  We introduce "local uncertainty relations" in thermal many body systems.
Using these relations, we derive basic bounds. These results include the
demonstration of universal non-relativistic speed limits (regardless of
interaction range), bounds on acceleration or force/stress, acceleration or
material stress rates, transport coefficients (including the diffusion constant
and viscosity), electromagnetic or other gauge field strengths, correlation
functions of arbitrary spatio-temporal derivatives, Lyapunov exponents, and
thermalization times. We further derive analogs of the Ioffe-Regel limit. These
bounds are relatively tight when compared to various experimental data. In the
$\hbar \to 0$ limit, all of our bounds either diverge (e.g., the derived speed
and acceleration limit) or vanish (as in, e.g., our viscosity and diffusion
constant bounds). Our inequalities hold at all temperatures and, as
corollaries, imply general power law bounds on response functions at both
asymptotically high and low temperatures. Our results shed light on how
apparent nearly instantaneous effective "collapse" to energy eigenstates may
arise in macroscopic interacting many body quantum systems. We comment on how
random off-diagonal matrix elements of local operators (in the eigenbasis of
the Hamiltonian) may inhibit their dynamics.
