05-27-2016, 09:15 PM

Ok, the von Neumann entropy \(-Tr \rho \ln \rho\) is anyway zero for a pure state. But the formula I have given there is what you get if you if you make a measurement in such a pure state, this would reduce the pure state into a density matrix

\[ \rho \to \sum \rho_a |a><a| \] with \(\rho_a\) being the probability of measuring a. And for this density matrix the von Neumann entropy is already

\[ S = \sum \rho_a \ln \rho_a.\] See https://en.wikipedia.org/wiki/Von_Neumann_entropy And in the particular case of position measurement this sum becomes an integral (which opens another can of worms, because one needs a prior, a measure on the space to define the entropy. For an Euclidean space one can, of course, use the standard measure \(d^3 x\) for this.) So, what I have given there is a particular variant of the von Neumann entropy, that one would obtain as the prediction for the result of a position measurement. Without reducing it to the pure state defined by the measurement result, which we don't know before.

So, there is a quite simple and well-defined connection between the two formulas. The one I have used is the one Valentini has used to define the relative entropy for dBB states outside the equilibrium. He has proven, with this notion of relative entropy, his subquantum H-theorem that the dBB probability distribution will approach quantum equilibrium. I wanted to explain this subquantum H-theorem, because this is what was really about something where dBB theory is different from QM, outside the quantum equilibrium.

I prefer not to talk about what I think about Motl, I do not want to have any personal attacks here, so I will not attack others too.

\[ \rho \to \sum \rho_a |a><a| \] with \(\rho_a\) being the probability of measuring a. And for this density matrix the von Neumann entropy is already

\[ S = \sum \rho_a \ln \rho_a.\] See https://en.wikipedia.org/wiki/Von_Neumann_entropy And in the particular case of position measurement this sum becomes an integral (which opens another can of worms, because one needs a prior, a measure on the space to define the entropy. For an Euclidean space one can, of course, use the standard measure \(d^3 x\) for this.) So, what I have given there is a particular variant of the von Neumann entropy, that one would obtain as the prediction for the result of a position measurement. Without reducing it to the pure state defined by the measurement result, which we don't know before.

So, there is a quite simple and well-defined connection between the two formulas. The one I have used is the one Valentini has used to define the relative entropy for dBB states outside the equilibrium. He has proven, with this notion of relative entropy, his subquantum H-theorem that the dBB probability distribution will approach quantum equilibrium. I wanted to explain this subquantum H-theorem, because this is what was really about something where dBB theory is different from QM, outside the quantum equilibrium.

I prefer not to talk about what I think about Motl, I do not want to have any personal attacks here, so I will not attack others too.