hw32 - q cl ( t ) = 0 when j ( t ) = 0. When solving...

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Homework Set No. 2, Physics 880.08 Deadline – Wednesday, April 27, 2011 1. (15 pts) Imagine that the full non-perturbative beta-function of QED were β ( α ) = α 2 3 π b 1 e 1 - 1 α B . Find the running QED coupling constant α ( Q 2 ) for such beta-function. Sketch α ( Q 2 ) as a function of Q 2 . Find the UV ±xed point and determine the large- Q 2 asymptotics of α ( Q 2 ), i.e., ±nd how it approaches the ±xed point. 2. a. (15 pts) Consider a harmonic oscillator in a background of a time-dependent external force (source) j ( t ). The Lagrangian is L = 1 2 m ˙ q 2 1 2 m ω 2 q 2 + q j ( t ) . Using quasi-classical method for evaluation of path integrals ±nd the time-evolution (Feyn- man) kernel U ( q f , t f ; q i , t i ) = S a q f ( t f ) | e - i ¯ h ˆ H ( t f - t i ) | q i ( t i ) A S = H a q f , t f | q i , t i A H = i [ D q ] exp i ¯ h t f i t i dt L ( t ) . You may use the result derived in class for the harmonic oscillator without the external force, though this time you also need to ±nd the classical action in terms of j ( t ). In evaluating the classical action assume that q i = q f = 0, or, more speci±cally, require that
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Unformatted text preview: q cl ( t ) = 0 when j ( t ) = 0. When solving classical EOM you may nd Fourier-integral decomposition q ( t ) = i- dE 2 e-i h E t q E useful. b. (10 pts) Use the result of part a to show that the two-point function for the harmonic oscillator without the external force is given by a | T q ( t 1 ) q ( t 2 ) | A = i h 2 m i- dE 2 e-i h E ( t 1-t 2 ) E 2 h 2 2 + i . (1) 1 c. (10 pts) Re-derive the two-point function in Eq. (1) by using creation and annihilation operators. For the simple harmonic oscillator (without the external force) write q ( t ) = r h 2 m b a e-i t + a e i t B and use commutation relations for a and a ([ a, a ] = 1, all other commutators are zero) to obtain Eq. (1). 2...
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This note was uploaded on 07/28/2011 for the course PHYSICS 880.08 taught by Professor Staff during the Fall '10 term at Ohio State.

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hw32 - q cl ( t ) = 0 when j ( t ) = 0. When solving...

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