Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1041the Liouville equation (6.64) can be expanded asi ∂ t 〈x + |ρ(t)|x − 〉 = (6.66)]}{−[∂ 2x 2 2m +− ∂x 2 −+ [V (x + ) − V (x − )] 〈x + |ρ(t)|x − 〉,while the Wigner function W (p, x, t) is now given byi ∂ t W (x, y, t) = H o W (x, y, t),withH o = 1 m p xp y + V (x + 1 2 y) − V (x − 1 y), (6.67)2and p x = −i∂ x , p y = −i∂ y .The new Hamiltonian H o (6.67) may be get from the corresponding LagrangianL o = mẋẏ − V (x + 1 2 y) + V (x − 1 y). (6.68)2In this way, Vitiello concluded that the density matrix and the Wignerfunction formalism required, even in the conservative case (with zero mechanicalresistance γ), the introduction of a ‘doubled’ set of coordinates,x ± , or, alternatively, x and y. One may understand this as related to theintroduction of the ‘couple’ of indices necessary to label the density matrixelements (6.66).Let us now consider the case of the particle interacting with a thermalbath at temperature T . Let f denote the random force on the particle atthe position x due to the bath. The interaction Hamiltonian between thebath and the particle is written asH int = −fx. (6.69)Now, in the Feynman–Vernon formalism (see [Feynman (1972)]), theeffective action A[x, y] for the particle is given byA[x, y] =∫ tfwith L o defined by (6.68) ande i I[x,y] = 〈(e − i R t ft it iL o (ẋ, ẏ, x, y) dt + I[x, y],f(t)x −(t)dt ) − (e i R t ft if(t)x +(t)dt ) + 〉, (6.70)where the symbol 〈.〉 denotes the average with respect to the thermal bath;‘(.) + ’ and ‘(.) − ’ denote time ordering and anti–time ordering, respectively;the coordinates x ± are defined as in (6.65). If the interaction between the