QftfT QtiQt2Gt3 Qtn WuU

Any time-ordered product can be obtained from a functional derivative of 2 with respect to J.

this case Minkowski space-time is transformed to Euclidean space-time. Or, instead of rotating the time axis, the identical effect is obtained if all energies are multiplied by the factor 1 — it. In either case the oscillating factors are converted to exponentials, which approach zero as T, —> -oo, with the ground state approaching zero least rapidly. Multiplying by exp(-i-EoTj) enables us to extract the ground state matrix element from (14.53):

because all other terms in the sum are damped by the exponential factor et(En-E0)Ti^ which approaches zero for Tj < 0 and En > E0.

This, then, provides a way to project out the ground state matrix element, even if we do not know the ground state wave function ipo(q) explicitly. Start with the matrix element {q',Tf\q,Ti)J, and insert a complete set of states,

(q',Tf\q,Ti)J = J dqf dq< (q\Tf\qf,t}}° (<?/, */|<7i, ij)J (quU^Ti)0

X J dq}dqii3n>{qf,tf)(<lf,tf\<li,ti}J Tl)n{qi,ti). (14.55)

Then, using Eq. (14.54) and the analogous relation for the final state gives e------ w ,Mf\q,iisJ

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