Mori–Zwanzig theory 599Geometrically, recall that the projection of a vectorbalong the direction of another
vectorais given by the formula
Pb=(
b·a
|a|)
a
|a|. (15.7.2)
Here,Pis an operator that gives the component ofbalong the direction ofa, with
a/|a|the unit vector along the direction ofa. An analog of this formula is used to
construct projection operators in phase space parallel and perpendicular to the vector
A. The projection operator must also eliminate or integrate out the bath degrees of
freedom. Thus, we define the operatorPthat both projects along the direction ofA
and integrates out the bath according to
P=〈...A†〉〈AA†〉−^1 A, (15.7.3)where the quantity on whichPacts replaces the dots, and〈...〉denotes an average
over a canonical ensemble.A†is the Hermitian conjugate ofA. The use of Hermitian
conjugates is introduced because of the close analogy between the Hilbert space for-
malism of quantum mechanics and the classical phase space propagator formalism we
will employ in the present derivation (see Section 3.10). The operator that projects
along the direction orthogonal toAis denotedQand is simplyI−P, whereIis the
phase space identity operator. The operatorsPandQcan be shown to be Hermitian
operators. Note that this definition of the projection operator issomewhat more gen-
eral than the simple geometric projector of eqn. (15.7.2) in that the quantities〈...A†〉
and〈AA†〉−^1 are matrices. These are multiplied together and then allowed to act on
A, ultimately producing another two-component vector. As expected for projection
operators, the actions ofPandQon the vectorAare
PA=A, QA= 0. (15.7.4)SincePA=A, it follows thatP^2 A=PA=A, andQ^2 A=−QQA= 0 =QA, which
also means thatPandQsatisfy
P^2 =P, Q^2 =Q. (15.7.5)This condition is known asidempotency, and the operatorsPandQare referred to as
idempotentoperators.
The projection operatorsPandQcan be used to analyze the dynamics of the
system variablesA. Recall that the time evolution of any quantity in the phase space is
determined by the action of the classical propagator exp(iLt). The vectorA, therefore,
evolves according to
A(t) = eiLtA(0). (15.7.6)
Differentiating both sides of this relation with respect to time yields
dA
dt= eiLtiLA(0), (15.7.7)whereiLis the classical Liouville operator of Section 3.10 We now use the projection
operators to separate this evolution equation forA(t) into components alongA(0)