Imaginary-time propagator, centroid

Before proceeding, it is necessary to define a centroid-constrained imaginary-time propagator [3] (i.e., the correlation function of quantum path fluctuations with respect to the position of the centroid variable... 

After obtaining the closed-form imaginary-time expression [e.g., Eq. (3.3)], one can continue the optimized LHO expression analytically for the centroid-constrained propagator C<.(t, q ) by replacing t with it and then use the resulting expression in, for example, Eq. (3.3). In the effective harmonic theory, the real-time version of Q(t, qj for each value of the centroid is given by... 

---