[en] We present the construction of an infinite dimensional Banach manifold of quantum mechanical states on a Hilbert space H using different types of small perturbations of a given Hamiltonian H₀. We provide the manifold with a flat connection, called the exponential connection, and comment on the possibility of introducing the dual mixture connection

[en] In-between the limits fixed by the available resources and the physical susceptibility of the geosphere, Physical Sciences have to play a part in building our common future, together with other sciences, as we focused during the first session of the school. During the second session of the school we considered the Sustainable Energy challenge taken up on a world-wide basis, giving rise to very diverse energy scenarios. This third session will start with a focus on the so-called EROI index (Energy Return on (Energy) Invested) of energy systems. Case studies point to the various perimeters that can be considered when calculating the Energy Invested. Besides, societies use a mix of energy sources having different EROIs indices, and the ways these sources contribute to a societal EROI will be addressed. A second part of the session will be devoted to a number of key-factors entering the calculation of EROIs: availability of resources, capacity of energy storage, structure and efficiency of energy grids. As an example, renewable energies have a high potential, but capturing, converting, storing and distributing these energies require important amounts of mineral resources. The associated amount of primary energy needed has to be evaluated. Besides entropy limits, technical, social and economic limits which make our use of scarce metals highly dispersive. These constraints have to be taken into account when designing future energy scenarios. A third part will be concerned with economic modeling incorporating biophysical constraints, such as energy availability and entropy production. Various recent approaches aiming at generating models with an intrinsic dynamics will be discussed. The third edition of this workshop at the Ecole de Physique des Houches aims at exposing and exploring the potential contributions of Physics to the energy challenge. Numerous roads of ongoing research will cross and cross-fertilize in the program, profiting a second objective of the workshop: design the knowledge basis for young physicists and engineers who will implement the energy transition. This vast scientific program cannot and will not be exhausted in a one week seminar. On the contrary, the workshop Energy Scenarios: Which Research in Physics? Is only the third one in an expected longer series. This document brings together the available presentations (slides): 1 - After Paris COP21: The New Climate Policy Momentum (Jean-Pascal van Ypersele); 2 - Challenges for climate science after Paris (Robert Vautard); 3 - The net EROI for solar PV: a case study for Spain (Pedro A. Prieto); 4 - History, applications, numerical values and problems with the calculation of EROI (Energy Return on energy) Investment (Charles A. S. Hall); 5 - Renewable Systems from pre-modern periods: intermittency issues and resources management (Mathieu Arnoux); 6 - Estimations of very long-term time series of fossil fuels global EROI (Victor Court, Florian Fizaine); 7 - Examining The Relation between Quality of Life and Biophysical vs Economic Conditions (Jessica G. Lambert); 8 - Suppose we agree how to calculate EROIs... so/now what? (Carey W. King); 9 - Grid, DG and demand control: operation vs. evolution (N. Hadjsaid); 10 - The use of intermittent sources for electricity production in Germany, Sweden, Europe (F. Wagner); 11 - Energy transmission networks at different scales: Method, results and limits (Johannes Dorfner); 12 - The EROIs of Power Plants - why are they so different? You never find two publications with the same results... (Daniel Weissbach); 13 - Overview of existing and innovative batteries, impact of the storage on the renewable electricity life cycle (Fabien Perdu); 14 - Power to gas to power, Solution or dead lock? (Georges Sapy); 15 - Smart Grids: A high-tech [r]evolution to facilitate the energy transition (Andrei Nekrassov); 16 - Natural Resources in a Monetary Macro-dynamics Work in progress (Gael Giraud); 17 - Hybrid LCA-I/O approach to estimate EROI, Application to wind power (Cyril Francois); 18 - Mineral resources availability and recycling limits: A constraint for ambitious and sustainable energy scenarios (Philippe Bihouix); 19 - Biophysical economics: Essential for the second half of the age of oil (Charles A. S. Hall, Kent Klitgaard); 20 - Transition Engineering Scenarios, What EROI tells us about the future (Susan Krumdieck); 21 - Dynamical Economic model in a finite world (Christophe Goupil, Eric Herbert, Yves D'Angelo, Quentin Couix); 22 - Long-term endogenous economic growth and energy transitions (Victor Court, Pierre-Andre Jouvet, Fredric Lantz); 23 - Growth, productivity and oil history - The Gordon knot (Michel Lepetit); 24 - Taking climate change seriously (Ivar Ekeland); 25 - Dedicating multi-scale approaches to low carbon prospective studies (Nadia Maizi); 26 - EROI and the zero lower bound (Michel Lepetit); 27 - Summing it up: A couple of implications for collective decision (Jean-Marc Jancovici)