arXiv subjects
Publications and source records attributed to Michał Wrochna.
In the setting of non-interacting Klein-Gordon fields on asymptotically anti-de Sitter spacetimes, we show that algebras of observables localized in a neighborhood of the boundary are subalgebras of a boundary algebra. The underlying mechanism is directly related to holography for classical fields. In particular, the proof relies on unique continuation theorems at the conformal boundary.
We consider the wave equation on asymptotically Minkowski spacetimes and the Klein-Gordon equation on even asymptotically de Sitter spaces. In both cases we show that the extreme difference of propagators (i.e. retarded propagator minus advanced, or Feynman minus anti-Feynman), defined as Fredholm inverses, induces a symplectic form on the space of solutions with wave front set confined to the radial sets. Furthermore, we construct isomorphisms between the solution spaces and symplectic spaces of asymptotic data. As an application of this result we obtain distinguished Hadamard two-point functions from asymptotic data. Ultimately, we prove that the corresponding Quantum Field Theory on asymptotically de Sitter spacetimes induces canonically a QFT beyond the future and past conformal boundary, i.e. on two even asymptotically hyperbolic spaces. Specifically, we show this to be true both at the level of symplectic spaces of solutions and at the level of Hadamard two-point functions.
In the setting of asymptotically Anti-de Sitter spacetimes, we consider Klein-Gordon fields subject to Dirichlet boundary conditions, with mass satisfying the Breitenlohner-Freedman bound. We introduce a condition on the b-wave front set of two-point functions of quantum fields, which locally in the bulk amounts to the usual Hadamard condition, and which moreover allows to estimate wave front sets for the holographically induced theory on the boundary. We prove the existence of two-point functions satisfying this condition, and show their uniqueness modulo terms that have smooth Schwartz kernel in the bulk and have smooth restriction to the boundary. Finally, using Vasy's propagation of singularities theorem, we prove an analogue of Duistermaat and Hörmander's theorem on distinguished parametrices.
We consider the Klein-Gordon equation on analytic spacetimes with an analytic Cauchy surface. In this setting, we prove the existence of pure analytic Hadamard states. The proof is based on considering an elliptic operator obtained by Wick rotating the Klein-Gordon operator in a neighborhood of a Cauchy hypersurface. The Cauchy data of Hadamard two-point functions are constructed as Calderón projectors (suitably generalized if the hypersurface is non-compact) for the elliptic operator.
We investigate linearized gauge theories on globally hyperbolic spacetimes in the BRST formalism. A consistent definition of the classical phase space and of its Cauchy surface analogue is proposed. We prove that it is isomorphic to the phase space in the subsidiary condition approach of Hack and Schenkel in the case of Maxwell, Yang-Mills, and Rarita-Schwinger fields. Defining Hadamard states in the BRST formalism in a standard way, their existence in the Maxwell and Yang-Mills case is concluded from known results in the subsidiary condition (or Gupta-Bleuler) formalism. Within our framework, we also formulate criteria for non-degeneracy of the phase space in terms of BRST cohomology and discuss special cases. These include an example in the Yang-Mills case, where degeneracy is not related to a non-trivial topology of the Cauchy surface.
We consider the massive Klein-Gordon equation on a class of asymptotically static spacetimes (in the long range sense) with Cauchy surface of bounded geometry. We prove the existence and Hadamard property of the in and out states constructed by scattering theory methods.
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 $M_{0}$.
We consider the massive Klein-Gordon equation on asymptotically Minkowski spacetimes, in the sense that the manifold is $R^{1+d}$ and the metric approaches that of Minkowski space at infinity in a short-range way (jointly in time and space variables). In this setup we define Feynman and anti-Feynman scattering data and prove the Fredholm property of the Klein-Gordon operator with the associated Atiyah-Patodi-Singer type boundary conditions at infinite times. We then construct a parametrix (with compact remainder terms) for the Fredholm problem and prove that it is also a Feynman parametrix in the sense of Duistermaat and Hörmander.
We systematically derive general properties of continuous and holomorphic functions with values in closed operators, allowing in particular for operators with empty resolvent set. We provide criteria for a given operator-valued function to be continuous or holomorphic. This includes sufficient conditions for the sum and product of operator-valued holomorphic functions to be holomorphic. Using graphs of operators, operator-valued functions are identified with functions with values in subspaces of a Banach space. A special role is thus played by projections onto closed subspaces of a Banach space, which depend holomorphically on a parameter.
We consider distributions on $\R^n\setminus{0}$ which satisfy a given set of partial differential equations and provide criteria for the existence of extensions to $\R^n$ that satisfy the same set of equations on $\R^n$. We use the results to construct distributions satisfying specific renormalisation conditions in the Epstein and Glaser approach to perturbative quantum field theory. Contrary to other approaches, we provide a unified apporach to treat Lorentz covariance, invariance under global gauge group and almost homogeneity, as well as discrete symmetries. We show that all such symmetries can be recovered by applying a linear map defined for all degrees of divergence. Using similar techniques, we find a relation between on-shell and off-shell time-ordered products involving higher derivatives of the fields.
We prove that the ground state for the Dirac equation on Minkowski space in static, smooth external potentials satisfies the Hadamard condition. We show that it follows from a condition on the support of the Fourier transform of the corresponding positive frequency solution. Using a Krein space formalism, we establish an analogous result in the Klein-Gordon case for a wide class of smooth potentials. Finally, we investigate overcritical potentials, i.e. which admit no ground states. It turns out, that numerous Hadamard states can be constructed by mimicking the construction of ground states, but this leads to a naturally distinguished one only under more restrictive assumptions on the potentials.
We systematically describe and classify 1-dimensional Schrödinger equations that can be solved in terms of hypergeometric type functions. Beside the well-known families, we explicitly describe 2 new classes of exactly solvable Schrödinger equations that can be reduced to the Hermite equation.
We present explicit formulae for Weber-Schafheitlin type integrals and give them an interpretation as the kernel of a physically relevant operator related to the hamiltonian of Aharanov and Bohm. In particular, we derive explicit formulae for Weber-Schafheitlin type integrals with exponent larger or equal 1, which are distributions on R_+. We discuss several special cases.