Mori–Zwanzig theory 601(s−iL)−^1 −(s−QiL)−^1 = (s−iL)−^1 (s−QiL−s+iL) (s−QiL)−^1= (s−iL)
− 1
(I−Q)iL(s−QiL)
− 1= (s−iL)−^1 PiL(s−QiL)−^1 , (15.7.14)so that
(s−iL)−^1 = (s−QiL)−^1 + (s−iL)−^1 PiL(s−QiL)−^1. (15.7.15)Inverting the Laplace transform of both sides, we obtain
eiLt= eQiLt+∫t0dτeiL(t−τ)PiLeQiLτ. (15.7.16)The second term in eqn. (15.7.8) can be evaluated by multiplying eqn. (15.7.16) on
the right byQiLA(0) to give
eiLtQiLA(0) = eQiLtQiLA(0) +∫t0dτeiL(t−τ)PiLeQiLτQiLA(0). (15.7.17)In the equation of motion dA/dt=iLAforA(t), the vectoriLAdrives the evolution.
We can therefore think ofiLAas a kind of general force driving the evolution ofA,
withiLA(0) being the initial value of this force. Indeed, the second component of this
vector is the initial physical force since ̇p=Ffrom Newton’s second law. The action of
QoniLA(0) projects the initial force onto a direction orthogonal toA. The evolution
operator exp(QiLt) acts as a classical propagator of a dynamics in which the forces
are orthogonal toA. Therefore,F(t)≡exp(QiLt)QiLA(0) is the time evolution of
the projected force in this orthogonal subspace. Of course, the propagator exp(QiLt)
and the true evolution operator exp(iLt) do not produce the same time evolution. The
dynamics generated by exp(QiLt) is generally not conservative and, therefore, not
straightforward to evaluate. However, we will see shortly that physically interesting
approximations are available for this dynamics in the case of a high frequency oscillator.
In order to complete the derivation of the GLE, we introduce the projected force
F(t) into eqn. (15.7.17) to obtain
eiLtQiLA(0) =F(t) +∫t0dτeiL(t−τ)PiLeQiLτQiLA(0)=F(t) +∫t0dτeiL(t−τ)PiLF(τ)=F(t) +∫t0dτeiL(t−τ)〈iLF(τ)A†〉〈AA†〉−^1 A(0)