Erratum to: Construction of KMS States in Perturbative QFT and Renormalized Hamiltonian Dynamics (Commun. Math. Phys., (2014) 332, (895-932), 10.1007/s00220-014-2141-7)

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Erscheinungsjahr:
2016
Medientyp:
Text
Schlagworte:
  • Quantum field theory
  • Space-time
  • Quantum fields
  • Gravitation
  • Black Holes (Astronomy)
  • Models
  • Quantum field theory
  • Space-time
  • Quantum fields
  • Gravitation
  • Black Holes (Astronomy)
  • Models
Beschreibung:
  • The formula for the ground state (ß = ∞) given in Theorem 3 of the paper is wrong. It is based on Theorem 2 and Proposition 3 which, however, are formulated and proved only for 0 <ß <8. In order to include the case ß = ∞ one may modify Theorem 2 in the following way. Theorem 2. Let ωß be a ß-KMS state (for 0 <ß <∞) or a ground state (for ß = ∞) on (Formula Presented.) with respect to αt . Then the following statements hold in the sense of formal power series in the interaction: • For (Formula Presented.) the function (Formula Presented.) can be extended to a continuous function on the closure of Tß n+2, which is analytic in the interior. • For 0 <ß <8 the linear functional (Formula Presented.) KMS state with respect to (Formula Presented.) • If for ß = ∞ the limit (Formula Presented.) exists uniformly on compact sets of (Formula Presented.) for all (Formula Presented.) then? (Formula Presented.) is a ground state with respect to (Formula Presented.). The proof for finite ß remains essentially the same. In the case of infinite ß one needs to have control on the behavior of the functions GA1,...,An as the arguments tend to infinity. In Proposition 3 one has to replace the formula by (Formula Presented.) (29) with (Formula Presented.) For finite ß this coincides with the original formula due to the KMS condition. For infinite ß one has to control the convergence of the integrals (Formula Presented.) If these integrals converge for all (Formula Presented.) uniformly in (Formula Presented.) on compact sets in (Formula Presented.), then (Formula Presented.) is a ground state. Proposition 4 was formulated and proven for finite ß. To extend it to the case of infinite ß requires additional work that goes beyond the purpose of this erratum. In Theorem 3, where the infinite volume limit is treated, one has to use the formula (29) from the revised Proposition 3. The results for the clustering of connected functions of the free massive field also yield the required convergence properties for the revised expression. We expect that the resulting state is Lorentz invariant and coincides with the vacuum state constructed in the seminal paper of Epstein and Glaser [1], but this remains to be shown. Acknowledgements. We are grateful to Pawel Duch from the Jagellonian University, Krakow, for communicating to us that there was an error in our formula for the ground state.
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