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Christian Gérard

Publications and source records attributed to Christian Gérard.

At least 19 recordsLinked to original sources
arXiv2026

IR-fixed Euclidean vacuum for linearized gravity on de Sitter space

We consider the Euclidean vacuum for linearized gravity on the global de Sitter space, obtained from the Euclidean Green's function on the 4-sphere. We use the notion of Calderón projectors to recover a quantum state for the Lorentzian theory on de Sitter space. We show that while the state is gauge invariant and Hadamard, it is not positive on the whole of the phase space. We show however that a suitable modification at low energies yields a well-defined Hadamard state on global de Sitter space.

math-ph↗
arXiv2024

Mourre theory for analytically fibered operators revisited

About 25 years ago our article "Mourre theory for analytically fibered operators" was published in J. of Functional Analysis. This article proposed a general construction of a conjugate operator for a wide class of self-adjoint analytically fibered hamiltonians, provided that one accepts a more accurate notion of threshold. It is only recently that Olivier Poisson mentionned us a problem with the statement that H 0 $\in$ C $\infty$ (A I). Actually even H 0 $\in$ C 2 (A I) or H 0 $\in$ C 1+0 (A I) , which is crucial for the full application of Mourre theory, is problematic with our initial construction. However the statement and the construction can be modified in order to make work all the theory. This is explained here.

math.SP↗
arXiv2023

Wick rotation of linearized gravity in Gaussian time and Calderón projectors

Motivated by the quantization of linearized gravity, we consider gauge-fixed linearized Einstein equations and their Wick rotation near a Cauchy surface. We show that Calderón projectors for the Wick-rotated equations induce Hadamard bi-solutions on the Lorentzian level. On the other hand, we find smoothing obstructions to gauge-invariance and positivity conditions needed in quantization. These obstructions are primarily due to boundary terms arising in the Wick-rotated theory and depend on the boundary conditions.

math-ph↗
arXiv2023

Microlocal Analysis of Quantum Fields on Curved Spacetimes

These lecture notes give an exposition of microlocal analysis methods in the study of Quantum Field Theory on curved spacetimes. We concentrate on free fields and the corresponding quasi-free states and mainly on Klein-Gordon fields.Updated version with some errors corrected. Discussion of mode expansion method added.

math.AP↗
arXiv2022

Hadamard states for quantized Dirac fields on Lorentzian manifolds of bounded geometry

We consider Dirac equations on even dimensional Lorentzian manifolds of bounded geometry with a spin structure. For the associated free quantum field theory, we construct pure Hadamard states using global pseudodifferential calculus on a Cauchy surface. We also give two constructions of Hadamard states for Dirac fields for arbitrary spacetimes with a spin structure.(2nd revised version). Will appear in Rev. Math. Phys.

math.AP↗
arXiv2021

The Hartle-Hawking-Israel state on stationary black hole spacetimes

We consider a free quantized Klein-Gordon field in a spacetime (M, g) containing a stationary black hole, more precisely a spacetime with a stationary bifurcate Killing horizon in the sense of Kay and Wald. We prove the existence of the Hartle-Hawking-Israel ground state, which is a pure state on the whole spacetime whose restriction to the exterior of the black hole is a thermal state at Hawking temperature.We show that the HHI state is a Hadamard state and is the unique Hadamard extension of the above thermal state to the whole spacetime. We construct the HHI state by Wick rotation in Killing time coordinates, using the notion of the Calderon projector for elliptic boundary value problems.

math-ph↗
arXiv2021

The Unruh state for massless fermions on Kerr spacetime and its Hadamard property

We give a rigorous definition of the Unruh state in the setting of massless Dirac fields on slowly rotating Kerr spacetimes. In the black hole exterior region, we show that it is asymptotically thermal at Hawking temperature on the past event horizon. Furthermore, we demonstrate that in the union of the exterior and interior regions, the Unruh state is pure and Hadamard. The main ingredients are the Häfner-Nicolas scattering theory, new microlocal estimates for characteristic Cauchy problems and criteria on the level of square-integrable solutions.

math-ph↗
arXiv2020

The Feynman problem for the Klein-Gordon equation

We report on the well-posedness of the Feynman problem for the Klein-Gordon equation on asymptotically Minkowski spacetimes. The main result is the invertibility of the Klein-Gordon operator with Feynman conditions at infinite times. Furthermore, the inverse is shown to coincide with the Duistermaat-Hörmander Feynman parametrix modulo smoothing terms.

math.AP↗
arXiv2018

The massive Feynman propagator on asymptotically Minkowski spacetimes II

We consider the massive Klein-Gordon equation on short-range asymptotically Minkowski spacetimes. Extending our results in [GW1], we show that the Klein-Gordon operator with Feynman type boundary conditions at infinite times is invertible and that its inverse, called the Feynman inverse, satisfies the microlocal conditions of Feynman parametrices in the sense of Duistermaat and Hörmander. This supplements the recent work of Vasy with more explicit techniques.

math.AP↗
arXiv2018

On the Hartle-Hawking-Israel states for spacetimes with static bifurcate Killing horizons

We revisit the construction by Sanders [S1] of the Hartle-Hawking-Israel state for a free quantum Klein-Gordon field on a spacetime with a static, bifurcate Killing horizon and a wedge reflection. Using the notion of the Calder{ó}n projector for elliptic boundary value problems and pseudodifferential calculus on manifolds, we give a short proof of its Hadamard property.

math-ph↗
arXiv2017

Analytic Hadamard states, Calderón projectors and Wick rotation near analytic Cauchy surfaces

We consider the Klein-Gordon equation on analytic spacetimes with an analytic Cauchy surface. In this setting, we prove the existence of pure analytic Hadamard states. The proof is based on considering an elliptic operator obtained by Wick rotating the Klein-Gordon operator in a neighborhood of a Cauchy hypersurface. The Cauchy data of Hadamard two-point functions are constructed as Calderón projectors (suitably generalized if the hypersurface is non-compact) for the elliptic operator.

math-ph↗
arXiv2017

Hadamard states for the Klein-Gordon equation on Lorentzian manifolds of bounded geometry

We consider the Klein-Gordon equation on a class of Lorentzian manifolds with Cauchy surface of bounded geometry, which is shown to include examples such as exterior Kerr, Kerr-de Sitter and Kerr-Kruskal spacetimes. In this setup, we give an approximate diagonalization and a microlocal decomposition of the Cauchy evolution using a time-dependent version of the pseu-dodifferential calculus on Riemannian manifolds of bounded geometry. We apply this result to construct all pure regular Hadamard states (and associated Feynman inverses), where regular refers to the state's two-point function having Cauchy data given by pseudodifferential operators. This allows us to conclude that there is a one-parameter family of elliptic pseudodifferential operators that encodes both the choice of (pure, regular) Hadamard state and the underlying spacetime metric.

math-ph↗
arXiv2017

Hadamard states for the linearized Yang-Mills equation on curved spacetime

We construct Hadamard states for the Yang-Mills equation linearized around a smooth, space-compact background solution. We assume the spacetime is globally hyperbolic and its Cauchy surface is compact or equal $\rr^d$. We first consider the case when the spacetime is ultra-static, but the background solution depends on time. By methods of pseudodifferential calculus we construct a parametrix for the associated vectorial Klein-Gordon equation. We then obtain Hadamard two-point functions in the gauge theory, acting on Cauchy data. A key role is played by classes of pseudodifferential operators that contain microlocal or spectral type low-energy cutoffs. The general problem is reduced to the ultra-static spacetime case using an extension of the deformation argument of Fulling, Narcowich and Wald. As an aside, we derive a correspondence between Hadamard states and parametrices for the Cauchy problem in ordinary quantum field theory.

math-ph↗
arXiv2017

On the adiabatic limit of Hadamard states

We consider the adiabatic limit of Hadamard states for free quantum Klein-Gordon fields, when the background metric and the field mass are slowly varied from their initial to final values. If the Klein-Gordon field stays massive, we prove that the adiabatic limit of the initial vacuum state is the (final) vacuum state, by extending to the symplectic framework the adiabatic theorem of Avron-Seiler-Yaffe. In cases when only the field mass is varied, using a abstract version of the mode decomposition method we can also consider the case when the initial or final mass vanishes, and the initial state is either a thermal state or a more general Hadamard state.

math-ph↗
arXiv2017

Construction of Hadamard states by characteristic Cauchy problem

We construct Hadamard states for Klein-Gordon fields in a spacetime $M_{0}$ equal to the interior of the future lightcone $C$ from a base point $p$ in a globally hyperbolic spacetime $(M, g)$. Under some regularity conditions at future infinity of $C$, we identify a boundary symplectic space of functions on $C$, which allows to construct states for Klein-Gordon quantum fields in $M_{0}$ from states on the CCR algebra associated to the boundary symplectic space. We formulate the natural microlocal condition on the boundary state on $C$ ensuring that the bulk state it induces in $M_{0}$ satisfies the Hadamard condition. Using pseudodifferential calculus on the cone $C$ we construct a large class of Hadamard boundary states on the boundary with pseudodifferential covariances, and characterize the pure states among them. We then show that these pure boundary states induce pure Hadamard states in $M_{0}$.

math-ph↗