By Peeters K.

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8) What we mean by the integral over paths will be made precise below. 5) where f ( x ) is a slowly varying function of x. In the region where g( x ) is rapidly varying, the oscillatory contributions will cancel out because the prefactor f ( x ) is almost constant. Therefore, the contributions to this integral will come from those regions where g( x ) is slowly varying as well. The dominant contribution will be in the region where dg( x )/dx ≈ 0 . 6) Expanding around such points gives the stationary phase approximation.

We do it by inserting N copies of unity in the form of Gaussian integrals, −δ 2π¯h 2( N −1) N −1 ∏ m =1 d4 αm exp − iδ 2 α 2¯h m = 1. 3 By shifting these new variables according to αn → µ µ αn − X˙ n (which we can do without changing the integral) the quadratic terms disappear, iδ 2¯h N ∑ n =1 iδ N µ µ − α2n + X˙ n X˙ n µ − m2 → 2αn X˙ n µ − (α2n + m2 ) . 13) µ All the Xn -dependence now sits in the first term inside the sum above. 11), these terms become i h¯ N ∑ α n ( X n − X n −1 ) = n =1 µ µ µ N −1 i µ i µ µ µ α N X N µ − α 1 X0 µ − ∑ α n +1 − α n X n µ .

The manipulations described above can be summarised by a new set of Feynman rules, formulated directly in momentum space. 1. Finally, you may wonder about the way in which all of these transition amplitudes depend on Planck’s constant h¯ . We have so far just carried them along, but in fact there is an important relation between the order in h¯ and the number of loops. To see this, note that every propagator carries a factor of h¯ , and every vertex carries a factor h¯ −1 . Thus, a diagram with E external lines, I internal lines and v vertices has a power h¯ E+ I −v .