Johnsonfreyd thesis

Page 80

Chapter 1: Feynman diagrams for quantum mechanics

72

where Îł is the family of almost-geodesics constructed in Proof 1.9.6. In particular, for q1 6∈ O0 and < 3 , the formal integral has no support, and for q1 ∈ O0 there is a unique point q0 ∈ O0 depending on at which the formal integral is supported. ij Moreover, q0i − q1i = a−1 (q1 , t0 ) bj (q1 , t0 ) + ∂g∂q−1 + O( 2 ), by tracing bounds in j 0 Proof 1.9.6. Using Fact 1.8.8, we have: ∂ 2 [−JÎł ] ∂ ∂v0j j = a (t , q ) v + b (t , q ) = a (t , q ) jk 0 0 k 0 0 jk 0 0 0 ∂q1l ∂q0k ∂q1l ∂q1l and therefore:

2

∂ [−JÎł ] ∂q1 ∂q0

−1 !kl

kj ∂q1l kl −1 = a (t , q ) + O( 2 ) = a (t0 , q0 ) 0 1 j ∂v0 −1

0 Here v0 = Îł(t Ë™ 0 ), q0 = Îł(t0 ), and q1 = Îł(t0 + ), for some classical path Îł. In ∂v , we hold ∂q1 ∂q1 q0 fixed and consider v0 as a function of q0 and q1 ; in ∂v0 we hold q0 fixed, considering q1 as a function of v0 and q0 . We observed already that q0 = q1 + v0 + O( 2 ). The estimate ∂q1l = δjl + O( 2 ) does not necessarily follow from this, but it does follow from integrating ∂v0j the equations of motion and noting that a bunch of terms are smooth and hence bounded on any compact domain. A similar argument gives:

Since

∂ 2 g−1 ∂q0

∂ 2 JÎł (∂q0 )2

−1 !kl

kl = a−1 (t0 , q1 ) + O( 2 )

is independent of , it follows that:

∂ 2 [JÎł − g−1 ] (∂q0 )2

−1 !kl

kl = a−1 (t0 , q1 ) + O( 2 )

Finally, note that very short classical paths have trivial Morse index. Combining these observations, Definition 1.2.14 gives: Z

formal

UÎł (t0 , q0 , t0 + , q1 ) g(q0 ) dVol(q0 ) √ dim N /2−dim N /2 √ −1 = 2Ď€~ −1 exp − ~ −1 JÎł t0 , q1 + a−1 b(t0 , q1 ) + O( 2 ), t0 + , q1 −1/2 1/2 Ă— det a−1 (t0 , q1 ) + O( 2 ) Ă— det a−1 (t0 , q1 ) + O( 2 ) Ă— g q1 + a−1 b(t0 , q1 ) + O( 2 ) Ă— 1 + O(~)

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