# Cast Caldeira-Leggett master equation to Lindblad form

Consider a Brownian motion particle, whose motion is described by

$$\frac{d}{dt}\rho_{S}=-\frac{i}{\hbar}[H_{S},\rho_{S}]+\sum_{i,j}a_{i,j}(F_{i}\rho_{S}F_{j}^{\dagger}-\frac{1}{2}\{F_{j}^{\dagger}F_{i},\rho_{S}\})$$ (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.