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Unformatted text preview: Quantum Field Theory: Example Sheet 1 Dr David Tong, October 2007 1. A string of length a , mass per unit length and under tension T is fixed at each end. The Lagrangian governing the time evolution of the transverse displacement y ( x, t ) is L = integraldisplay a dx bracketleftBigg 2 parenleftbigg y t parenrightbigg 2 T 2 parenleftbigg y x parenrightbigg 2 bracketrightBigg (1) where x identifies position along the string from one end point. By expressing the displacement as a sine series Fourier expansion in the form y ( x, t ) = radicalbigg 2 a summationdisplay n =1 sin parenleftBig nx a parenrightBig q n ( t ) (2) show that the Lagrangian becomes L = summationdisplay n =1 bracketleftbigg 2 q 2 n T 2 parenleftBig n a parenrightBig 2 q 2 n bracketrightbigg . (3) Derive the equations of motion. Hence show that the string is equivalent to an infinite set of decoupled harmonic oscillators with frequencies n = radicalbigg T parenleftBig n a parenrightBig . (4) 2. Show directly that if ( x ) satisfies the Klein-Gordon equation, then ( 1 x ) also satisfies this equation for any Lorentz transformation . 3. The motion of a complex field ( x ) is governed by the Lagrangian L = m 2 2 ( ) 2 . (5) Write down the Euler-Lagrange field equations for this system. Verify that the La- grangian is invariant under the infinitesimal transformation = i ,...
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This note was uploaded on 09/30/2009 for the course QFT EXAMPL n/a taught by Professor Davidtong during the Spring '09 term at Cambridge.
- Spring '09