1,719 research outputs found
On boundary conditions for linearised Einstein's equations
Full text linkWe investigate the properties of a fairly large class of boundary conditions for the linearised Einstein equations in the Riemannian setting, ones which generalise the linearised counterpart of boundary conditions proposed by Anderson. Through the prism of the quest to quantise gravitational waves in curved spacetimes, we study their properties from the point of view of ellipticity, gauge invariance, and the existence of a spectral gap
Diagonalization of elliptic systems via pseudodifferential projections
Full text linkConsider an elliptic self-adjoint pseudodifferential operator A acting on m-columns of half-densities on a closed manifold M, whose principal symbol is assumed to have simple eigenvalues. Relying on a basis of pseudodifferential projections commuting with A, we construct an almost-unitary pseudodifferential operator that diagonalizes A modulo an infinitely smoothing operator. We provide an invariant algorithm for the computation of its full symbol, as well as an explicit closed formula for its subprincipal symbol. Finally, we give a quantitative description of the relation between the spectrum of A and the spectrum of its approximate diagonalization, and discuss the implications at the level of spectral asymptotics
Hadamard States for Quantum Abelian Duality
No full textAbelian duality is realized naturally by combining differential cohomology and locally covariant quantum field theory. This leads to a (Formula presented.)-algebra of observables, which encompasses the simultaneous discretization of both magnetic and electric fluxes. We discuss the assignment of physically well-behaved states on this algebra and the properties of the associated GNS triple. We show that the algebra of observables factorizes as a suitable tensor product of three (Formula presented.)-algebras: the first factor encodes dynamical information, while the other two capture topological data corresponding to electric and magnetic fluxes. On the former factor and in the case of ultra-static globally hyperbolic spacetimes with compact Cauchy surfaces, we exhibit a state whose two-point correlation function has the same singular structure of a Hadamard state. Specifying suitable counterparts also on the topological factors, we obtain a state for the full theory, ultimately implementing Abelian duality transformations as Hilbert space isomorphisms
Global propagator for the massless Dirac operator and spectral asymptotics
Full text linkAbstract: We construct the propagator of the massless Dirac operator W on a closed Riemannian 3-manifold as the sum of two invariantly defined oscillatory integrals, global in space and in time, with distinguished complex-valued phase functions. The two oscillatory integrals—the positive and the negative propagators—correspond to positive and negative eigenvalues of W, respectively. This enables us to provide a global invariant definition of the full symbols of the propagators (scalar matrix-functions on the cotangent bundle), a closed formula for the principal symbols and an algorithm for the explicit calculation of all their homogeneous components. Furthermore, we obtain small time expansions for principal and subprincipal symbols of the propagators in terms of geometric invariants. Lastly, we use our results to compute the third local Weyl coefficients in the asymptotic expansion of the eigenvalue counting functions of W
Global and microlocal aspects of Dirac operators: propagators and Hadamard states
Full text linkWe propose a geometric approach to construct the Cauchy evolution operator
for the Lorentzian Dirac operator on Cauchy-compact globally hyperbolic
4-manifolds. We realise the Cauchy evolution operator as the sum of two
invariantly defined oscillatory integrals -- the positive and negative Dirac
propagators -- global in space and in time, with distinguished complex-valued
geometric phase functions. As applications, we relate the Cauchy evolution
operators with the Feynman propagator and construct Cauchy surfaces covariances
of quasifree Hadamard states.Comment: 40 pages, 2 pictures -- accepted in Advances in Differential
Equation
High-contrast random systems of PDEs: homogenisation and spectral theory
Full text linkWe develop a qualitative homogenization and spectral theory for elliptic systems of partial differential equations in divergence form with highly contrasting (i.e. non-uniformly elliptic) random coefficients. The focus of this paper is on the behavior of the spectrum as the heterogeneity parameter tends to zero; in particular, we show that in general one does not have Hausdorff convergence of spectra. The theoretical analysis is complemented by several explicit examples, showcasing the wider range of applications and physical effects of systems with random coefficients, when compared with systems with periodic coefficients or with scalar operators (both random and periodic)
Invariant subspaces of elliptic systems I: Pseudodifferential projections
Full text linkConsider an elliptic self-adjoint pseudodifferential operator A acting on m-columns of half-densities on a closed manifold M, whose principal symbol is assumed to have simple eigenvalues. We show existence and uniqueness of m orthonormal pseudodifferential projections commuting with the operator A and provide an algorithm for the computation of their full symbols, as well as explicit closed formulae for their subprincipal symbols. Pseudodifferential projections yield a decomposition of into invariant subspaces under the action of A modulo . Furthermore, they allow us to decompose A into m distinct sign definite pseudodifferential operators. Finally, we represent the modulus and the Heaviside function of the operator A in terms of pseudodifferential projections and discuss physically meaningful examples
Invariant subspaces of elliptic systems II: Spectral theory
Full text linkConsider an elliptic self-adjoint pseudodifferential operator A acting on m-columns of half-densities on a closed manifold M M, whose principal symbol is assumed to have simple eigenvalues.We show that the spectrum of A decomposes, up to an error with superpolynomial decay, into m distinct series, each associated with one of the eigenvalues of the principal symbol of A. These spectral results are then applied to the study of propagation of singularities in hyperbolic systems. The key technical ingredient is the use of the carefully devised pseudodifferential projections introduced in the first part of this work, which decompose L2(M) into almost-orthogonal almost-invariant subspaces under the action of both A and the hyperbolic evolution
Spectral asymptotics for linear elasticity: the case of mixed boundary conditions
Full text linkWe establish two-term spectral asymptotics for the operator of linear elasticity with mixed boundary conditions on a smooth compact Riemannian manifold of arbitrary dimension. We illustrate our results by explicit examples in dimension two and three, thus verifying our general formulae both analytically and numerically
Spacetime diffeomorphisms as matter fields
Full text linkWe work on a 4-manifold equipped with Lorentzian metric g and consider a volume-preserving diffeomorphism that is the unknown quantity of our mathematical model. The diffeomorphism defines a second Lorentzian metric h, the pullback of g. Motivated by elasticity theory, we introduce a Lagrangian expressed algebraically (without differentiations) via our pair of metrics. Analysis of the resulting nonlinear field equations produces three main results. First, we show that for Ricci-flat manifolds, our linearized field equations are Maxwell's equations in the Lorenz gauge with exact current. Second, for Minkowski space, we construct explicit massless solutions of our nonlinear field equations; these come in two distinct types, right-handed and left-handed. Third, for Minkowski space, we construct explicit massive solutions of our nonlinear field equations; these contain a positive parameter that has the geometric meaning of quantum mechanical mass and a real parameter that may be interpreted as electric charge. In constructing explicit solutions of nonlinear field equations, we resort to group-theoretic ideas: We identify special four-dimensional subgroups of the Poincaré group and seek diffeomorphisms compatible with their action in a suitable sense
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