arXiv subjects
Publications and source records attributed to Alain Bachelot.
We give a rigorous construction of the Dirac Sea for the fermionic quantization in the non-separable Hilbert spaces. These CAR-representations depend on the Axiom of Choice, hence are not unique, nevertheless they are unitarily equivalent to the classic Fock representation.
We investigate the linear and semilinear massive Klein-Gordon equations in geometrical frameworks of type "Conformal Cyclic Cosmology" of R. Penrose, or "Singular Bouncing Scenario" as well. We give sufficient conditions on the decay of the mass to the fields be able to propagate across the Big-Bang.
We investigate the propagation of the scalar waves in the FLRW universes beginning with a Big Bang and ending with a Big Crunch, a Big Rip, a Big Brake or a Sudden Singularity. We obtain the sharp description of the asymptotics for the solutions of the linear Klein-Gordon equation, and similar results for the semilinear equation with a subcritical exponent. We prove that the number of cosmological particle creation is finite under general assumptions on the initial Big Bang and the final Big Crunch or Big Brake.
We investigate the wave propagation on a compact 3-manifold of constant positive curvature with a non trivial topology, the Poincaré dodecahedral space, when the scale factor is exponentially increasing. We prove the existence of a limit state as t tends to infinity and we get its analytic expression. The deep sky is described by this asymptotic profile thanks to the Sachs-Wolfe formula. We transform the Cauchy problem into a mixed problem posed on a fundamental domain determined by the quaternionic calculus. We perform an accurate scheme of computation: we employ a variational method using a space of second order finite elements that is invariant under the action of the binary icosahedral group.
We investigate the propagation of the scalar waves in the Witten space-time called "bubble of nothing" and in its remarkable sub-manifold, the Lorentzian Hawking wormhole. Due to the global hyperbolicity, the global Cauchy problem is well-posed in the functional framework associated with the energy. We perform a complete spectral analysis that allows to get an explicit form of the solutions in terms of special functions. If the effective mass is non zero, the profile of the waves is asymptotically almost periodic in time. In contrast, the massless case is dispersive. We develop the scattering theory, classical as well as quantum. The quantized scattering operator leaves invariant the Fock vacuum: there is no creation of particles. The resonances can be defined in the massless case and they are purely imaginary.
In this paper we investigate the Klein-Gordon equation in the past causal domain of a De Sitter brane imbedded in a Anti-de Sitter bulk. We solve the global mixed hyperbolic problem. We prove that any finite energy solution can be expressed as a Kaluza-Klein tower that is a superposition of free fields in the Steady State Universe, of which we study the asymptotic behaviours. We show that the leading term of a gravitational fluctuation is a massless graviton, i.e. the De Sitter brane is linearly stable.
This paper deals with the propagation of the gravitational waves in the Poincaré patch of the 5-dimensional Anti-de Sitter universe. We construct a large family of unitary dynamics with respect to some high order energies that are conserved and positive. These dynamics are associated with asymptotic conditions on the conformal time-like boundary of the universe. This result does not contradict the statement of Breitenlohner-Freedman that the hamiltonian is essentially self-adjoint in L2 and thus accordingly the dynamics is uniquely determined. The key point is the introduction of a new Hilbert functional framework that contains the massless graviton which is not normalizable in L2. Then the hamiltonian is not essentially self-adjoint in this new space and admits a lot of positive self-adjoint extensions.
This paper deals with the Klein-Gordon equation on the Poincaré chart of the 5-dimensional Anti-de Sitter universe. When the mass $μ$ is larger than $-{1}{4}$, the Cauchy problem is well posed despite the loss of global hyperbolicity due to the time-like horizon. We express the finite energy solutions in the form of a continuous Kaluza-Klein tower and we deduce a uniform decay as $\mid t\mid^{-{3}{2}}$. We investigate the case $μ={ν^2-1}{2}$, $ν\in\NN^*$, which encompasses the gravitational fluctuations, $ν=4$, and the electromagnetic waves, $ν=2$. The propagation of the wave front set shows that the horizon acts like a perfect mirror. We establish that the smooth solutions decay as $\mid t\mid^{-2-\sqrt{μ+{1}{4}}}$, and we get global $L^p$ estimates of Strichartz type. When $ν$ is even, there appears a lacuna and the equipartition of the energy occurs at finite time for the compactly supported initial data, although the Huygens principle fails. We adress the cosmological model of the negative tension Minkowski brane, on which a Robin boundary condition is imposed. We prove the hyperbolic mixed problem is well-posed and the normalizable solutions can be expanded into a discrete Kaluza-Klein tower. We establish some $L^2-L^{\infty}$ estimates in suitable weighted Sobolev spaces.
We study the wave equation for the gravitational waves in the Randal-Sundrum brane cosmology model. We solve the global Cauchy problem and we establish that the solutions are the sum of a slowly decaying massless wave localized near the brane, and a superposition of massive dispersive waves. We compute the kernel of the truncated resolvent. We prove some $L^1-L^{\infty}$, $L^2-L^{\infty}$ decay estimates and global $L^p$ Strichartz type inequalities. We develop the complete scattering theory : existence and asymptotic completeness of the wave operators, computation of the scattering matrix, determination of the resonances on the logarithmic Riemann surface.
We investigate the global solutions of the Dirac equation on the Anti-de-Sitter Universe. Since this space is not globally hyperbolic, the Cauchy problem is not, {\it a priori}, well-posed. Nevertheless we can prove that there exists unitary dynamics, but its uniqueness crucially depends on the ratio beween the mass $M$ of the field and the cosmological constant $Λ>0$ : it appears a critical value, $Λ/12$, which plays a role similar to the Breitenlohner-Freedman bound for the scalar fields. When $M^2\geq Λ/12$ there exists a unique unitary dynamics. In opposite, for the light fermions satisfying $M^2<Λ/12$, we construct several asymptotic conditions at infinity, such that the problem becomes well-posed. In all the cases, the spectrum of the hamiltonian is discrete. We also prove a result of equipartition of the energy.