


81T20 Quantum field theory on curved space or space-time backgrounds
83C47 Methods of quantum field theory in general relativity and gravitational theory
Full Text: DOI arXiv


[1] Allen, B., Vacuum states in de Sitter space, Phys. Rev. D, 32, 3136, (1985)
[2] Akhizer, N. I., The classical moment problem and some related questions in analysis, (1965), Oliver and Boyd
[3] Araki, H.; Yamagami, S., On the quasi-equivalence of quasifree states of the canonical commutation relations, Publ. RIMS, Kyoto U., 18, (1982) · Zbl 0505.46052
[4] Barnich, G.; Brandt, F.; Henneaux, M., Local BRST cohomology in gauge theories, Phys. Rep., 338, 439, (2000) · Zbl 1097.81571
[5] Bär, C.; Ginoux, N.; Pfäffle, F., Wave equations on Lorenzian manifolds and quantization, Eur. Math. Soc., 194, (2007) · Zbl 1118.58016
[6] Bardeen, J. M.; Carter, B.; Hawking, S. W., The four laws of black hole mechanics, Comm. Math. Phys., 31, 161, (1973) · Zbl 1125.83309
[7] Bernal, Antonio N.; Sánchez, Miguel, On smooth Cauchy hypersurfaces and geroch’s splitting theorem, Comm. Math. Phys., 243, 461, (2003) · Zbl 1085.53060
[8] De Bievre, S.; Merkli, M., The Unruh effect revisited, Classical Quantum Gravity, 23, 6525, (2006) · Zbl 1123.83012
[9] Bogoliubov, N. N.; Shirkov, D. V., Introduction to the theory of quantized fields, (1980), Wiley · Zbl 0925.81002
[10] Boulware, D. G., Quantum field theory in Schwarzschild and Rindler spaces, Phys. Rev. D, 11, 1404, (1975)
[11] Bratteli, O.; Robinson, D. W., Operator algebras and quantum statistical mechanics 2: equilibrium states. models in quantum statistical mechanics, (2002), Springer
[12] Bros, J.; Moschella, U., Two point functions and quantum fields in de Sitter universe, Rev. Math. Phys., 8, 327, (1996) · Zbl 0858.53054
[13] Bros, J.; Moschella, U.; Gazeau, J. P., Quantum field theory in the de Sitter universe, Phys. Rev. Lett., 73, 1746, (1994) · Zbl 1020.81736
[14] Brunetti, R.; Duetsch, M.; Fredenhagen, K., Perturbative algebraic quantum field theory and the renormalization groups, Adv. Theor. Math. Phys., 13, 1541, (2009) · Zbl 1201.81090
[15] Brunetti, R.; Fredenhagen, K.; Kohler, M., The microlocal spectrum condition and Wick polynomials of free fields on curved space-times, Comm. Math. Phys., 180, 633, (1996) · Zbl 0923.58052
[16] Brunetti, R.; Fredenhagen, K., Microlocal analysis and interacting quantum field theories: renormalization on physical backgrounds, Comm. Math. Phys., 208, 623, (2000) · Zbl 1040.81067
[17] R. Brunetti, K. Fredenhagen, K. Rejzner, Quantum gravity from the point of view of locally covariant quantum field theory, arXiv:1306.1058 [math-ph]. · Zbl 1346.83001
[18] Brunetti, R.; Fredenhagen, K.; Verch, R., The generally covariant locality principle: A new paradigm for local quantum field theory, Comm. Math. Phys., 237, 31, (2003) · Zbl 1047.81052
[19] DeWitt, B. S.; Brehme, R. W., Radiation damping in a gravitational field, Ann. Physics, 9, 220, (1960) · Zbl 0092.45003
[20] Dafermos, M.; Rodnianski, I., The red-shift effect and radiation decay on black hole spacetimes, Comm. Pure Appl. Math., 62, 859, (2009) · Zbl 1169.83008
[21] 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
[22] Dappiaggi, C.; Hack, T.-P.; Pinamonti, N., The extended algebra of observables for Dirac fields and the trace anomaly of their stress-energy tensor, Rev. Math. Phys., 21, 1241, (2009) · Zbl 1231.81062
[23] Dimock, J., Algebras of local observables on a manifold, Comm. Math. Phys., 77, 219, (1980) · Zbl 0455.58030
[24] Duetsch, M.; Fredenhagen, K., Algebraic quantum field theory, perturbation theory, and the loop expansion, Comm. Math. Phys., 219, 5, (2001) · Zbl 1019.81041
[25] Duetsch, M.; Fredenhagen, K., A local (perturbative) construction of observables in gauge theories: the example of QED, Comm. Math. Phys., 203, 71, (1999) · Zbl 0938.81028
[26] Duetsch, M.; Fredenhagen, K., Causal perturbation theory in terms of retarded products, and a proof of the action Ward identity, Rev. Math. Phys., 16, 1291, (2004) · Zbl 1084.81054
[27] Epstein, H.; Glaser, V., The role of locality in perturbation theory, Annales Poincare Phys. Theor. A, 19, 211, (1973) · Zbl 1216.81075
[28] Fewster, C. J.; Pfenning, M. J., A quantum weak energy inequality for spin one fields in curved space-time, J. Math. Phys., 44, 4480, (2003) · Zbl 1062.81115
[29] Fewster, C. J.; Hunt, D. S., Quantization of linearized gravity in cosmological vacuum spacetimes, Rev. Math. Phys., 25, 1330003, (2013) · Zbl 1266.83024
[30] Fewster, C. J., A general worldline quantum inequality, Classical Quantum Gravity, 17, 1897, (2000) · Zbl 1079.81555
[31] F. Finster, A. Strohmaier, Gupta-Bleuler quantization of the Maxwell field in globally hyperbolic space-times, arXiv:1307.1632 [math-ph]. · Zbl 1321.81063
[32] Ford, L. H.; Roman, T. A., Restrictions on negative energy density in flat space-time, Phys. Rev. D, 55, 2082, (1997)
[33] Fredenhagen, K.; Haag, R., On the derivation of Hawking radiation associated with the formation of a black hole, Comm. Math. Phys., 127, 273, (1990) · Zbl 0692.53040
[34] K. Fredenhagen, F. Lindner, Construction of KMS states in perturbative QFT and renormalized Hamiltonian dynamics, arXiv:1306.6519 [math-ph]. · Zbl 1305.82012
[35] Fredenhagen, K.; Haag, R., Generally covariant quantum field theory and scaling limits, Comm. Math. Phys., 108, 91, (1987) · Zbl 0626.46063
[36] Fredenhagen, K.; Rejzner, K., Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory, Comm. Math. Phys., 317, 697, (2013) · Zbl 1263.81245
[37] Fulling, S. A.; Narcowich, F. J.; Wald, R. M., Singularity structure of the two point function in quantum field theory in curved space-time. II, Ann. Physics, 136, 243, (1981) · Zbl 0495.35054
[38] C. Gerard, M. Wrochna, Construction of Hadamard states by pseudo-differential calculus, arXiv:1209.2604 [math-ph]. · Zbl 1298.81214
[39] Glimm, J.; Jaffe, A., Quantum physics: A functional integral point of view, (1987), Springer Berlin
[40] Haag, R., (Local Quantum Physics: Fields, Particles, Algebras, Texts and Monographs in Physics, (1992), Springer Berlin, Germany), 356
[41] Hawking, S. W.; Hawking, S. W., Particle creation by black holes, Comm. Math. Phys., Comm. Math. Phys., 46, 206, (1976), (erratum) · Zbl 1378.83040
[42] Higuchi, A.; Marolf, D.; Morrison, I. A., On the equivalence between Euclidean and in-in formalisms in de Sitter QFT, Phys. Rev. D, 83, 084029, (2011)
[43] Hollands, S., Massless interacting quantum fields in desitter spacetime, Ann. Henri Poincaré, 13, 1039, (2012) · Zbl 1332.81156
[44] Hollands, S., Correlators, Feynman diagrams, and quantum no-hair in desitter spacetime, Comm. Math. Phys., 319, 1, (2013) · Zbl 1278.81136
[45] Hollands, S., Renormalized quantum Yang-Mills fields in curved spacetime, Rev. Math. Phys., 20, 1033, (2008) · Zbl 1161.81022
[46] Hollands, S., The operator product expansion for perturbative quantum field theory in curved spacetime, Comm. Math. Phys., 273, 1, (2007) · Zbl 1142.81016
[47] J. Holland, S. Hollands, Recursive construction of operator product expansion coefficients, arXiv:1401.3144 [math-ph].
[48] Hollands, S.; Kopper, C., The operator product expansion converges in perturbative field theory, Comm. Math. Phys., 313, 257, (2012) · Zbl 1332.81119
[49] Hollands, S.; Ruan, W., The state space of perturbative quantum field theory in curved space-times, Ann. Henri Poincaré, 3, 635, (2002) · Zbl 1158.81348
[50] Hollands, S.; Wald, R. M., Local Wick polynomials and time ordered products of quantum fields in curved space-time, Comm. Math. Phys., 223, 289, (2001) · Zbl 0989.81081
[51] Hollands, S.; Wald, R. M., Existence of local covariant time ordered products of quantum fields in curved space-time, Comm. Math. Phys., 231, 309, (2002) · Zbl 1015.81043
[52] Hollands, S.; Wald, R. M., On the renormalization group in curved space-time, Comm. Math. Phys., 237, 123, (2003) · Zbl 1059.81138
[53] Hollands, S.; Wald, R. M., Conservation of the stress tensor in interacting quantum field theory in curved spacetimes, Rev. Math. Phys., 17, 227, (2005) · Zbl 1078.81062
[54] Hollands, S.; Wald, R. M., Axiomatic quantum field theory in curved spacetime, Comm. Math. Phys., 293, 85, (2010) · Zbl 1193.81076
[55] Hormander, L., The analysis of linear partial differential operators I: distribution theory and Fourier analysis, (1990), Springer-Verlag · Zbl 0712.35001
[56] Iyer, V.; Wald, R. M., Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D, 50, 846, (1994)
[57] Junker, W.; Schrohe, E., Adiabatic vacuum states on general space-time manifolds: definition, construction, and physical properties, Annales Poincare Phys. Theor., 3, 1113, (2002) · Zbl 1038.81052
[58] Kay, B. S.; Wald, R. M., Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on space-times with a bifurcate Killing horizon, Phys. Rep., 207, 49, (1991) · Zbl 0861.53074
[59] Kay, B. S., The Casimir effect without MAGIC, Phys. Rev. D, 20, 3052, (1979)
[60] Keller, G.; Kopper, C., Perturbative renormalization of composite operators via flow equations. 2. short distance expansion, Comm. Math. Phys., 153, 245, (1993) · Zbl 0793.47061
[61] Kodama, H.; Sasaki, M., Cosmological perturbation theory, Progr. Theoret. Phys. Suppl., 78, 1, (1984)
[62] Kopper, C.; Muller, V. F., Renormalization proof for massive \(\phi_4^4\) theory on Riemannian manifolds, Comm. Math. Phys., 275, 331, (2007) · Zbl 1139.81049
[63] Marolf, D.; Morrison, I. A., The IR stability of de Sitter QFT: results at all orders, Phys. Rev. D, 84, 044040, (2011)
[64] Mukhanov, V. F.; Feldman, H. A.; Brandenberger, R. H., Theory of cosmological perturbations. part 1. classical perturbations. part 2. quantum theory of perturbations. part 3. extensions, Phys. Rep., 215, 203, (1992)
[65] Radzikowski, M. J., Micro-local approach to the Hadamard condition in quantum field theory on curved space-time, Comm. Math. Phys., 179, 529, (1996) · Zbl 0858.53055
[66] Radzikowski, M. J., A local to global singularity theorem for quantum field theory on curved space-time, Comm. Math. Phys., 180, 1, (1996) · Zbl 0874.58079
[67] Reed, M.; Simon, B., Fourier analysis and self-adjointness, (1975), Academic Press
[68] Sahlmann, H.; Verch, R., Passivity and microlocal spectrum condition, Comm. Math. Phys., 214, 705, (2000) · Zbl 1010.81046
[69] Sanders, K., Equivalence of the (generalised) Hadamard and microlocal spectrum condition for (generalised) free fields in curved spacetime, Comm. Math. Phys., 295, 485, (2010) · Zbl 1192.53072
[70] K. Sanders, On the construction of Hartle-Hawking-Israel states across a static bifurcate Killing horizon, arXiv:1310.5537 [gr-qc]. · Zbl 1325.81129
[71] K. Sanders, C. Dappiaggi, T.-P. Hack, Electromagnetism, local covariance, the Aharonov-Bohm effect and Gauss’ law, arXiv:1211.6420 [math-ph]. · Zbl 1293.81034
[72] Schomblond, C.; Spindel, P., Unicity conditions of the scalar field propagator delta(1) (x, y) in de Sitter universe, Annales Poincare Phys. Theor., 25, 67, (1976)
[73] Streater, R. F.; Wightman, A. S., PCT, spin and statistics, and all that, 207, (2000), Princeton Univ. Pr Princeton, USA · Zbl 1026.81027
[74] Unruh, W. G., Notes on black hole evaporation, Phys. Rev. D, 14, 870, (1976)
[75] Unruh, W. G.; Wald, R. M., What happens when an accelerating observer detects a Rindler particle, Phys. Rev. D, 29, 1047, (1984)
[76] van Daele, A.; Verbeure, A., Unitary equivalence of Fock representations on the Weyl algebra, Comm. Math. Phys., 20, 268, (1971) · Zbl 0208.38301
[77] Verch, R., Local definiteness, primarity and quasiequivalence of quasifree Hadamard quantum states in curved space-time, Comm. Math. Phys., 160, 507, (1994) · Zbl 0790.53077
[78] Wald, R. M., On particle creation by black holes, Comm. Math. Phys., 45, 9, (1975)
[79] Wald, R. M., Quantum field theory in curved spacetime and black hole thermodynamics, (1994), University of Chicago Press Chicago · Zbl 0842.53052
[80] Wald, R. M., The thermodynamics of black holes, Liv. Rev. Rel., 4, 6, (2001), arXiv:9912119 [gr-qc] · Zbl 1060.83041
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.