# A question in Weinberg's QFT.

1. Oct 15, 2013

### ndung200790

In &9.1 writing:
<q',t'/T{O$_{A}$(P(tA),Q(tA)),O$_{B}$(P(tB),Q(tB))...}/q,t>=

$\int$$\Pi$dq$_{a}$($\tau$)dp$_{b}$($\tau$)O$_{A}$O$_{B}$x

exp[i$\int$d$\tau${$\sum$q$^{.}$$\tau$p$_{a}$($\tau$)-H(q($\tau$)p($\tau$)}].(9.1.38)

Where T is the time order product.
It is q$_{a}$,p$_{a}$ are not constrained to obey the equation of classical Hamintonian dynamics:
q$^{.}$$_{a}$($\tau$)-$\delta$H/$\delta$p$_{a}$($\tau$)=0.(9.1.39)

p$^{.}$$_{a}$($\tau$)-$\delta$H/$\delta$q$_{a}$($\tau$)=0.(9.1.40)
Nevertheless,there is a limited sense in which path integrals do respect these equations of motion.Suppose that one of the functions in (9.1.38) say O$_{A}$(p(tA),q(tA)),happens to be the left-hand side of either Eq.(9.1.39),(9.1.40).We note that(for t<tA<t')

(q$^{.}$$_{a}$(tA)-$\delta$H(q(tA),p(tA))/$\delta$p$_{a}$(tA))exp(iI[q,p]=-i$\delta$exp(iI[q,p])/$\delta$p$_{a}$(tA)

p$^{.}$(tA)-$\delta$H(q(tA),p(tA))/$\delta$q$_{a}$(tA)exp(iI[q,p])=i$\delta$exp(iI[q,p])/$\delta$q$_{a}$(tA).

where iI is the argument of the exponential in Eq.(9.1.38).
As long as tA does not approach t or t', the integrations over q$_{a}$(tA) and p$_{a}$(tA) are unconstrained,and so with reasonable assumtions about the convergence of these integrals,the integral of such variational derivatives must vanish.Hence the path integral(9.1.38) vanish if O$_{A}$(p,q)is taken to be the left-hand side of either of the equations of motion (9.1.39) or (9.1.40).

Then my argue is:
Why does unconstrained on the integrations over p and q with condition of convergence lead to the integral must vanish? If it is not vanish,is it divergence,but why?