In this paper the semilinear equation of the spin-$\frac{1}{2}$ fields in the de Sitter space is investigated. We prove the existence of the global in time small data solution in the expanding de Sitter universe. Then, under the Lochak-Majorana condition, we prove the existence of the global in time solution with large data. The sufficient conditions for the solutions to blow up in finite time are given for large data in the expanding and contracting de Sitter spacetimes. The influence of the Hubble constant on the lifespan is estimated.
We study the resolvent of the massive Dirac operator in the Schwarzschild-Anti-de Sitter space-time. After separation of variables, we use standard one dimensional techniques to obtain an explicit formula. We then make use of this formula to extend the resolvent meromorphically accross the real axis.
We show asymptotic completeness for linear massive Dirac fields on the Schwarzschild-Anti-de Sitter spacetime. The proof is based on a Mourre estimate. We also construct an asymptotic velocity for this field.
We prove the existence of exponentially accurate quasimodes using the square of the Dirac operator on the Schwarzschild-Anti-de Sitter spacetime and the Agmon estimates. We then deduce a logarithmic lower bound for the local energy decay of the Dirac propagator.
We provide the full asymptotic description of the quasi-normal modes (resonances) in any strip of fixed width for Dirac fields in slowly rotating Kerr-Newman-de Sitter black holes. The resonances split in a way similar to the Zeeman effect. The method is based on the extension to Dirac operators of techniques applied by Dyatlov 2011, 2012 to the (uncharged) Kerr-de Sitter black holes. We show that the mass of the Dirac field does not have effect on the two leading terms in the expansions of resonances.
本文研究了反德西特宇宙中Dirac方程的整体解。由于这个空间不是全局双曲的，柯西问题不是，{ \it a priori}，适定的。尽管如此，我们还是可以证明存在幺正动力学，但它的唯一性关键取决于场的质量$M$与宇宙学常数$Λ>0$的比值：它出现一个临界值$Λ/12$，它的作用类似于标量场的Breitenlohner-Freedman界。当$M^2\geq Λ/12$时，存在唯一的幺正动力学.相反，对于满足$M^2<Λ/12$的轻费米子，我们构造了几个在无穷远处的渐近条件，使得问题成为适定的。在所有情况下，哈密顿量的谱都是离散的。我们还证明了能量均分的一个结果。
We give an expansion of the solution of the evolution equation for the massless Dirac fields in the outer region of de Sitter-Reissner-Nordström black hole in terms of resonances. By means of this method we describe the decay of local energy for compactly supported data. The proof uses the cut-off resolvent estimates for the semi-classical Schrödinger operators from Bony and Häfner, 2008. The method extends to the Dirac operators on spherically symmetric asymptotically hyperbolic manifolds.
Symmetry operators and separability of massive Klein-Gordon and Dirac equations in the general 5-dimensional Kerr-(anti-)de Sitter black hole background
We present the ${\mathscr P}(φ)_2$ model on de Sitter space in the canonical formulation. We discuss the role of the Noether theorem and we provide explicit expressions for the energy-stress tensor of the interacting model.
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 recently constructed the $R$-Poincaré algebra from an appropriate deformed Poisson brackets which reproduce the Fock coordinate transformation. We showed then that the spacetime of this transformation is the de Sitter one. In this paper, we derive in the $R$-Minkowski spacetime the Dirac equation and show that this is none other than the Dirac equation in the de Sitter spacetime given by its conformally flat metric. Furthermore, we propose a new approach for solving Dirac's equation in the de Sitter spacetime using the Schrödinger picture.
The quasi-normal modes for black holes are the resonances for the scattering of incoming waves by black holes. Here we consider scattering of massless uncharged Dirac fields propagating in the outer region of de Sitter-Reissner-Nordstr{ö}m black hole, which is spherically symmetric charged exact solution of the Einstein-Maxwell equations. Using the spherical symmetry of the equation and restricting to a fixed harmonic the problem is reduced to a scattering problem for the 1D massless Dirac operator on the line. The resonances for the problem are related to the resonances for a certain semiclas…
In 1975 Figari, Høegh-Krohn and Nappi constructed the P(phi)_2 model on the de Sitter space. Here we complement their work with new results, as well as by connecting this model to various areas of mathematics. In particular, i.) we discuss the causal structure of de Sitter space and the induces representations of the Lorentz group. We show that the UIRs of SO_0(1,2) for both the principal and the complementary series can be formulated on Hilbert spaces whose functions are supported on a Cauchy surface, ii.) we describe the free classical dynamical system in both its covariant and canonical for…
Exact solutions of the Dirac general relativistic equation (DE) that describe the dynamics of a massive, electrically charged particle with half-integer spin in the curved spacetime geometry of an electrically charged, rotating Kerr-Newman-(anti) de Sitter black hole (BH) are investigated. We first, derive the DE in the Kerr-Newman-de Sitter (KNdS) BH background using a generalised Kinnersley null tetrad in the Newman-Penrose formalism. In this frame, we prove the separation of the DE into ordinary differential equations for the radial and angular parts. Under specific transformations of the i…
We examine the solutions of the semilinear wave equation, and, in particular, of the $φ^p$ model of quantum field theory in the curved space-time. More exactly, for $1< p<4$ we prove that solution of the massless self-interacting scalar field equation in the Einstein-de Sitter universe has finite lifespan.
