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# ps05 - Physics 651 Problem Set 5 NO CLASS OCT 1 Wednesday 1...

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Physics 651 – Problem Set 5 (due 10/06/03) NO CLASS OCT 1, Wednesday 1. Do problem 2.1. 2. (This problem was done in class sometime ago.) Show that the Feynman prop- agator D F ( x - y ) is a Green’s function of the Klein–Gordon operator, i.e., that ( + m 2 ) D F ( x - y ) = - (4) ( x - y ). Then take the Fourier transform of both sides of this equation and derive the expression for the momentum-space propagator D F ( p ) defined by D F ( x - y ) = d 4 p (2 π ) 4 e - ip · ( x - y ) D F ( p ) . 3. Consider the most general renormalizable (that is, no coupling parameters with negative mass dimensions) theory of N real scalar fields φ i ( i = 1 , 2 , ..., N ) with V ( φ i ) = V ( i φ 2 i ) that is, V is a function of the particular combination i φ 2 i only. How many parameters does the theory have ? Do the scalar fields have the same mass m ? For N = 2, show that this theory is identical to the complex scalar field theory. Show that the internal symmetry of the theory is SO (2) = U (1). What is the conserved current corresponding to this symmetry ? (Recall last homework and discussions in class.)

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ps05 - Physics 651 Problem Set 5 NO CLASS OCT 1 Wednesday 1...

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