  We discuss our recent study of local quantum mechanical uncertainty relations
in quantum many body systems. These lead to fundamental bounds for quantities
such as the speed, acceleration, relaxation times, spatial gradients and the
Lyapunov exponents. We additionally obtain bounds on various transport
coefficients like the viscosity, the diffusion constant, and the thermal
conductivity. Some of these bounds are related to earlier conjectures, such as
the bound on chaos by Maldacena, Shenker and Stanford while others are new. Our
approach is a direct way of obtaining exact bounds in fairly general settings.
We employ uncertainty relations for local quantities from which we strip off
irrelevant terms as much as possible, thereby removing non-local terms. To
gauge the utility of our bounds, we briefly compare their numerical values with
typical values available from experimental data. In various cases, approximate
simplified variants of the bounds that we obtain can become fairly tight, i.e.,
comparable to experimental values. These considerations lead to a minimal time
for thermal equilibrium to be achieved. Building on a conjectured relation
between quantum measurements and equilibration, our bounds, far more
speculatively, suggest a minimal time scale for measurements to stabilize to
equilibrium values.
