Adiabatic Vacuum States on General Spacetime Manifolds: Definition, Construction, and Physical Properties

Abstract

Adiabatic vacuum states are a well-known class of physical states for linear quantum fields on Robertson-Walker spacetimes. We extend the definition of adiabatic vacua to general spacetime manifolds by using the notion of the Sobolev wavefront set. This definition is also applicable to interacting field theories. Hadamard states form a special subclass of the adiabatic vacua. We analyze physical properties of adiabatic vacuum representations of the Klein-Gordon field on globally hyperbolic spacetime manifolds (factoriality, quasiequivalence, local definiteness, Haag duality) and construct them explicitly, if the manifold has a compact Cauchy surface. W. Junker and E. SchroheAnn. Henri Poincaré can be extended to interacting fields, in close analogy to the Hadamard states. It turned out that the notion of the Sobolev (or H s -) wavefront set is the appropriate mathematical tool for this purpose. In Appendix B we review this notion and the calculus related to it.After an introduction to the structure of the algebra of observables of the Klein-Gordon quantum field on a globally hyperbolic spacetime manifold (M, g) in Section 2 we present our definition of adiabatic states of order N (Definition 3.2) in Section 3. It contains the Hadamard states as a special case: They are adiabatic states "of infinite order". To decide which order of adiabatic vacuum is physically admissible we investigate the algebraic structure of the corresponding GNS-representations. Haag, Narnhofer & Stein [23] suggested as a criterion for physical representations that they should locally generate von Neumann factors that have all the same set of normal states (in other words, the representations are locally primary and quasiequivalent). We show in Section 4.1 (Theorem 4.5 and Theorem 4.7) that this is generally the case if N > 5/2. For the case of pure states on a spacetime with compact Cauchy surface, which often occurs in applications, we improve the admissible order to N > 3/2. In addition, in Section 4.2 we show that adiabatic vacua of order N > 5/2 satisfy the properties of local definiteness (Corollary 4.13) and those of order N > 3/2 Haag duality (Theorem 4.15). These results extend corresponding statements for adiabatic vacuum states on Robertson-Walker spacetimes due to Lüders & Roberts [35], and for Hadamard states due to Verch [49]; for their discussion in the framework of algebraic quantum field theory we refer to [21]. In Section 5 we explicitly construct pure adiabatic vacuum states on an arbitrary spacetime manifold with compact Cauchy surface (Theorem 5.10). In Section 6 we show that our adiabatic states are indeed a generalization of the well-known adiabatic vacua on Robertson-Walker spaces: Theorem 6.3 states that the adiabatic vacua of order n (according to the definition of [35]) on a Robertson-Walker spacetime with compact spatial section are adiabatic vacua of order 2n in the sense of our microlocal Definition 3.2. We conclude in Section 7 by summarizing the physical interpretation of our mathematical analysis and calcula...

Quantum Fields in Curved Spacetime

“…Existence of a large class of Hadamard state on any globally hyperbolic spacetime can be established by a deformation argument [39] combined with microlocal techniques, or by methods from the theory of pseudo-differential operators [40,60].…”

Quantum Energy Inequalities and Stability Conditions in Quantum Field Theory

“…We conclude that the real linear scalar field obeys a QEI with respect to the class of weights delineated by (3) and the class of Hadamard states. The same argument would apply to a suitable class of adiabatic states [32] in which one replaces the smooth wave-front set by a wave-front set modulo Sobolev regularity. Note that this QEI applies to the normal ordered stress-energy tensor, rather than the renormalised tensor.…”

Erratum to "Hadamard States, Adiabatic Vacua and the Construction of Physical States for Scalar Quantum Fields on Curved Spacetime"

“…(80) and (83) are correct due to the particular properties of the Fourier integral operators E + and E − . For the precise proof we refer to Lemma 5.6 in [2], but the essential idea is that even when P is only in…”

“…A similar argument justifies Eq. (82) (see the proof of Lemma 5.6 in [2], here the additional assumption enters that Q has a real-valued principal symbol). For the same reasons the operators Q in the proof of Theorem 3.18, Q n in Eq.…”

“…E2 we explain the origin of this error and prove the weaker statement that in the spatially compact case adiabatic vacua and Hadamard states generate unitarily equivalent GNS-representations. For a microlocal characterization of adiabatic vacuum states on arbitrary spacetimes and a clarification of their precise relation to Hadamard states in terms of wavefront sets we refer to [2]. …”