Cast Caldeira-Leggett master equation to Lindblad form Consider a Brownian motion particle, whose motion is described by
$\frac{d}{dt}\rho_{S}=-\frac{i}{\hbar}[H_{S},\rho_{S}] - \frac{i \gamma}{2\hbar}[x p + p x, \rho_S]+\sum_{i,j}a_{i,j}(F_{i}\rho_{S}F_{j}^{\dagger}-\frac{1}{2}\{F_{j}^{\dagger}F_{i},\rho_{S}\})$   (1)
where the matrix $a$ is
$(a_{i,j})=\begin{pmatrix} 
4m\gamma k_{B}T/\hbar^{2}  & -i\gamma/\hbar \\
  i\gamma/\hbar           & 0
\end{pmatrix}
$   (2)
and $F_{1}=x$, $F_{2}=p$.
Obviously, the above master equation cannot be brought into Lindblad form since matrix $a$ is negative.
However, one new term $a_{2,2}=\gamma/4mk_{B}T$, according to the book The Theory of Open Quantum Systems written by Breuer, can be added to make matrix $a$ non-negative in the high temperature limit.(In this limit, the added new term is very small). After this adding, the Brownian master equation can be written in Lindblad form.
So my question is: Is there any reason to solidify such an adding? Since Eq(1) is derived from firsthand principle, I simply cannot see whether it's tenable to add this new term.
 A: Quoting Breuer, "One might ask the question of whether the generator of the Brownian motion
master equation can be brought into Lindblad form. The answer to this
question is negative.", so in principle that's the answer, it is not a Lindblad equation.
That being said, it's in some sense almost a Lindblad equation. In equation 3.396 we have the assumption that $\hbar \omega_0 \ll \mathrm{Min}\{\hbar\Omega, 2 \pi k_B T\}$. Usually the cutoff frequency is assumed to be very high, especially since that determines how fast the bath correlation decays, therefore the more difficult assumption to cover is $\hbar \omega_0 \ll  2 \pi k_B T$, which is equivalent to $\hbar/(k_B T) \ll \omega_0^{-1}$. If this assumption is granted then we can see that if you add the term you suggested then it suddenly becomes a Lindblad equation, therefore, although the equation is not a Lindblad equation, if the new term is added, the error will be small, and it will be Lindblad. "Is there any reason to solidify such an adding?" The reason is because this term is small basically, with (practically) the only new requirement being that you assume high temperature.
Also interesting to note is that these equations with negative rates appear also in exactly solvable models, the difference being that in those models there is a transient time dependence in the dissipator, so although the long time limit of the dissipator has negative rates (and wouldn't be CPTP), the whole evolution is CPTP.
