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Unformatted text preview: Physics 509: Relativistic Quantum Field Theory Problem Set 1 Due Friday, 30 September 2011 1. SHO Green’s Function from Euclidean Time Consider the SHO with Lagrangian L = 1 2 ˙ φ 2- 1 2 m 2 φ 2 . It is known that G ( t ) = | T φ ( t ) φ (0) | = 1 2 m e- im | t | by using the Heisenberg equation ¨ φ ( t ) + m 2 φ ( t ) = 0 to derive ( ∂ 2 t + m 2 ) G ( t ) =- iδ ( t ) and by arguing that the form of G ( t ) given above satisfies the correct boundary conditions for a vacuum expectation value (VEV) of fields. a) By inserting a complete set of states into the definition of G ( t ) as a time-ordered VEV of fields, show that G ( t ) → 0 when Im t → -∞ , Re t > 0, and also when Im t → + ∞ , Re t < 0. This is the boundary condition that determines G ( t ). b) Consider Fourier transform of G ( t ) G ( t ) = ∞-∞ dω 2 π ie- iωt ω 2- m 2 + i1 . (1 . 1) Shift the contour of integration by ω → e iθ ω and simultaneously make the change t → e- iθ t . Due to the pole prescription make sure that you will not encounter any singularity for 0 ≤ θ < π . c) Consider θ = π 2 . Effectively it corresponds to the change t → - iτ . This is called Wick rotation; τ is referred to as Euclidian time. Writing the obtained integral as G ( τ ) = ∞-∞ dω 2 π G ( ω ) e- iωτ do the ω-integral to obtain G ( τ ) and show that it falls of rapidly as τ → ±∞ . d) Show that G ( τ ) satisfies the Euclidian equation of motion (- ∂ 2 τ + m 2 ) G ( τ ) = δ ( τ ). Thus, Feynman Green function can be obtained by analytical continuation of G ( τ ) which is the unique Green function of operator- ∂ 2 τ + m 2 that goes to zero at Euclidean infinity. e) Consider Green functions of the form | T( φ ( t 1 ) φ ( t 2 ) ...φ ( t n )) | in some general theory (e.g. SHO with anharmonicity). By inserting the complete set of states, assuming that the spectrum is bounded from below and H | = 0 show that they can be brought to the form dE k ρ ( E 1 ,...,E n- 1 ) e- i ∑ n- 1 k =1 E k ( t k- t k +1 ) . 1 This form of the Green functions guarantees that they can be analytically continued to imaginary time. Notice that they go to zero as t k- t k +1 → - i ∞ . Recall that high- energy states contribution in that case are suppressed by Boltzmann factor which is related to the fact that Euclidean Green functions compute thermal expectation of the fields. 2. Interaction Picture and Forced SHO In this problem we will generalize the interaction picture representation to the case when the interaction is explicitly time dependent and apply the result to study the forced SHO. Consider a one dimensional quantum mechanical system with coordinate φ S and conjugate momentum π S , where the subscript S marks these as time-independent Schr¨odinger picture operators. Assume the Schr¨odinger picture Hamiltonian is of the form H ( φ S ,π S ,t ) = H ( φ S ,π S ) + V ( φ S ,π S ,t ) and define time evolution in the Schr¨odinger picture by...
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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.
- Fall '09
- Quantum Field Theory