~ H / dqi dqi' (2 ^ofein; qj, t^Hiiq^Koiqj, tf, qy, tr)

K2 = {-if f dqdq' [" dt [ tfKo(qn,tn\q,t)Hi(q)K0(q,trf,t') J Jto Jto x HI(q')Ko(q',t'-,q0,t0) , (14.39)

where t is associated with the slab at time tj and t' at time ty. The second order contributions from a single time slab are ignored because they are negligibly small compared to the contributions from two different times. To see this, note that there are n -1 contributions from a single slab, but (n -1 )(n - 2)/2 contributions from two slabs, so that the error in neglecting the single slab contributions goes like ~ 2/n —» 0 as n —» oo. Note that the time-ordered structure of (14.39) emerges automatically in an almost trivial fashion; it comes from the fact that one of the Hi's must necessarily follow the other. Now to find the second order ¿'-matrix element implied by K2, exploit the fact that K0 is the free propagator to carry out the following reduction:

(S/3a)2 =(-î)2 j dqn dq0 dqi dq2 J^ dt2 (j>g{qn,tn) K0{qn,tn\q2, t2) HI{q2)

x / dti Ko(q2,t2;qi,ti)Hi{qi)Ko(qi,h;qo,to)<f>a{qo,to) Jt0

Using the definition of Kq and the completeness of the position eigenstates gives

(S/3a)2 =(-i)2/ dqidq2dq[dq'2 J^ dt2(0\q2,t2}oo{q2,t2\H!(Q(t2))\q'2,t2)o rti x / dti 0(q'2,t2\q[,h)00(q[,tl\HI(Q{tl))\qi,t1)00(ql,t1\a)

=(-z)2 fndt2 P dh (p\H,(Q(t2))HI(Q(t1))\a) Jtn Jtn s:

Again recognize the familiar second order result, Eq. (3.25). The structure of the other terms in the series is now apparent and leads to the suggestive diagrammatic representation shown in Fig. 14.3.

In preparation for application of these ideas to field theory, we next discuss time-ordered products and the generating function.

The condition that the paths do not double back on themselves was implicit in our construction of the path integral, and this constraint is implemented through the requirement that the propagator for backward propagation in time be zero,

Therefore, the right-hand side of (14.43) is equal to (q;, tf\Q{t2)Q(ti)\qi, t%) if t2 > ii, and hence, in general, (14.43) is the time-ordered product of two operators, dqdp

(qfMTmMWM

This result clearly generalizes to

It is convenient to introduce a generating function from which an arbitrary time-ordered product can be determined. To this end, introduce the function t[J\ = J

[ 2tt i V' dt[pq-H(p,q) + J(t)q(t)} j s Jt' ={Qf,tfW'U) • (14.

Be careful not to confuse (qj, tf\qlt U) with (q/,tf\qu ¿¿); they are very different objects, but z[0) = <<?/, tf\qi,ti)° = (qj, tj\qi, ti) (14.48)

is the propagator over the finite time interval [tj,ti]. If the functional derivative is defined by the relation s (i) (F If M+-,)] -n/wi) =im then

. f \dqdp It follows immediately that i ['' dtlpq~H(p,q) + J(t)q(t)}

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