Ivancevic_Applied-Diff-Geom

1042 Applied Differential Geometry: A Modern Introductionbath and the coordinate x (6.69) were turned off, then the operator f ofthe bath would develop in time according to f(t) = e iHγt/ fe −iHγt/ , whereH γ is the Hamiltonian of the isolated bath (decoupled from the coordinatex). f(t) is then the force operator of the bath to be used in (6.70).The interaction I[x, y] between the bath and the particle has been evaluatedin [Srivastava et al. (1995)] for a linear passive damping due tothermal bath by following Feynman–Vernon and Schwinger [Feynman andHibbs (1965)]. The final result from [Srivastava et al. (1995)] is:I[x, y] = 1 2+ i2∫ tfdt [x(t)F retyt i∫ tf∫ tft it i(t) + y(t)F adv (t)]dtds N(t − s)y(t)y(s),where the retarded force on y, Fyret , and the advanced force on x, Fxadv , aregiven in terms of the retarded and advanced Green functions G ret (t − s)and G adv (t − s) byF rety (t) =∫ tft ids G ret (t − s)y(s), F advx (t) =∫ tft ixds G adv (t − s)x(s),respectively. In (6.71), N(t − s) is the quantum noise in the fluctuatingrandom force given by: N(t − s) = 1 2〈f(t)f(s) + f(s)f(t)〉.The real and the imaginary part of the action are given respectively byRe (A[x, y]) =∫ tfL = mẋẏ −[V (x + 1 2 y) − V (x − 1 ]2 y) + 1 [xFrety2t iL dt, (6.71)+ yFxadv ], (6.72)and Im (A[x, y]) = 1 ∫ tf∫ tfN(t − s)y(t)y(s) dtds. (6.73)2 t i t iEquations (6.71–6.73), are exact results for linear passive damping dueto the bath. They show that in the classical limit ‘ → 0’ nonzero y yields an‘unlikely process’ in view of the large imaginary part of the action implicitin (6.73). Nonzero y, indeed, may lead to a negative real exponent in theevolution operator, which in the limit → 0 may produce a negligiblecontribution to the probability amplitude. On the contrary, at quantumlevel nonzero y accounts for quantum noise effects in the fluctuating random