Annales Henri Poincaré

, Volume 16, Issue 8, pp 1837–1868 | Cite as

Gupta–Bleuler Quantization of the Maxwell Field in Globally Hyperbolic Space-Times

  • Felix Finster
  • Alexander StrohmaierEmail author


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.


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Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität RegensburgRegensburgGermany
  2. 2.Department of Mathematical SciencesLoughborough UniversityLoughboroughUK

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