Scaled Caldeira-Leggett Dynamics: Deterministic Trajectories and the Classical Limit
The dynamics of an open quantum system in the high-temperature regime of the Caldeira-Leggett model using a Bohmian language is investigated. By expressing the density matrix in polar form, generalized continuity and Hamilton-Jacobi equations are obtained. Thermal effects appear explicitly in the former, while they influence the latter only indirectly through an effective potential. Remarkably, the effective potential does not vanish in the classical limit and cannot, in general, be decomposed into simple additive quantum and thermal contributions. Instead, it retains a nontrivial structure leading to a residual temperature dependence in the resulting dynamics. As a consequence, the resulting temperature-dependent-classical trajectories remain deterministic even in the presence of a thermal environment. This behavior contrasts with the stochastic dynamics observed in the Langevin description and reflects the ensemble-based nature of the present approach. To further explore the quantum-to-classical transition, a scaled version of the Caldeira-Leggett equation is introduced by scaling both the density matrix and Planck constant through a quantumness parameter. This parameter takes the value one in the fully quantum regime and smoothly approaches zero in the classical limit. Applications to Gaussian states demonstrate a gradual suppression of coherence and interference, governed jointly by thermal effects and the transition parameter.