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The Caldeira-Leggett model for a one-dimensional system with coordinate$X$ and momentum$P$ is given by the Hamiltonian
$$\frac{P^2}{2m} + V(X) + \sum_i \frac 1 2 p_i^2 + \frac 1 2 \omega_i^2 \left(q_i - \frac{c_i X}{\omega_i^2} \right)^2$$
where the bath oscillator masses have been set to unity. The frequencies$\omega_i$ and couplings$c_i$ are chosen based on the type of bath (e.g. an ohmic bath or a Debye bath).
However, what if the system is two dimensional – i.e., we have coordinates$(X,Y)$ and momenta$(P_x, P_y)$. Will we just couple a separate set of oscillators to$X$ and to$Y$? And will these oscillators have the same$c_i$ and$\omega_i$? What if the environment is anisotropic in some way?
