This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: Physics 509: Relativistic Quantum Field Theory Problem Set 2 Due Friday, 07 October 2011 7. Convergence of Perturbation Theory in a 0-Dimensional QFT Much insight into QFT can be gained by studying the integral Z ( j ) = ∞-∞ dxe- 1 2 x 2- λ 4! x 4 + jx , (1) where λ ≥ 0 and j ∈ R . This integral can be viewed as the functional integral represen- tation for the generating functional of a Euclidian QFT in d = 0 spacetime dimensions. Note that the integral is rapidly convergent. a) Set λ = 0 and compute Z ( j ) in this case. Use this to calculate the free “Green’s functions” x n = ∞-∞ dxx n e- 1 2 x 2 ∞-∞ dxe- 1 2 x 2 , (2) where n ≥ 1. Give a Feynman diagram interpretation and show that for n = 2 , 4 , 6 , 8 the Green’s functions are correctly given by their symmetry factor alone. b) Now consider Z (0) for λ > 0. Assuming λ is sufficiently small, expand it in a pertur- bation series Z (0) = ∞ n =0 λ n Z n and evaluate Z n . Show that Z n grows factorially as n → ∞ and hence that the perturbation series has zero radius of convergence. Explain this result in terms of the original integral representation for Z (0). The conclusion of b) seems to put a catastrophic dent in our confidence in perturbation theory. In fact, this situation is generic in both quantum mechanics and QFT, for reasonstheory....
View Full Document
This note was uploaded on 02/02/2012 for the course PHY 509 taught by Professor Alexanderm.polyakov during the Fall '09 term at Princeton.
- Fall '09
- Quantum Field Theory