Re: Hamiltonian vs. Energy

Chris H. Fleming schrieb:
On Mar 11, 1:17 pm, Arnold Neumaier <Arnold.Neuma...@xxxxxxxxxxxx>
wrote:

The correct quantization of dissipative systems is in terms of
Lindblad master equations, or, in more mathematical terms, completely
positive semigroups.

The Lagrangian and Hamiltonian approach is most certainly still
applicable.
Only one must model the dissipative environment and then trace out
it's degrees of freedom from the master equation. When that is done,
Lindblad terms usually only arise in some kind of bandwidth limit
where the system energies are much less than the environment energies
(a high temperature reservoir for instance).

If you model the dissipaticve environment explicitly, you have a
bigger conservative system, not a dissipative system. Of course, the conservative system is Hamiltonian, but it does not describe the
dissipative system alone. Already a simple damped harmonic oscillator becomes a huge and unwieldy dynamical system which is no longer equivalent to the damped harmonic oscillator, but includes unwanted memory terms.

Wen you contract it to the degrees of freedoms of the original system, you get an integro-differential equation with memory, which is no longer
described by a Hamiltonian or Lagrangian framework.

If you then remove the memory by employing the Markov approximation
you get again a differential equation, which defines the Lindblad (or, classicallally, the Focker-Planck) dynamics. Aghain, this is no longer
described by a Hamiltonian or Lagrangian framework.


One can rewrite dissipative systems in a way that they look
conservative, at the expense of destroying the physical interpretation
of the Hamiltonian. There are quite a number of publications in
this direction. But I haven't seen any treatment where this
pseudo-conservative reformulation produced anything of value beyond
what was already put into the description.


Arnold Neumaier

.