A Canonical Complex Structure and the Bosonic Signature Operator for Scalar Fields in Globally Hyperbolic Spacetimes
arXiv:2112.11036 · doi:10.1007/s00023-022-01236-3
Abstract
The bosonic signature operator is defined for Klein-Gordon fields and massless scalar fields on globally hyperbolic Lorentzian manifolds of infinite lifetime. The construction is based on an analysis of families of solutions of the Klein-Gordon equation with a varying mass parameter. It makes use of the so-called bosonic mass oscillation property which states that integrating over the mass parameter generates decay of the field at infinity. We derive a canonical decomposition of the solution space of the Klein-Gordon equation into two subspaces, independent of observers or the choice of coordinates. This decomposition endows the solution space with a canonical complex structure. It also gives rise to a distinguished quasi-free state. Taking a suitable limit where the mass tends to zero, we obtain corresponding results for massless fields. Our constructions and results are illustrated in the examples of Minkowski space and ultrastatic spacetimes.
arXiv comment20 pages, LaTeX, many smaller improvements (published version)
References in corpus (25)
- On smooth Cauchy hypersurfaces and Geroch's splitting theorem
- Rigorous steps towards holography in asymptotically flat spacetimes
- Feynman Propagator for a Free Scalar Field on a Causal Set
- Quantum Field Theory on Curved Backgrounds -- A Primer
- Scalar Field Theory on a Causal Set in Histories Form
- On a Recent Construction of "Vacuum-like" Quantum Field States in Curved Spacetime
- Construction of Hadamard states by pseudo-differential calculus
- "Vacuum-like" Hadamard states for quantum fields on curved spacetimes
- A Non-Perturbative Construction of the Fermionic Projector on Globally Hyperbolic Manifolds I - Space-Times of Finite Lifetime
- Approximate KMS states for scalar and spinor fields in Friedmann-Robertson-Walker spacetimes
- Construction of Hadamard states by characteristic Cauchy problem
- The Fermionic Projector in a Time-Dependent External Potential: Mass Oscillation Property and Hadamard States
- A Non-Perturbative Construction of the Fermionic Projector on Globally Hyperbolic Manifolds II -- Space-Times of Infinite Lifetime
- Radiative observables for linearized gravity on asymptotically flat spacetimes and their boundary induced states
- An Evolution Equation Approach to the Klein-Gordon Operator on Curved Spacetime
- The Fermionic Signature Operator and Quantum States in Rindler Space-Time
- The Fermionic Signature Operator in De Sitter Spacetime
- Complex Structures for Klein-Gordon Theory on Globally Hyperbolic Spacetimes
- Self-Adjointness in Klein-Gordon Theory on Globally Hyperbolic Spacetimes
- Lorentzian Spectral Geometry for Globally Hyperbolic Surfaces
- A new class of Fermionic Projectors: Møller operators and mass oscillation properties
- The Fermionic Signature Operator in the Exterior Schwarzschild Geometry
- The Fermionic Signature Operator and Space-Time Symmetries
- The Fermionic Signature Operator and Hadamard States in the Presence of a Plane Electromagnetic Wave
- The Hamiltonians of Linear Quantum Fields: I. Existence Theory