Conservation laws arising in the study of forward-forward Mean-Field Games

We consider forward-forward Mean Field Game (MFG) models that arise in numerical approximations of stationary MFGs. First, we establish a link between these models and a class of hyperbolic conservation laws as well as certain nonlinear wave equations. Second, we investigate existence and long-time behavior of solutions for such models

Constraints on neutrino decay lifetime using long-baseline charged and neutral current data

We investigate the status of a scenario involving oscillations and decay for charged and neutral current data from the MINOS and T2K experiments. We first present an analysis of charged current neutrino and anti-neutrino data from MINOS in the framework of oscillation with decay and obtain a best fit for non-zero decay parameter $\alpha_3$. The MINOS charged and neutral current data analysis results in the best fit for $|\Delta m_{32}^2| = 2.34\times 10^{-3}$~eV$^2$, $\sin^2 \theta_{23} = 0.60$ and zero decay parameter, which corresponds to the limit for standard oscillations. Our combined MINOS and T2K analysis reports a constraint at the 90\% confidence level for the neutrino decay lifetime $\tau_3/m_3 > 2.8 \times 10^{-12}$~s/eV. This is the best limit based only on accelerator produced neutrinos

Back to Parmenides

After a brief introduction to issues that plague the realization of a theory of quantum gravity, I suggest that the main one concerns a quantization of the principle of relative simultaneity. This leads me to a distinction between time and space, to a further degree than that present in the canonical approach to general relativity. With this distinction, one can make sense of superpositions as interference between alternative paths in the relational configuration space of the entire Universe. But the full use of relationalism brings us to a timeless picture of Nature, as it does in the canonical approach (which culminates in the Wheeler-DeWitt equation). After a discussion of Parmenides and the Eleatics' rejection of time, I show that there is middle ground between their view of absolute timelessness and a view of physics taking place in timeless configuration space. In this middle ground, even though change does not fundamentally exist, the illusion of change can be recovered in a way not permitted by Parmenides. It is recovered through a particular density distribution over configuration space which gives rise to 'records'. Incidentally, this distribution seems to have the potential to dissolve further aspects of the measurement problem that can still be argued to haunt the application of decoherence to Many-Worlds quantum mechanics. I end with a discussion indicating that the conflict between the conclusions of this paper and our view of the continuity of the self may still intuitively bother us. Nonetheless, those conclusions should be no more challenging to our intuition than Derek Parfit's thought experiments on the subject.Comment: 25 pages, 1 figure. Winner of the essay contest: "Space-time after quantum gravity" (University of Illinois and Universit\'e de Geneve). To be published in special editio

Atom-field transfer of coherence in a two-photon micromaser assisted by a classical field

We investigate the transfer of coherence from atoms to a cavity field initially in a statistical mixture in a two-photon micromaser arrangement. The field is progressively modified from a maximum entropy state (thermal state) towards an almost pure state (entropy close to zero) due to its interaction with atoms sent across the cavity. We trace over the atomic variables, i.e., the atomic states are not collapsed by a detector after they leave the cavity. We find that by applying an external classical driving field it is possible to substantially increase the field purity without the need of previously preparing the atoms in a superposition of their energy eigenstates. We also discuss some of the nonclassical features of the resulting field.Comment: 10 pages, 7 figures, LaTe

The Hessian Riemannian flow and Newton's method for Effective Hamiltonians and Mather measures

Effective Hamiltonians arise in several problems, including homogenization of Hamilton--Jacobi equations, nonlinear control systems, Hamiltonian dynamics, and Aubry--Mather theory. In Aubry--Mather theory, related objects, Mather measures, are also of great importance. Here, we combine ideas from mean-field games with the Hessian Riemannian flow to compute effective Hamiltonians and Mather measures simultaneously. We prove the convergence of the Hessian Riemannian flow in the continuous setting. For the discrete case, we give both the existence and the convergence of the Hessian Riemannian flow. In addition, we explore a variant of Newton's method that greatly improves the performance of the Hessian Riemannian flow. In our numerical experiments, we see that our algorithms preserve the non-negativity of Mather measures and are more stable than {related} methods in problems that are close to singular. Furthermore, our method also provides a way to approximate stationary MFGs.Comment: 24 page