Hadamard States for the Klein–Gordon Equation on Lorentzian Manifolds of Bounded Geometry

Abstract

We consider the Klein-Gordon equation on a class of Lorentzian manifolds with Cauchy surface of bounded geometry, which is shown to include examples such as exterior Kerr, Kerr-de Sitter spacetime and the maximal globally hyperbolic extension of the Kerr outer region. In this setup, we give an approximate diagonalization and a microlocal decomposition of the Cauchy evolution using a time-dependent version of the pseudodifferential calculus on Riemannian manifolds of bounded geometry. We apply this result to construct all pure regular Hadamard states (and associated Feynman inverses), where regular refers to the state's two-point function having Cauchy data given by pseudodifferential operators. This allows us to conclude that there is a oneparameter family of elliptic pseudodifferential operators that encodes both the choice of (pure, regular) Hadamard state and the underlying spacetime metric.2010 Mathematics Subject Classification. 81T20, 35S05, 35L05, 58J40, 53C50.The detailed results are stated in Thm. 7.8 and 7.10, see also Prop. 7.6 for the arguments that allow to get two-point functions for the original Klein-Gordon equation on the full spacetime (M, g) rather than for the reduced equation (1.3) on I × Σ.Since one can get many regular states out of a given one by applying suitable Bogoliubov transformations as in [GW1], Thm. 1.2 yields in fact a large class of Hadamard states.1.3. From quantum fields to spacetime geometry. In our approach, microlocal splittings are obtained by settingis defined by formula (1.6) with b ± (t) constructed for t ∈ I as approximate solutions (i.e. modulo smoothing terms) of the operatorial equation (1.7)∂ t + ib ± + r • ∂ t − ib ± = ∂ 2 t + r∂ t + a, and satisfying some additional conditions, see Sect. 6 (in particular Thm. 6.1) for details. We note that the approximate factorization (1.7) was already used by Junker in his construction of Hadamard states [Ju1, Ju2]. h 1,sϕ ∈ S 0 KdS , resp. ∈ S −1,0 K

Hadamard Property of the in and out States for Klein–Gordon Fields on Asymptotically Static Spacetimes

“…The C ∞ (b) cases are proved in [GOW,Thm. 5.12], by checking that the general framework of [ALNV] applies to these two situations.…”

“…We state below a particular case of Seeley's theorem for real powers, partly proved in [GOW,Sect. 5] and based on a general result from [ALNV].…”

“…A redefinition of b(t) involving a cutoff in low frequencies as in [GW2,GOW] gives then control of the norm sufficient to obtain in addition…”

“…3.5 for the manifold Σ × R, because multi-commutators ad γ l A appear in (A.1). However we can still apply the abstract framework in [ALNV] to this situation, see [GOW,Subsect. 5.3.4] for a concise summary.…”

“…Among several equivalent definitions of manifolds of bounded geometry (see [Sh1,GOW]), the one below is particularly useful in applications.…”