The quantum development of an asymptotically Euclidean Cauchy hypersurface

Cauchy 超曲面上量子场论的几何风味。第二部分：量子化与演化方法

Geometric flavours of Quantum Field theory on a Cauchy hypersurface. Part II: Methods of quantization and evolution

本文利用无限维高斯分析，严格建立标量场 Cauchy 数据空间的几何量子化框架，并以 Khäler 几何描述纯态空间，提出保持几何结构且无范数损失的量子联络演化方程。

Some criteria for Wind Riemannian completeness and existence of Cauchy hypersurfaces

Recently, a link between Lorentzian and Finslerian Geometries has been carried out, leading to the notion of wind Riemannian structure (WRS), a generalization of Finslerian Randers metrics. Here, we further develop this notion and its applications to spacetimes, by introducing some characterizations and criteria for the completeness of WRS's. As an application, we consider a general class of spacetimes admitting a time function $t$ generated by the flow of a complete Killing vector field (generalized standard stationary spacetimes or, more precisely, SSTK ones) and derive simple criteria ensur…

On the global "two-sided" characteristic Cauchy problem for linear wave equations on manifolds

The global characteristic initial value problem for linear wave equations on globally hyperbolic Lorentzian manifolds is examined, for a class of smooth initial value hypersurfaces satisfying favourable global properties. First it is shown that, if geometrically well-motivated restrictions are placed on the supports of the (smooth) initial datum and of the (smooth) inhomogeneous term, then there exists a continuous global solution which is smooth "on each side" of the initial value hypersurface. A uniqueness result in Sobolev regularity $H^{1/2+\varepsilon}_\mathrm{loc}$ is proved among soluti…

Cauchy超曲面上量子场论的几何风味：Hamilton形式体系的Gaussian分析及其在宇宙学中的应用

Geometric Flavours of Quantum Field Theory on a Cauchy Hypersurface: Gaussian Analysis for the Hamiltonian Formalism and Applications to Cosmology

本文以几何Hamilton与预量子化方法研究弯曲时空标量场QFT，统一多种量子表示并引入协变时间导数，最终在宇宙学自由模型中导出粒子产生效应。

Globally Hyperbolic Flat Spacetimes

We consider (flat) Cauchy-complete GH spacetimes, i.e., globally hyperbolic flat lorentzian manifolds admitting some Cauchy hypersurface on which the ambient lorentzian metric restricts as a complete riemannian metric. We define a family of such spacetimes - model spacetimes - including four subfamilies: translation spacetimes, Misner spacetimes, unipotent spacetimes, and Cauchy-hyperbolic spacetimes (the last family - undoubtfull the most interesting one - is a generalization of standart spacetimes defined by G. Mess). We prove that, up to finite coverings and (twisted) products by euclidean …

Existence of Bound States for Layers Built Over Hypersurfaces of Euclidean Space

In this paper, we study the bound states of quantum layers. We prove that for the quantum layer built over a parabolic manifold which is not totally geodesic, if the second fundamantal form decays sufficiently fast, then the bound states exist. In the 2d case, we prove that the quantum layer over a convex surface whose second fundamental form tends to zero at infinity must have bound states.

States of Low Energy in Homogeneous and Inhomogeneous, Expanding Spacetimes

We construct states on the algebra of the Klein-Gordon field that minimize the energy density in homogeneous and in inhomogeneous spacetimes, both with compact Cauchy hypersurfaces. The energy density is measured by geodesic observers and smeared over a spacelike slab of spacetime, entirely containing a Cauchy hypersurface and extended in time. We further show that these states are Hadamard states. The present construction generalizes the construction of States of Low Energy in Robertson-Walker spacetimes presented by Olbermann (2007).

Instability results for the wave equation in the interior of Kerr black holes

We prove that a large class of smooth solutions $ψ$ to the linear wave equation $\Box_gψ=0$ on subextremal rotating Kerr spacetimes which are regular and decaying along the event horizon become singular at the Cauchy horizon. More precisely, we show that assuming appropriate upper and lower bounds on the energy along the event horizon, the solution has infinite (non-degenerate) energy on any spacelike hypersurfaces intersecting the Cauchy horizon transversally. Extrapolating from known results in the Reissner--Nordström case, the assumed upper and lower bounds required for our theorem are conj…

Dirac operators on hypersurfaces as large mass limits

We show that the eigenvalues of the intrinsic Dirac operator on the boundary of a Euclidean domain can be obtained as the limits of eigenvalues of Euclidean Dirac operators, either in the domain with a MIT-bag type boundary condition or in the whole space, with a suitably chosen zero order mass term.

Integral Formula for the Characteristic Cauchy Problem on a curved Background

We give a local integral formula, valid on general curved space-times, for the characteristic Cauchy problem for the Dirac equation with arbitrary spin using the method developed by Friedlander in his book "the wave equation on a curved spacetime" (1975). The results obtained by Penrose in the flat case in "Null hypersurface initial data for classical fields of arbitrary spin for general relativity" (Gen. Rel. Grav 1980) are recovered directly. It is expected that this method can be used to obtain sharp estimates for the characteristic Cauchy problem for the Dirac equation.