G. Burdet

Time-dependent quantum systems and chronoprojective geometry

Chronoprojective transformations in the framework of five-dimensional Schrödinger formalism are used to construct the solution of the Schrödinger equation with a time-dependent harmonic potential from the solution of a free Schrödinger equation.

Cartan structures on Galilean manifolds: The chronoprojective geometry

A new geometry is constructed over Galilean manifolds expressing the compatibility requirement between the conformal equivalence notion of two Galilean structures and the projective equivalence notion of two affine connections. It is shown that it is the very geometry of the Newtonian cosmology (chronoprojective flatness is equivalent to isotropy of Newtonian cosmological models); moreover, it also explains various ‘‘accidental’’ symmetries in classical mechanics.

Gravitational waves without gravitons

The radiation connection induced on an isotropic hypersurface of a Lorentz manifold is described. Consequences for the energy tensor in Einstein field equation are analyzed. A cosmological interpretation is proposed.

Exact solutions of the nonlinear Schrödinger equation in time-dependent parabolic density profiles

By using covariance properties of an extended Schrödinger formalism, exact soliton-like solutions of the nonlinear Schrödinger equation in time-dependent inhomogeneous media (parabolic density profiles) are constructed.

Chronoprojective invariance of the five-dimensional Schrodinger formalism

The geometry which underlies the invariance properties of the five-dimensional Schrodinger formalism is presented as a reduction of the O(5,2) conformal geometry. Various applications are given.

Spéculation sur la cosmologie quantique primordiale

An old-fashioned axiomatic formalism for the quantization of the gravitational field is adapted to a recently introduced notion of cosmological gravitational wave. Dynamical equations for the quantum field are proposed. An outline for the primordial cosmological scenario is suggested.

LORENTZIAN MANIFOLDS ADMITTING ISOTROPIC HYPERSURFACES SOLUTIONS OF EINSTEIN’S FIELD EQUATIONS

All spaces solutions of Einstein’s field equations, admitting an isotropic (null) hypersurface (hereafter referred under the acronymus “ishyps”) are determined in a geometric way. We consider in more details two sub-cases, the generalized Robinson-Bertotti, and the pp-waves spaces.

Space-time metrics inducing the radiation connection

It is shown that Lorentzian manifolds giving rise to the radiation connection, i.e. such that the canonical connection induces a unique connection on an isotropic hypersurface, are of type D or conformally flat in Petrov classification, generalizing the Robinson-Bertotti electromagnetic universe with cosmological constant. Under an holonomy hypothesis it is deduced that the underlying space-time manifold is a direct product of two two-dimensional spaces, one of them being space-like, and the other one time-like.

A classical analogue of Yukawa coupling: Presymplectic formalism

A unified treatment of Yang-Mills and Higgs fields in classical gauge theory is carried out in a general relativistic context. A presymplectic formalism for a spinless test particle dwelling in this background geometry is described. The mass of this particle is found to depend specifically upon its generalized isospin and the Higgs field. This mass generating process is very much reminiscent of the so-called Yukawa coupling in the (electro-weak) standard model. The space of motions (phase space) is constructed together with a set of generalized Wong equations. Comparison with the Marsden-Weinstein symplectic reduction procedure is achieved.

Structural invariance of the Schrödinger equation and chronoprojective geometry

We describe an extension of the chronoprojective geometry and show how its automorphisms are related to the invariance properties of the Schrodinger equation describing a quantum test particle in any Newton–Cartan structure.