Abstract
We prove that the ground state for the Dirac equation on Minkowski space in static, smooth external potentials satisfies the Hadamard condition. We show that it follows from a condition on the support of the Fourier transform of the corresponding positive frequency solution. Using a Klein space formalism, we establish an analogous result in the Klein–Gordon case for a wide class of smooth potentials. Finally, we investigate overcritical potentials, i.e. which admit no ground states. It turns out, that numerous Hadamard states can be constructed by mimicking the construction of ground states, but this leads to a naturally distinguished one only under more restrictive assumptions on the potentials.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Arendt W., Batty C.J.K., Hieber M., Neubrander F.: Vector-Valued Laplace Transforms and Cauchy Problems. Springer, Berlin (2011)
Albeverio S., Gottschalk H.: Scattering theory for quantum fields with indefinite metric. Commun. Math. Phys. 216, 491–513 (2001)
Avron J., Herbst I., Simon B.: Schrödinger operators with magnetic fields. I: general interactions. Duke Math. J. 45, 847–883 (1978)
Araki H., Shiraishi M.: On quasifree states of the canonical commutation relations (I). Publ. Res. Inst. Math. Sci. 7(1), 105–120 (1971)
Bachelot A.: Superradiance and scattering of the charged Klein–Gordon field by a step-like electrostatic potential. J. Math. Pure Appl. 83(10), 1179–1239 (2004)
Bahns, D.: Schwinger functions in noncommutative quantum field theory. Ann. Henri Poincaré 11, 1273–1283 (2010), 0908.4537
Brunetti, R., Fredenhagen, K.: Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds. Commun. Math. Phys. 208, 623–661 (2000). ArXiv:math-ph/9903028
Bär, C., Fredenhagen, K. (eds.): Quantum Field Theory on Curved Spacetimes. Lecture Notes in Physics, vol. 786. Springer, Berlin (2009)
Binz E., Honegger R., Rieckers A.: Construction and uniqueness of the C*-Weyl algebra over a general pre-symplectic space. J. Math. Phys. 45, 2885–2907 (2004)
Bognar J.: Indefinite Inner Product Spaces. Ergebnisse Mathematik und Grenz Geb. Springer, Berlin (1974)
Bratteli O., Robinson D.W.: Operator Algebras and Quantum Statistical Mechanics. Equilibrium States. Statistical Mechanics, vol. 2. Springer, Berlin (1997)
Broadbridge P.: Existence theorems for Segal quantization via spectral theory in Krein space. Austral. Math. Soc. Ser. B 24, 439–460 (1983)
Baez J., Segal I., Zhuo Z.: Introduction to Algebraic and Constructive Quantum Field Theory. Princeton University Press, Princeton (1992)
Chernoff P.: Essential self-adjointness of powers of generators of hyperbolic equations. J. Funct. Anal. 12(4), 401–414 (1973)
Davies E.B.: The functional calculus. J. London Math. Soc. 2, 166–176 (1995)
Dereziński, J., Gérard, C.: Mathematics of Quantization and Quantum Fields (In preparation)
Dereziński J., Gérard C.: Energy quantization of linear dynamics. Banach Center Publ. 89, 75–104 (2010)
Dimock J.: Dirac quantum fields on a manifold. Trans. AMS 269, 133–147 (1982)
Dosch H.G., Müller V.F.: Renormalization of quantum electrodynamics in an arbitrarily strong time independent external field. Fortschr. Phys. 23(11–12), 661–689 (1975)
Dappiaggi C., Moretti V., Pinamonti N.: Distinguished quantum states in a class of cosmological spacetimes and their Hadamard property. J. Math. Phys. 50, 062304–062304-38 (2009)
Fradkin E.S., Gitman D.M., Shvartsman S.M.: Quantum Electrodynamics With Unstable Vacuum. Springer, Berlin (1991)
Fulling S.A.: Aspects of Quantum Field Theory in Curved Spacetime. Cambridge University Press, Cambridge (1989)
Feshbach H., Villars F.: Elementary relativistic wave mechanics of spin 0 and spin 1/2 particles. Rev. Mod. Phys. 30, 24–45 (1958)
Gérard, C.,: Scattering theory for Klein–Gordon equations with non-positive energy. Ann. Henri Poincaré. 13. ISSN:1424-0637 (2011)
Greiner, W., Müller, B., Rafelski, J.: Quantum Electrodynamics of Strong Fields. Lecture Notes in Physics, vol. 440. Springer, Berlin (1985)
Hack, T.-P.: On the backreaction of scalar and spinor quantum fields in curved spacetimes—from the basic foundations to cosmological applications. PhD thesis, DESY-THESIS-2010-042 (2010)
Hollands, S.: The Hadamard condition for Dirac fields and adiabatic states on Robertson–Walker spacetimes. Commun. Math. Phys. 216, 635–661 (2001). ArXiv:gr-qc/9906076
Hörmander L.: The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis. Springer, Berlin (1983)
Hollands, S., Wald, R. M.: Conservation of the stress tensor in perturbative interacting quantum field theory in curved spacetimes. Rev. Math. Phys. 17, 227–311 (2005). ArXiv:gr-qc/0404074
Jin, W. M.: Quantization of Dirac fields in static spacetime. Classical Quant. Grav. 17, 2949–2964 (2000). ArXiv:gr-qc/0009010
Jonas P.: On a class of selfadjoint operators in Krein space and their compact perturbations. Integr. Equat. Operat. Theor. 11, 351–384 (1988). doi:10.1007/BF01202078
Kluger Y., Eisenberg J.M., Svetitsky B., Cooper F., Mottola E.: Pair production in a strong electric field. Phys. Rev. Lett. 67, 2427–2430 (1991)
Kratzert K.: Singularity structure of the two point function of the free Dirac field on a globally hyperbolic spacetime. Ann. Phys. 9, 475–498 (2000)
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, 49–136 (1991)
Langer, H.: Spectral functions of definitizable operators in Krein spaces. In: Butkovic, D., Kraljevic, H., Kurepa, S. (eds.) Functional Analysis. Lecture Notes in Mathematics, vol. 948, pp. 1–46. Springer, Berlin (1982). doi:10.1007/BFb0069840
Langer H., Najman B., Tretter C.: Spectral theory of the Klein–Gordon equation in Pontryagin spaces. Commun. Math. Phys. 267, 159–180 (2006)
Langer H., Najman B., Tretter C.: Spectral theory of the Klein–Gordon equation in Krein spaces. Proc. Edinburgh Math. Soc. 51(03), 711–750 (2008)
Manogue C.A.: The Klein paradox and superradiance. Ann. Phys. 181, 261–283 (1988)
Marecki, P.: Quantum electrodynamics on background external fields. PhD thesis DESY-THESIS-2004-002 (2003)
Moretti, V.: Comments on the stress-energy tensor operator in curved spacetime. Commun. Math. Phys. 232, 189–221 (2003). ArXiv:gr-qc/0109048
Mohr P.J., Plunien G., Soff G.: QED corrections in heavy atoms. Phys. Rep. 293, 227–369 (1998)
Maz’ya, V., Shubin, M.: Discreteness of spectrum and positivity criteria for Schrödinger operators. Ann. Math. 162, 919–942 (2005). ArXiv:math/0305278
Mühlhoff, R.: Cauchy problem and Green’s functions for first order differential operators and algebraic quantization. J. Math. Phys. 52(2), 022303 (2011)
Radzikowski M.J.: Micro-local approach to the Hadamard condition in quantum field theory on curved space–time. Commun. Math. Phys. 179, 529–553 (1996)
Ringwald, A.: Fundamental physics at an X-ray free electron laser. Phys. Lett. B 510, 107 (2001). ArXiv:hep-ph/0112254
Ruijsenaars, S.~N.~M.: Charged particles in external fields II. The quantized Dirac and Klein–Gordon theories. Commun. Math. Phys. 52, 267–294 (1977). doi:10.1007/BF01609487
Ruffini, R., Vereshchagin, G., Xue, S.-S.: Electron–positron pairs in physics and astrophysics: from heavy nuclei to black holes. Phys. Rep. 487, 1–140 (2010)
Sanders K.: The locally covariant Dirac field. Rev. Math. Phys. 22, 381–430 (2010)
Scharf G.: Finite Quantum Electrodynamics: The Causal Approach. Texts and Monographs in Physics, 2nd edn. Springer, Berlin (1995)
Shigekawa I.: Spectral properties of Schrödinger operators with magnetic fields for a spin 1/2 particle. J. Funct. Anal. 101(2), 255–285 (1991)
Schroer B., Swieca J.A.: Indefinite metric and stationary external interactions of quantized fields. Phys. Rev. 2, 2938–2943 (1970)
Sahlmann, H., Verch, R.: Passivity and microlocal spectrum condition. Commun. Math. Phys. 214, 705–731 (2000). ArXiv:math-ph/0002021
Sahlmann, H., Verch, v: Microlocal spectrum condition and Hadamard form for vector-valued quantum fields in curved spacetime. Rev. Math. Phys. 13, 1203–1246 (2001). ArXiv:math-ph/0008029
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). ArXiv:math-ph/0202003
Thaller B.: The Dirac Equation. Texts and Monographs in Physics. Springer, Berlin (1992)
Verch R.: Continuity of symplectically adjoint maps and the algebraic structure of Hadamard vacuum representations for quantum fields on curved spacetime. Rev. Math. Phys. 9, 635–674 (1997)
Veselić K.: A spectral theory for the Klein–Gordon equation with an external electrostatic potential. Nucl. Phys. 147, 215–224 (1970)
Wald R.M.: The back reaction effect in particle creation in curved spacetime. Commun. Math. Phys. 54, 1–19 (1977)
Wald R.M.: Quantum Field Theory In Curved Spacetime and Black Hole Thermodynamics. University of Chicago Press, Chicago (1994)
Zahn J.: Divergences in quantum field theory on the noncommutative two-dimensional Minkowski space with Grosse–Wulkenhaar potential. Ann. Henri Poincaré 12, 777–804 (2011)
Acknowledgements
It is a pleasure to thank D. Bahns for many useful remarks and careful reading of the manuscript, as well as J. Zahn for helpful discussions and valuable comments. The author is grateful to J. Dereziński and C. Gérard for making the manuscript of the book [16] available before publication. Financial support of the RTG 1493 is gratefully acknowledged.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Klaus Fredenhagen.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Wrochna, M. Quantum Field Theory in Static External Potentials and Hadamard States. Ann. Henri Poincaré 13, 1841–1871 (2012). https://doi.org/10.1007/s00023-012-0173-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-012-0173-0