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Unformatted text preview: Physics 509: Relativistic Quantum Field Theory Problem Set 2 Due Friday, 07 October 2011 7. Convergence of Perturbation Theory in a 0Dimensional 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....
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 Fall '09
 ALEXANDERM.POLYAKOV
 Quantum Field Theory, real scalar fields, perturbation series, local phase rotations

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