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# hw32 - q cl t = 0 when j t = 0 When solving classical EOM...

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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 bracketleftBig 1 e 1 - 1 α bracketrightBig . 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 fixed point and determine the large- Q 2 asymptotics of α ( Q 2 ), i.e., find how it approaches the fixed 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 find the time-evolution (Feyn- man) kernel U ( q f , t f ; q i , t i ) = S ( q f ( t f ) | e - i ¯ h ˆ H ( t f - t i ) | q i ( t i ) ) S = H ( q f , t f | q i , t i ) H = integraldisplay [ D q ] exp i ¯ h t f integraldisplay 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 find the classical action in terms of j ( t ). In evaluating the classical action assume that q i = q f = 0, or, more specifically, 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 π 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 ˆ q ( t 1 ) ˆ q ( t 2 ) | A = i ¯ h 2 m ∞ i-∞ dE 2 π e-i ¯ h E ( t 1-t 2 ) E 2 − ¯ h 2 ω 2 + i ǫ . (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 ˆ q ( t ) = r ¯ h 2 m ω b ˆ a e-i ω t + ˆ a † e i ω t B and use commutation relations for ˆ a and ˆ a † ([ˆ a, ˆ a † ] = 1, all other commutators are zero) to obtain Eq. (1). 2...
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hw32 - q cl t = 0 when j t = 0 When solving classical EOM...

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