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A question in Weinberg's QFT.

  1. Oct 15, 2013 #1
    In &9.1 writing:
    <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?
     
  2. jcsd
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