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hw4 - Homework Set No 4 Physics 880.08 Deadline Wednesday...

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Homework Set No. 4, Physics 880.08 Deadline – Wednesday, December 8, 2010 at NOON! 1. Time-ordered product of real scalar fields is defined by T φ ( x ) φ ( y ) = θ ( x 0 y 0 ) φ ( x ) φ ( y ) + θ ( y 0 x 0 ) φ ( y ) φ ( x ) , where φ ’s are operators in Heisenberg picture. a. (5 pts) In a free scalar field theory with mass m use Klein-Gordon equation along with the canonical commutation relations to show that [ 2 + m 2 ] T φ ( x ) φ ( y ) = 4 ( x y ) where the derivative squared (the D’Alembertian) is taken with respect to 4-coordinates x . b. (10 pts) Similar to what we did in class for retarded Green function, find an explicit expression for the Feynman propagator in coordinate space in a massless ( m = 0) theory by performing the following integral D F ( x y ) = integraldisplay d 4 k (2 π ) 4 e i k · ( x y ) i k 2 + . Is Feynman propagator causal? 2. Time-ordered product of Dirac spinors is defined by T ψ α ( x ) ¯ ψ β ( y ) = θ ( x 0 y 0 ) ψ α ( x ) ¯ ψ β ( y ) θ ( y 0 x 0 ) ¯ ψ β ( y ) ψ α ( x ) where α,β = 1 , 2 , 3 , 4 are Dirac indices. ψ and ¯ ψ are operators in Heisenberg picture.
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