@gandalf61 why not write this into an answer?

I notice perhaps this question is a non-question. I think for a sphere, the electric field inside is just zero, something which can be proven precisely this way. And for more complicated geometries, the condition of constant charge may fail (or if I impose this to be true, then this automatically implies the field is 0?)

What you're saying is essentially effective field theory

Please check carefully what they talk about. It has been experimentally shown (in fact, in the very paper linked by PM2Ring, but also previously) that group velocities can exceed the speed of light, without violating SR.

Can you link the source about this saw tooth pattern? Im an amateur at this stuff and would like to learn a bit more.

@Filippo I seem to have misunderstood/misinterpreted your question just as Giorgio did, as my sign is wrong. But nonetheless, even with a positive sign in front of the gradient, it leads to unphysical states. (Or there simply may never be a stable state.)

That's not what you said. You claim: " If the system is prepared in the state x∈S, then ∇ρ(x) is the direction in which the system is most likely moving". I.e. see my answer soon TM

The ground state is a gaussian (centered around the minimum). The squared is thus also a gaussian. In particular, it does not evolve in time.

The statement is wrong. Consider harmonic oscillator in ground state. S being R

IBP in the first term of (3) twice gives you the result. Note that the second one doesnt need IBP (or the fact its an integral) since you actually use * as an operator on the algebra of operators. Thats why the order of Psi and H also change in your second method (IBP = integration by parts)

I dont understand why this is closed. Cant vote to open though. But what prevents you from constructing the rhs by definition? Even if you want a to be linear, just check for linearity and you are done.

The source [1] confuses me. If two systems are in detailed balance, or thermodynamical equilibrium, entropy does NOT increase. So the calculation with the sun radiatiating and our earth's surface having constant temperature does not seem to work out in such a way. But then again, it also states our earth is (and this is true) a non-equilibrium system.

You are correct. But is this a formal series you are trying to do? Since the radius of convergence is not infinity for square root. So the other expansion "has nothing to do with the original lagrangian"