# Brownian Motion and Path Integrals

1. Dec 4, 2011

### Anaxandridas

I am reading "Quantum Mechanics and Path Integrals" (by Feynman and Hibbs) and working out some of the problems... as a hobby of sorts.

I have run into a problem in section 12-6 Brownian Motion. On page 339 (of the emended edition), the authors demonstrate, by example, a method for calculating a probability distribution using a path integral.

The text states that "the integral

P(D,$\theta$)=$\int exp \left\{ -\frac{1}{2R} \int^{T}_{0} \ddot{x}^{2}\left(t\right) dt \right\} D x\left(t\right)$

is gaussian and becomes an extremum for the path

$\frac{d^{4}\bar{x}}{dt^{4}}$ = 0."

I am having difficulty recognizing from where this path condition arises. Please let me know if you some insight regarding this problem.