I am reading Feynman and Hibbs on Path Integrals. In section 3.5, they show that the kernel for a lagrangian of the form L=a(t)˙x2+b(t)˙xx+c(t)x2+d(t)˙x+e(t)x+f(t) is K(b,a)=eiℏScl[b,a]F(ta,tb). In general, how do I calculate the factor F(ta,tb). In the problems after the section, I have calculated the classical action, for the particle in a magnetic field, and the forced harmonic oscillator. But I don't know how to calculate the prefactors. For e.g. this is the problem 3-11 from Feynman and Hibbs asks you to calculate the kernel of the harmonic oscillator driven bby an external force f(t). The Lagrangian is L=m2˙x2−mω22x2+f(t)x. The answer is K=√mω2πiℏsinωTeiℏScl
where T=tf−ti and Scl is the classical action. How can I see that the above is the factor multiplying the exponent directly or via a calculation.
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