  The effectiveness of the variational approach a la Feynman is proved in the
spin-boson model, i.e. the simplest realization of the Caldeira-Leggett model
able to reveal the quantum phase transition from delocalized to localized
states and the quantum dissipation and decoherence effects induced by a heat
bath. After exactly eliminating the bath degrees of freedom, we propose a
trial, non local in time, interaction between the spin and itself simulating
the coupling of the two level system with the bosonic bath. It stems from an
Hamiltonian where the spin is linearly coupled to a finite number of harmonic
oscillators whose frequencies and coupling strengths are variationally
determined. We show that a very limited number of these fictitious modes is
enough to get a remarkable agreement, up to very low temperatures, with the
data obtained by using an approximation-free Monte Carlo approach, predicting:
1) in the Ohmic regime, a Beretzinski-Thouless-Kosterlitz quantum phase
transition exhibiting the typical universal jump at the critical value; 2) in
the sub-Ohmic regime ($s \leq 0.5$), mean field quantum phase transitions, with
logarithmic corrections for $s=0.5$.
