"You're stepping on dangerous grounds, young man."
- Berthold-Georg Englert
That's Berge's comment on some things in my project, during the second presentation that just took place yesterday.
Okay, okay. It's not as bad as it sounds. Anyone who knows Englert will know that he has killer eyes. But then again, anyone who knows him will also know that he is seldom harsh to students (at least undergraduates), even if he disagrees with what's being presented. In any case, he did say it with a tinge of humour in his voice (you know, the kind whereby a sagely grandmaster advises a young apprentice).
So indeed, he's a bit skeptical when I presented a master equation for quantum Brownian motion that has negative coefficients for the dissipators. The main issue he has with these negative coefficients is that they violate a sacred condition called complete positivity, which roughly implies that certain probabilities cannot be negative (naturally).
But you see, the situation is not as simple as stated above. The coefficients can be negative if, in a vague sense, they have been positive for sufficiently long periods prior to that. Specifically, there must be enough time when the coefficients are positive such that the eigenvalues of the matrix formed by these coefficients can be positive as well, even though certain matrix elements are negative. This will imply positive probabilities.
The thing is that this is a very delicate issue and difficult to evaluate for sure, which is his comment on "dangerous grounds". But then again, I didn't create this master equation which has negative coefficients. This comes from some published articles, and in fact this equation has been used by several authors.
Well, Yeo Ye is telling me to go ahead, because it's not too big an issue. Still, it's not very reassuring to know that my project (at least a part of it) is build on shaky grounds.
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