On the global Hadamard parametrix in QFT and the signed squared geodesic distance defined in domains larger than convex normal neighbourhoods

We consider the global Hadamard condition and the notion of Hadamard parametrix whose use is pervasive in algebraic QFT in curved spacetime (see references in the main text). We point out the existence of a technical problem in the literature concerning well-definedness of the global Hadamard parametrix in normal neighbourhoods of Cauchy surfaces. We discuss in particular the definition of the (signed) geodesic distance $$\sigma $$ σ and related structures in an open neighbourhood of the diagonal of $$M\times M$$ M × M larger than $$U\times U$$ U × U , for a normal convex neighbourhood U, where (M, g) is a Riemannian or Lorentzian (smooth Hausdorff paracompact) manifold. We eventually propose a quite natural solution which slightly changes the original definition by Kay and Wald and relies upon some non-trivial consequences of the paracompactness property. The proposed re-formulation is in agreement with Radzikowski’s microlocal version of the Hadamard condition.

The Legendre–Hadamard condition in Cosserat elasticity theory

Summary The Legendre–Hadamard necessary condition for energy minimizers is derived in the framework of Cosserat elasticity theory.

PROPAGATION OF SINGULARITIES ON AS SPACETIMES FOR GENERAL BOUNDARY CONDITIONS AND THE HOLOGRAPHIC HADAMARD CONDITION

We consider the Klein–Gordon equation on asymptotically anti-de-Sitter spacetimes subject to Neumann or Robin (or Dirichlet) boundary conditions and prove propagation of singularities along generalized broken bicharacteristics. The result is formulated in terms of conormal regularity relative to a twisted Sobolev space. We use this to show the uniqueness, modulo regularizing terms, of parametrices with prescribed $\text{b}$ -wavefront set. Furthermore, in the context of quantum fields, we show a similar result for two-point functions satisfying a holographic Hadamard condition on the $\text{b}$ -wavefront set.

The art of the state

Quantum field theory (QFT) on curved spacetimes lacks an obvious distinguished vacuum state. We review a recent no-go theorem that establishes the impossibility of finding a preferred state in each globally hyperbolic spacetime, subject to certain natural conditions. The result applies in particular to the free scalar field, but the proof is model-independent and therefore of wider applicability. In addition, we critically examine the recently proposed “SJ states”, that are determined by the spacetime geometry alone, but which fail to be Hadamard in general. We describe a modified construction that can yield an infinite family of Hadamard states, and also explain recent results that motivate the Hadamard condition without direct reference to ultra-high energies or ultra-short distance structure.

Modes of Stability Loss of Materials

Three problems of stability loss are investigated and corresponded buckling modes are discussed. The first one is the stability loss of a compressed transversely isotropic linearly elastic medium. The standard analysis based on the Hadamard condition is conducted to solve this problem. The critical compression could be uniquely defined from the bifurcation equations but not a wave length. So, the buckling mode remains generally indefinite. The second considered problem is the stability loss of a compressed half-space with a free surface. It could be shown that the waviness is localized near the free plane surface but as for an entire space the wave length and the buckling mode are indefinite. These problems are treated in linear and nonlinear statement. In linear approach the pre-buckling deformations are ignored. It is shown that for some values of parameters the linear approach leads not only to the numerical error but also to qualitatively incorrect results. The thisrd problem under investifation is the stability loss of an uniformly compressed plate lying on a soft elastic half-space. In this problem the wave length is uniquely defined. Using the nonlinear post-critical analysis it is shown that the buckling mode could be fully defined and is has a chessboard-like character.

MICROLOCAL SPECTRUM CONDITION AND HADAMARD FORM FOR VECTOR-VALUED QUANTUM FIELDS IN CURVED SPACETIME

Some years ago, Radzikowski has found a characterization of Hadamard states for scalar quantum fields on a four-dimensional globally hyperbolic spacetime in terms of a specific form of the wavefront set of their two-point functions (termed "wavefront set spectrum condition"), thereby initiating a major progress in the understanding of Hadamard states and the further development of quantum field theory in curved spacetime. In the present work, we extend this important result on the equivalence of the wavefront set spectrum condition with the Hadamard condition from scalar fields to vector fields (sections in a vector bundle) which are subject to a wave-equation and are quantized so as to fulfill the covariant canonical commutation relations, or which obey a Dirac equation and are quantized according to the covariant anti-commutation relations, in any globally hyperbolic spacetime having dimension three or higher. In proving this result, a gap which is present in the published proof for the scalar field case will be removed. Moreover we determine the short-distance saling limits of Hadamard states for vector-bundle valued fields, finding them to coincide with the corresponding flat-space, massless vacuum states.