Construction of Hadamard states by characteristic Cauchy problem

Abstract

We construct Hadamard states for Klein-Gordon fields in a spacetime $M_{0}$ equal to the interior of the future lightcone $C$ from a base point $p$ in a globally hyperbolic spacetime $(M, g)$. Under some regularity conditions at future infinity of $C$, we identify a boundary symplectic space of functions on $C$, which allows to construct states for Klein-Gordon quantum fields in $M_{0}$ from states on the CCR algebra associated to the boundary symplectic space. We formulate the natural microlocal condition on the boundary state on $C$ ensuring that the bulk state it induces in $M_{0}$ satisfies the Hadamard condition. Using pseudodifferential calculus on the cone $C$ we construct a large class of Hadamard boundary states on the boundary with pseudodifferential covariances, and characterize the pure states among them. We then show that these pure boundary states induce pure Hadamard states in $M_{0}$.Comment: 34 p.; v3, references added, matches published versio

Conformal extension of the Bunch-Davies state across the de Sitter boundary

“…The use of conformal methods, pioneered by Penrose [31,32], proved to be particularly successful in this respect. On asymptotically flat spacetimes, their mathematically rigorous implementation led to successive developments including the description of the symplectic space of solutions at null infinity [2,9], the interpretation of asymptotic data in terms of a characteristic Cauchy (or Goursat) problem [15,25,22], and showing that the naturally arising decomposition into positive and negative frequencies defines a canonical, pure Hadamard state [9,27,28,17]. The techniques were generalized in various ways to other settings including Schwarzschild spacetime [11,30], and possibly massive Klein-Gordon fields on a class of cosmological spacetimes including the cosmological chart of de Sitter space [10].…”

Microlocal Analysis of Quantum Fields on Curved Spacetimes

“…Let us introduce some assumptions from [GW2], which avoid this problem and are a version of the notion of asymptotical flatness (with past time infinity). We will come back to this notion in Subsect.…”

“…One first proves that (13.11) C : C ∞ 0 (M ) → H C continuously. This can be easily deduced from [GW2,Lemma 2.8]…”

“…13.6 and the fact that in the coordinates (f, θ) on C, T * C ± is given by {±σ > 0}. We refer the reader to [GW2] for details.…”

“…in energy spaces, by adapting a method due to Hörmander [H6]. We refer again the reader to [GW2]. The restriction to n ≥ 4 comes from the use of a Hardy's type inequality on the cone C. P 13.5.…”