Multi-instantons in quantum mechanics (QM)

In general, a linear combination of instanton solutions is not a solution of the imaginary-time equations of motion, because the equations not linear. Moreover, in quantum mechanics (QM), all solutions of the classical equations can depend only on one time collective coordinate (in this respect, in field theory, the situation is different). However, a linear combination of largely separated instantons (a multi-instanton configuration) renders the action almost stationary, because each instanton solution differs, at large distances, from a constant solution by only exponentially small corrections (in field theory this is only true if the theory is massive). A situation where multi-instantons play a role is provided by large order behaviour estimates of perturbation theory for potentials with degenerate minima. When one starts from a situation in which the minima are almost degenerate, one obtains, in the degenerate limit, a contribution of the superposition of two, infinitely separated, instantons, but with an infinite multiplicative coefficient. Indeed, in this limit, the fluctuations which tend to change the distance between the instanton and the anti-instanton induce a vanishingly small variation of the action. To correctly determine the limit, one has to introduce a second collective coordinate which describes these fluctuations. The determination, at leading order, of all many-instanton contributions has led to conjecture the exact form of the semi-classical expansion for potentials with degenerate minima, generalizing the exact Bohr-Sommerfeld quantization condition.

Simulating magnetic monopole-defect dynamics

We present simulations of one magnetic monopole interacting with multiple magnetic singularities. Three-dimensional plots of the energy density are constructed from explicit solutions to the Bogomolny equation obtained by Blair, Cherkis, and Durcan. Animations follow trajectories derived from collective coordinate mechanics on the multi-centered Taub-NUT monopole moduli space. We supplement our numerical results with a complete analytic treatment of the single-defect case.

On the evolution of compact binary black holes

Based on the consideration of potential energy of the di-black-hole as a function of mass asymmetry (transfer) collective coordinate, the possibility of matter transfer between the black holes in a binary system is investigated. The sensitivity of the calculated results is studied to the value of the total mass of binary system. The conditions for the merger of two black holes are analyzed in the context of gravitational wave emission.

Individuals physically interacting in a group rapidly coordinate their movement by estimating the collective goal

How can a human collective coordinate, for example to move a banquet table, when each person is influenced by the inertia of others who may be inferior at the task? We hypothesized that large groups cannot coordinate through touch alone, accruing to a zero-sum scenario where individuals inferior at the task hinder superior ones. We tested this hypothesis by examining how dyads, triads and tetrads, whose right hands were physically coupled together, followed a common moving target. Surprisingly, superior individuals followed the target accurately even when coupled to an inferior group, and the interaction benefits increased with the group size. A computational model shows that these benefits arose as each individual uses their respective interaction force to infer the collective’s target and enhance their movement planning, which permitted coordination in seconds independent of the collective’s size. By estimating the collective’s movement goal, its individuals make physical interaction beneficial, swift and scalable.