Ps3 - 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

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Unformatted text preview: 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 + V , where H is the free ( λ = 0) part of the Hamiltonian, and V is the anharmonic perturbation. Let | be the ground state of H with energy E , while | Ω is the ground state of H with energy E Ω . a) Show that as T → (1- iε ) ∞ for ε an infinitesimal positive quantity, we have | e- iHT | = Ze- iE Ω T , (1) where Z = | | Ω | 2 is the wavefunction renormalization. Also show that in the same limit | T e- i T/ 2- T/ 2 dtV I ( t ) | = Ze- i ( E Ω- E ) 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 due to interactions and the wavefunction renormalization Z...
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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.

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Ps3 - 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

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