Abstract
We give a complete framework for the Gupta–Bleuler quantization of the free electromagnetic field on globally hyperbolic space-times. We describe one-particle structures that give rise to states satisfying the microlocal spectrum condition. The field algebras in the so-called Gupta–Bleuler representations satisfy the time-slice axiom, and the corresponding vacuum states satisfy the microlocal spectrum condition. We also give an explicit construction of ground states on ultrastatic space-times. Unlike previous constructions, our method does not require a spectral gap or the absence of zero modes. The only requirement, the absence of zero-resonance states, is shown to be stable under compact perturbations of topology and metric. Usual deformation arguments based on the time-slice axiom then lead to a construction of Gupta–Bleuler representations on a large class of globally hyperbolic space-times. As usual, the field algebra is represented on an indefinite inner product space, in which the physical states form a positive semi-definite subspace. Gauge transformations are incorporated in such a way that the field can be coupled perturbatively to a Dirac field. Our approach does not require any topological restrictions on the underlying space-time.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Change history
28 December 2017
In Section 5.1 in [1] it is incorrectly claimed that condition (A) is equivalent to the vanishing of the operator B in the expansion.
References
Ashtekar A., Sen A.: On the role of space-time topology in quantum phenomena: superselection of charge and emergence of nontrivial vacua. J. Math. Phys. 21(3), 526–533 (1980)
Atiyah M.F., Patodi V.K., Singer I.M.: Spectral asymmetry and Riemannian geometry. I. Math. Proc. Camb. Philos. Soc. 77, 43–69 (1975)
Bär, C., Ginoux, N., Pfäffle, F.: Wave equations on lorentzian manifolds and quantization. In: ESI Lectures in Mathematics and Physics. European Mathematical Society (EMS), Zürich (2007)
Barnich, G., Brandt, F., Henneaux, M.: Local BRST cohomology in gauge theories. Phys. Rep. 338, 439–569 (2000). arXiv:hep-th/0002245
Benini, M., Dappiaggi, C., Hack, T.-P., Schenkel, A.: A C *-algebra for quantized principal U(1)-connections on globally hyperbolic lorentzian manifolds (2013). arXiv:1307.3052 [math-ph]
Benini, M., Dappiaggi, C., Schenkel, A.: Quantized abelian principal connections on lorentzian manifolds (2013). arXiv:1303.2515 [math-ph]
Bernal, A.N., Sánchez, M.: On smooth Cauchy hypersurfaces and Geroch’s splitting theorem. Commun. Math. Phys. 243(3), 461–470 (2003). arXiv:gr-qc/0306108
Bleuler K.: Eine neue Methode zur Behandlung der longitudinalen und skalaren Photonen. Helv. Phys. Acta 23, 567–586 (1950)
Brunetti, R., Fredenhagen, K.: Interacting quantum fields in curved space: renormalizability of \({\phi^4}\). In: Operator Algebras and Quantum Field Theory (Rome, 1996), pp. 546–563. Int. Press, Cambridge (1997). arXiv:gr-qc/9701048
Brunetti, R., Fredenhagen, K., Köhler, M.: The microlocal spectrum condition and Wick polynomials of free fields on curved spacetimes. Commun. Math. Phys. 180(3), 633–652 (1996). arXiv:gr-qc/9510056
Carron G.: Théorèmes de l’indice sur les variétés non-compactes. J. Reine Angew. Math. 541, 81–115 (2001)
Carron G.: Le saut en zéro de la fonction de décalage spectral. J. Funct. Anal. 212(1), 222–260 (2004)
Dappiaggi, C., Lang, B.: Quantization of Maxwell’s equations on curved backgrounds and general local covariance. Lett. Math. Phys. 101(3), 265–287 (2012). arXiv:1104.1374v2 [gr-qc]
Dappiaggi, C., Siemssen, D.: Hadamard states for the vector potential on asymptotically flat spacetimes. arXiv:1106.5575 [gr-qc] (2011)
Dencker N.: On the propagation of polarization sets for systems of real principal type. J. Funct. Anal. 46(3), 351–372 (1982)
Dimock J.: Algebras of local observables on a manifold. Commun. Math. Phys. 77(3), 219–228 (1980)
Dimock J.: Quantized electromagnetic field on a manifold. Rev. Math. Phys. 4(2), 223–233 (1992)
Dütsch, M., Fredenhagen, K.: A local (perturbative) construction of observables in gauge theories: the example of QED. Commun. Math. Phys. 203(1), 71–105 (1999). arXiv:hep-th/9807078
Fewster, C.J., Pfenning, M.J.: A quantum weak energy inequality for spin-one fields in curved space-time. J. Math. Phys. 44(10), 4480–4513 (2003). arXiv:gr-qc/0303106
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(2), 243–272 (1981)
Furlani E.P.: Quantization of the electromagnetic field on static space-times. J. Math. Phys. 36(3), 1063–1079 (1995)
Greiner, W., Reinhardt, J.: Field Quantization. Springer, Berlin (1996)
Gupta S.N.: Theory of longitudinal photons in quantum electrodynamics. Proc. Phys. Soc. Sect. A. 63, 681–691 (1950)
Henneaux M., Teitelboim C.: Quantization of Gauge Systems. Princeton University Press, Princeton (1992)
Hollands, S.: Renormalized quantum Yang–Mills fields in curved spacetime. Rev. Math. Phys. 20(9), 1033–1172 (2008). arXiv:0705.3340 [gr-qc]
Hollands, S., Wald, R.M.: Local Wick polynomials and time ordered products of quantum fields in curved spacetime. Commun. Math. Phys. 223(2), 289–326 (2001). arXiv:gr-qc/0103074
Hollands, S., Wald, R.M.: Existence of local covariant time ordered products of quantum field in curved spacetime. Commun. Math. Phys. 231(2), 309–345 (2002). arXiv:gr-qc/0111108
Hörmander, L.: The analysis of linear partial differential operators. III. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 274. Springer, Berlin (1985)
John, F.: Partial differential equations. In: Applied Mathematical Sciences, 4th edn. vol. 1. Springer, New York, (1991)
Lautrup B.: Canonical quantum electrodynamics in covariant gauges. Mat. Fys. Medd. Dan. Vid. Selsk. 35, 11 (1966)
Melrose, R.B.: The Atiyah–Patodi–Singer index theorem. In: Research Notes in Mathematics, vol. 4. A K Peters Ltd., Wellesley (1993)
Melrose R.B.: Geometric Scattering Theory. Stanford Lectures. Cambridge University Press, Cambridge (1995)
Müller, O.: Asymptotic flexibility of globally hyperbolic manifolds. C. R. Math. Acad. Sci. Paris 350(7–8), 421–423 (2012). arXiv:1110.1037 [math.DG]
Müller W.: Eta invariants and manifolds with boundary. J. Differ. Geom. 40(2), 311–377 (1994)
Müller, W., Strohmaier, A.: Scattering at low energies on manifolds with cylindrical ends and stable systoles. Geom. Funct. Anal. 20(3), 741–778 (2010). arXiv:0907.3517 [math.AP]
Peskin, M.E., Tonomura, A.: The Aharonov–Bohm effect. In: Lecture Notes in Physics, vol. 340. Springer, Berlin (1989)
Radzikowski M.J.: Micro-local approach to the Hadamard condition in quantum field theory on curved space-time. Commun. Math. Phys. 179(3), 529–553 (1996)
Sanders, K.: Equivalence of the (generalised) Hadamard and microlocal spectrum condition for (generalised) free fields in curved spacetime. Commun. Math. Phys. 295(2), 485–501 (2010). arXiv:0903.1021 [math-ph]
Sanders, K., Dappiaggi, C., Hack, T.-H.: Electromagnetism, local covariance, the Aharonov–Bohm effect and Gauss’ law (2012). arXiv:1211.6420 [math-ph]
Streater R.F.: Spontaneous breakdown of symmetry in axiomatic theory. Proc. R. Soc. Ser. A 287, 510–518 (1965)
Strocchi F.: Gauge problem in quantum field theory. Phys. Rev. 162(2), 1429–1438 (1967)
Strocchi F., Wightman A.S.: Proof of the charge superselection rule in local relativistic quantum field theory. J. Math. Phys. 15, 2198–2224 (1974)
Strocchi, F., Wightman, A.S. Erratum: “proof of the charge superselection rule in local relativistic quantum field theory” (J. Math. Phys. 15, 2198–2224 (1974)). J. Math. Phys. 17(10), 1930–1931 (1976)
Strohmaier, A., Verch, R., and Wollenberg, M.: Microlocal analysis of quantum fields on curved space-times: analytic wave front sets and Reeh–Schlieder theorems. J. Math. Phys. 43(11), 5514–5530 (2002). arXiv:math-ph/0202003
Wang X.P.: Asymptotic expansion in time of the Schrödinger group on conical manifolds. Ann. Inst. Fourier (Grenoble) 56(6), 1903–1945 (2006)
Weinberg, S.: The Quantum Theory of Fields, vol. I. Cambridge University Press, Cambridge (1996). Foundations, corrected reprint of the 1995 original
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Karl-Henning Rehren.
F.F. was supported by the Scheme 4 Grant #41122 of the London Mathematical Society.
A correction to this article is available online at https://doi.org/10.1007/s00023-017-0632-8.
Rights and permissions
About this article
Cite this article
Finster, F., Strohmaier, A. Gupta–Bleuler Quantization of the Maxwell Field in Globally Hyperbolic Space-Times. Ann. Henri Poincaré 16, 1837–1868 (2015). https://doi.org/10.1007/s00023-014-0363-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-014-0363-z