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ps3 - Physics 509 Relativistic Quantum Field Theory Problem...

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Physics 509: Relativistic Quantum Field Theory Problem Set 3 Due Friday, 14 October 2011 Reading: Peskin 2.4, 3.1-3.2, 4.1-4.4 11. Anharmonic SHO and Feynman Diagrams I Consider, as usual, the anharmonic oscillator L = 1 2 ˙ φ 2 - 1 2 m 2 φ 2 - λ 4! φ 4 , where λ is a small, positive coupling constant. Write H = H 0 + V , where H 0 is the free ( λ = 0) part of the Hamiltonian, and V is the anharmonic perturbation. Let | 0 be the ground state of H 0 with energy E 0 , while | Ω is the ground state of H with energy E Ω . a) Show that as T (1 - i ε ) for ε an infinitesimal positive quantity, we have 0 | e - iHT | 0 = Ze - iE Ω T , (1) where Z = | 0 | Ω | 2 is the wavefunction renormalization. Also show that in the same limit 0 | T e - i T/ 2 - T/ 2 dt V I ( t ) | 0 = Ze - i ( E Ω - E 0 ) T . (2) b) Using the result in (2) and your ability to calculate SHO Green’s functions (Feynman diagrams), find the shift in the ground state energy E Ω - E 0 due to interactions and the wavefunction renormalization Z up to O ( λ 2 ).
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