2013/09/30 by William C. C. Lima
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Classical mechanics #Cosmology and Gravitation Theories #Hamiltonian (control theory) #Hilbert space #Mathematical physics #Mathematics #Operator (biology) #Physics #Quantization (signal processing) #Quantum #Quantum Electrodynamics and Casimir Effect #Quantum field theory #Quantum gravity #Quantum mechanics #Scalar (mathematics) #Scalar field #Scalar field theory #Scalar potential #Spacetime #gr-qc #hep-th #math-ph #math.MP
paper · pdf · doi:10.1103/physrevd.88.124005
published as Phys. Rev. D 88, 124005 (2013); Erratum 94, 129901(E) (2016) · 14 pages. Rectification of the statement at the end of the Sec. III B of the published version. All other results unchanged
openalex publication_date 2013/12/02 · arxiv created 2017/01/11 · arxiv updated 2017/01/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We discuss the quantization of an unstable field through the construction of a ``one-particle Hilbert space.'' The system considered here is a neutral scalar field evolving over a globally hyperbolic static spacetime and subject to a stationary external scalar potential. In order to prove our results we assume spacetimes without horizons and that the theory possess a ``mass gap.'' Our strategy consists in building a complex structure, which arises from a suitable positive bilinear form defined over the space of classical solutions of the field equation. Once the space of states of the quantum field has been set, it is possible to study the effect of the time translation symmetry on it. From the time translation operator we obtain an expression for the Hamiltonian operator associated with the unstable sector of the field. This last result coincides with findings from long ago showing that the unstable degrees of freedom of the field behave as nonrelativistic particles in a parabolic potential barrier.