Searcharxiv⌕ Search

arXiv subjects

Christian Gérard

Publications and source records attributed to Christian Gérard.

36 records · Page 2Linked to original sources
arXiv2016

The massive Feynman propagator on asymptotically Minkowski spacetimes

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.

math-ph↗
arXiv2016

Feynman propagators and Hadamard states from scattering data for the Klein-Gordon equation on asymptotically Minkowski spacetimes

We consider the massive Klein-Gordon equation on a class of asymptotically static spacetimes. We prove the existence and Hadamard property of the in and out states constructed by scattering theory methods. Assuming in addition that the metric approaches that of Minkowski space at infinity in a short-range way, jointly in time and space variables, we define Feynman scattering data and prove the Fredholm property of the Klein-Gordon operator with the associated Atiyah-Patodi-Singer boundary conditions. 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 H{ö}rmander.

math-ph↗
arXiv2015

Asymptotic Completeness in Quantum Field Theory: Translation Invariant Nelson Type Models Restricted to the Vacuum and One-Particle Sectors

Time-dependent scattering theory for a large class of translation invariant models, including the Nelson and Polaron models, restricted to the vacuum and one-particle sectors is studied. We formulate and prove asymptotic completeness for these models. The translation invariance imply that the Hamiltonians considered are fibered with respect to the total momentum. On the way to asymptotic completeness we determine the spectral structure of the fiber Hamiltonians, establish a Mourre estimate and derive a geometric asymptotic completeness statement as an intermediate step.

math-ph↗
arXiv2014

Construction of Hadamard states by pseudo-differential calculus

We give a new construction based on pseudo-differential calculus of quasi-free Hadamard states for Klein-Gordon equations on a class of space-times whose metric is well-behaved at spatial infinity. In particular we construct all pure Hadamard states and study their covariance under symplectic transformations. Using our results we give a new construction of Hadamard states on arbitrary globally hyperbolic space-times.

math-ph↗
arXiv2013

Hawking effect for a toy model of interacting fermions

We consider a toy model of interacting Dirac fermions in a 1+1 dimensional space time describing the exterior of a star collapsing to a blackhole. In this situation we give a rigorous proof of the {\em Hawking effect}, namely that under the associated quantum evolution, an initial vacuum state will converge when $t\to +\infty$ to a thermal state at Hawking temperature. We establish this result both for observables falling into the blackhole along null characteristics, and for static observables. We also consider the case of an interaction localized near the star boundary, obtaining similar results. We hence extend to an interacting model previous results of Bachelot and Melnyk, obtained for free Dirac fields.

math-ph↗
arXiv2013

A criterion for asymptotic completeness in local relativistic QFT

We formulate a generalized concept of asymptotic completeness and show that it holds in any Haag-Kastler quantum field theory with an upper and lower mass gap. It remains valid in the presence of pairs of oppositely charged particles in the vacuum sector, which invalidate the conventional property of asymptotic completeness. Our result can be restated as a criterion characterizing a class of theories with complete particle interpretation in the conventional sense. This criterion is formulated in terms of certain asymptotic observables (Araki-Haag detectors) whose existence, as strong limits of their approximating sequences, is our main technical result. It is proven with the help of a novel propagation estimate, which is also relevant to scattering theory of quantum mechanical dispersive systems.

math-ph↗
arXiv2013

Boundary values of resolvents of self-adjoint operators in Krein spaces

We prove in this paper resolvent estimates for the boundary values of resolvents of selfadjoint operators on a Krein space: if $H$ is a selfadjoint operator on a Krein space $\cH$, equipped with the Krein scalar product $\langle \cdot| \cdot \rangle$, $A$ is the generator of a $C_{0}-$group on $\cH$ and $I\subset \rr$ is an interval such that: \begin{itemize} \item[]1) $H$ admits a Borel functional calculus on $I$, \item[]2) the spectral projection $\one_{I}(H)$ is positive in the Krein sense, \item[]3) the following {\em positive commutator estimate} holds: \[ \Re \langle u| [H, ıA]u\rangle\geq c \langle u| u\rangle, \ u \in {\rm Ran}\one_{I}(H), \ c>0. \] \end{itemize} then assuming some smoothness of $H$ with respect to the group $\e^{ıt A}$, the following resolvent estimates hold: \[ \sup_{z\in I\pm ı]0, ν]}\| \langle A\rangle ^{-s}(H-z)^{-1}\langle A\rangle^{-s}\| <\infty, \ s>\12. \] As an application we consider abstract Klein-Gordon equations \[ \p_{t}^{2}ϕ(t)- 2 ık ϕ(t)+ hϕ(t)=0, \] and obtain resolvent estimates for their generators in {\em charge spaces} of Cauchy data.

math-ph↗
arXiv2013

Resolvent and propagation estimates for Klein-Gordon equations with non-positive energy

We study in this paper an abstract class of Klein-Gordon equations: \[ \p_{t}^{2}ϕ(t)- 2ık \p_{t}ϕ(t)+ h ϕ(t)=0, \] where $ϕ: \rr\to \cH$, $\cH$ is a (complex) Hilbert space, and $h$, $k$ are self-adjoint, resp. symmetric operators on $\cH$. We consider their generators $H$ (resp. $K$) in the two natural spaces of Cauchy data, the energy (resp. charge) spaces. We do not assume that the dynamics generated by $H$ or $K$ has any positive conserved quantity, in particular these operators may have complex spectrum. Assuming conditions on $h$ and $k$ which allow to use the theory of selfadjoint operators on Krein spaces, we prove weighted estimates on the boundary values of the resolvents of $H$, $K$ on the real axis. From these resolvent estimates we obtain corresponding propagation estimates on the behavior of the dynamics for large times. Examples include wave or Klein-Gordon equations on asymptotically euclidean or asymptotically hyperbolic manifolds, minimally coupled with an external electro-magnetic field decaying at infinity.

math-ph↗
arXiv2012

Towards asymptotic completeness of two-particle scattering in local relativistic QFT

We consider the problem of existence of asymptotic observables in local relativistic theories of massive particles. Let $\ti p_1$ and $\ti p_2$ be two energy-momentum vectors of a massive particle and let $\De$ be a small neighbourhood of $\ti p_1+\ti p_2$. We construct asymptotic observables (two-particle Araki-Haag detectors), sensitive to neutral particles of energy-momenta in small neighbourhoods of $\ti p_1$ and $\ti p_2$. We show that these asymptotic observables exist, as strong limits of their approximating sequences, on all physical states from the spectral subspace of $\De$. Moreover, the linear span of the ranges of all such asymptotic observables coincides with the subspace of two-particle Haag-Ruelle scattering states with total energy-momenta in $\De$. The result holds under very general conditions which are satisfied, for example, in $\la ϕ^4_2$. The proof of convergence relies on a variant of the phase-space propagation estimate of Graf.

math-ph↗
arXiv2012

Binding condition for a general class of quantum field Hamiltonians

We consider a system of a quantum particle interacting with a quantum field and an external potential $V(\bx)$. The Hamiltonian is defined by a quadratic form $H^V = H^0 + V(\bx)$, where $H^0$ is a quadratic form which preserves the total momentum. $H^0$ and $H^V$ are assumed to be bounded from below. We give a criterion for the positivity of the binding energy $E_\mathrm{bin} = E^0-E^V$, where $E^0$ and $E^V$ are the ground state energies of $H^0$ and $H^V$. As examples of the result, the positivity of the binding energy of the semi-relativistic Pauli-Fierz model and Nelson type Hamiltonian is proved.

math-ph↗
arXiv2011

Scattering theory for Klein-Gordon equations with non-positive energy

We study the scattering theory for charged Klein-Gordon equations: \[\{{array}{l} (\p_{t}- ıv(x))^{2}ϕ(t,x) ε^{2}(x, D_{x})ϕ(t,x)=0,[2mm] ϕ(0, x)= f_{0}, [2mm] ı^{-1} \p_{t}ϕ(0, x)= f_{1}, {array}. \] where: \[ε^{2}(x, D_{x})= \sum_{1\leq j, k\leq n}(\p_{x_{j}} ıb_{j}(x))A^{jk}(x)(\p_{x_{k}} ıb_{k}(x))+ m^{2}(x),\] describing a Klein-Gordon field minimally coupled to an external electromagnetic field described by the electric potential $v(x)$ and magnetic potential $\vec{b}(x)$. The flow of the Klein-Gordon equation preserves the energy: \[ h[f, f]:= \int_{\rr^{n}}\bar{f}_{1}(x) f_{1}(x)+ \bar{f}_{0}(x)ε^{2}(x, D_{x})f_{0}(x) - \bar{f}_{0}(x) v^{2}(x) f_{0}(x) \d x. \] We consider the situation when the energy is not positive. In this case the flow cannot be written as a unitary group on a Hilbert space, and the Klein-Gordon equation may have complex eigenfrequencies. Using the theory of definitizable operators on Krein spaces and time-dependent methods, we prove the existence and completeness of wave operators, both in the short- and long-range cases. The range of the wave operators are characterized in terms of the spectral theory of the generator, as in the usual Hilbert space case.

math-ph↗
arXiv2011

Infrared problem for the Nelson model on static space-times

We consider the Nelson model with variable coefficients and investigate the problem of existence of a ground state and the removal of the ultraviolet cutoff. Nelson models with variable coefficients arise when one replaces in the usual Nelson model the flat Minkowski metric by a static metric, allowing also the boson mass to depend on position. A physical example is obtained by quantizing the Klein-Gordon equation on a static space-time coupled with a non-relativistic particle. We investigate the existence of a ground state of the Hamiltonian in the presence of the infrared problem, i.e. assuming that the boson mass tends to 0 at infinity.

math-ph↗
arXiv2010

Absence of ground state for the Nelson model on static space-times

We consider the Nelson model on some static space-times and investigate the problem of absence of a ground state. Nelson models with variable coefficients arise when one replaces in the usual Nelson model the flat Minkowski metric by a static metric, allowing also the boson mass to depend on position. We investigate the absence of a ground state of the Hamiltonian in the presence of the infrared problem, i.e. assuming that the boson mass $m(x)$ tends to $0$ at spatial infinity. Using path space techniques, we show that if $m(x)\leq C |x|^{-μ}$ at infinity for some $C>0$ and $μ>1$ then the Nelson Hamiltonian has no ground state.

math-ph↗
arXiv2009

Positive energy quantization of linear dynamics

The abstract mathematical structure behind the positive energy quantization of linear classical systems is described. It is separated into 3 stages: the description of a classical system, the algebraic quantization and the Hilbert space quantization. 4 kinds of systems are distinguished: neutral bosonic, neutral bosonic, charged bosonic and charged fermionic. The formalism that is described follows closely the usual constructions employed in quantum physics to introduce non-interacting quantum fields.

math-ph↗
arXiv2009

Spectral and scattering theory of charged $P(φ)_2$ models

We consider in this paper space-cutoff charged $P(φ)_{2}$ models arising from the quantization of the non-linear charged Klein-Gordon equation: \[ (\p_{t}+ıV(x))^{2}ϕ(t, x)+ (-Δ_{x}+ m^{2})ϕ(t,x)+ g(x)\p_{\overline{z}}P(ϕ(t,x), \overlineϕ(t,x))=0, \] where $V(x)$ is an electrostatic potential, $g(x)\geq 0$ a space-cutoff and $P(λ, \overlineλ)$ a real bounded below polynomial. We discuss various ways to quantize this equation, starting from different CCR representations. After describing the construction of the interacting Hamiltonian $H$ we study its spectral and scattering theory. We describe the essential spectrum of $H$, prove the existence of asymptotic fields and of wave operators, and finally prove the {\em asymptotic completeness} of wave operators. These results are similar to the case when V=0.

math-ph↗
arXiv2008

Spectral and scattering theory for some abstract QFT Hamiltonians

We introduce an abstract class of bosonic QFT Hamiltonians and study their spectral and scattering theories. These Hamiltonians are of the form $H=\d\G(ω)+ V$ acting on the bosonic Fock space $\G(\ch)$, where $ω$ is a massive one-particle Hamiltonian acting on $\ch$ and $V$ is a Wick polynomial $\Wick(w)$ for a kernel $w$ satisfying some decay properties at infinity. We describe the essential spectrum of $H$, prove a Mourre estimate outside a set of thresholds and prove the existence of asymptotic fields. Our main result is the {\em asymptotic completeness} of the scattering theory, which means that the CCR representations given by the asymptotic fields are of Fock type, with the asymptotic vacua equal to the bound states of $H$. As a consequence $H$ is unitarily equivalent to a collection of second quantized Hamiltonians.

math-ph↗
arXiv2008

Spectral and scattering theory for space-cutoff $P(φ)_{2}$ models with variable metric

We consider space-cutoff $P(φ)_{2}$ models with a variable metric of the form \[ H= \d\G(ω)+ \int_{\rr}g(x):P(x, φ(x)):\d x, \] on the bosonic Fock space $L^{2}(\rr)$, where the kinetic energy $ω= h^{\12}$ is the square root of a real second order differential operator \[ h= Da(x)D+ c(x), \] where the coefficients $a(x), c(x)$ tend respectively to 1 and $m_{\infty}^{2}$ at $\infty$ for some $m_{\infty}>0$. The interaction term $\int_{\rr}g(x):P(x, φ(x)):\d x$ is defined using a bounded below polynomial in $λ$ with variable coefficients $P(x, λ)$ and a positive function $g$ decaying fast enough at infinity. We extend in this paper the results of \cite{DG} where $h$ had constant coefficients and $P(x, λ)$ was independent of $x$. We describe the essential spectrum of $H$, prove a Mourre estimate outside a set of thresholds and prove the existence of asymptotic fields. Our main result is the {\em asymptotic completeness} of the scattering theory, which means that the CCR representation given by the asymptotic fields is of Fock type, with the asymptotic vacua equal to bound states of $H$. As a consequence $H$ is unitarily equivalent to a collection of second quantized Hamiltonians. An important role in the proofs is played by the {\em higher order estimates}, which allow to control powers of the number operator by powers of the resolvent. To obtain these estimates some conditions on the eigenfunctions and generalized eigenfunctions of $h$ are necessary. We also discuss similar models in higher space dimensions where the interaction has an ultraviolet cutoff.

math-ph↗