In &9.1 writing:(adsbygoogle = window.adsbygoogle || []).push({});

<q',t'/T{O[itex]_{A}[/itex](P(tA),Q(tA)),O[itex]_{B}[/itex](P(tB),Q(tB))...}/q,t>=

[itex]\int[/itex][itex]\Pi[/itex]dq[itex]_{a}[/itex]([itex]\tau[/itex])dp[itex]_{b}[/itex]([itex]\tau[/itex])O[itex]_{A}[/itex]O[itex]_{B}[/itex]x

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

Where T is the time order product.

It is q[itex]_{a}[/itex],p[itex]_{a}[/itex] are not constrained to obey the equation of classical Hamintonian dynamics:

q[itex]^{.}[/itex][itex]_{a}[/itex]([itex]\tau[/itex])-[itex]\delta[/itex]H/[itex]\delta[/itex]p[itex]_{a}[/itex]([itex]\tau[/itex])=0.(9.1.39)

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

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

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# A question in Weinberg's QFT.

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