ps1 - Physics 509: Relativistic Quantum Field Theory...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Physics 509: Relativistic Quantum Field Theory Problem Set 1 Due Friday, 30 September 2011 1. SHO Greens Function from Euclidean Time Consider the SHO with Lagrangian L = 1 2 2- 1 2 m 2 2 . It is known that G ( t ) = | T ( t ) (0) | = 1 2 m e- im | t | by using the Heisenberg equation ( t ) + m 2 ( t ) = 0 to derive ( 2 t + m 2 ) G ( t ) =- i ( 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 ) = - d 2 ie- it 2- m 2 + i1 . (1 . 1) Shift the contour of integration by e i and simultaneously make the change t e- i t . Due to the pole prescription make sure that you will not encounter any singularity for 0 < . c) Consider = 2 . Effectively it corresponds to the change t - i . This is called Wick rotation; is referred to as Euclidian time. Writing the obtained integral as G ( ) = - d 2 G ( ) e- i do the -integral to obtain G ( ) and show that it falls of rapidly as . d) Show that G ( ) satisfies the Euclidian equation of motion (- 2 + m 2 ) G ( ) = ( ). Thus, Feynman Green function can be obtained by analytical continuation of G ( ) which is the unique Green function of operator- 2 + m 2 that goes to zero at Euclidean infinity. e) Consider Green functions of the form | T( ( t 1 ) ( t 2 ) ... ( 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 ( 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 S and conjugate momentum S , where the subscript S marks these as time-independent Schrodinger picture operators. Assume the Schrodinger picture Hamiltonian is of the form H ( S , S ,t ) = H ( S , S ) + V ( S , S ,t ) and define time evolution in the Schrodinger picture by...
View Full Document

Page1 / 7

ps1 - Physics 509: Relativistic Quantum Field Theory...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online