From Special Relativity to Feynman Diagrams.pdf

Which is consistent with 11131 if instead y x we

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which is consistent with ( 11.131 ). If instead y 0 > x 0 we close the contour in the upper half-plane along the semi-circle C ( + ) , see Fig. 11.3 , and obtain:
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11.4 Invariant Commutation Rules and Causality 391 Fig.11.3 The y 0 > x 0 case Fig.11.4 Prescription for the Feynman propagator: Shifting the poles in the p 0 plane C F f ( p 0 , p ) dp 0 = C F + C ( + ) f ( p 0 , p ) dp 0 = 2 π i Res − ¯ p 0 ( f ) = 2 ¯ p 0 e i ( ¯ p 0 ( x 0 y 0 ) + p · ( x y ) ) . Inserting the above result in ( 11.132 ) and changing the integration variable from p to p , we find d 3 p ( 2 π ) 3 C F dp 0 f ( p 0 , p ) = c d 3 p ( 2 π ) 3 e i p · ( x y ) 2 E p = D F ( x y ), which completes the proof of ( 11.132 ). Summarizing, the Feynman propagator is defined by the integral over the four-momentum space of f ( p 0 , p ), with the prescrip- tion that the integral over p 0 be computed along C F . Such prescription is equivalent to integrating over the real p 0 axis and shifting at the same time the poles to ± ( ¯ p 0 i )), where is an infinitesimal displacement, as shown in Fig. 11.4 . The denominator p 2 m 2 c 2 of f ( p 0 , p ), with this prescription, changes into p 2 m 2 c 2 + i and ( 11.132 ) can be also written as follows D F ( x y ) = i 2 C F d 4 p ( 2 π ) 4 e i p · ( x y ) p 2 m 2 c 2 = d 4 p ( 2 π ) 4 e i p · ( x y ) D F ( p ), (11.133)
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392 11 Quantization of Boson and Fermion Fields where D F ( p ) i 2 p 2 m 2 c 2 + i , (11.134) Equation ( 11.134 ) is the expression of the Feynman propagator in momentum space. We now show that D F ( x y ) is a Green’s function for the Klein–Gordon equa- tion. Let us briefly recall the notion of Green’s function for a linear differential equation. Consider the problem of finding the function f ( x μ ) which satisfies the inhomogeneous differential equation L ( x ) f ( x ) = g ( x ), (11.135) L ( x ) being a local differential operator. If there exists a unique solution for each g ( x ), there must exist an inverse operator L 1 such that, formally: f ( x ) = L 1 g ( x ). Computing the operator L 1 , however, is more than just taking the inverse to L , since it denotes that operation plus the boundary conditions. By definition the Green’s function G ( x , y ) is the solution to ( 11.135 ) where g ( x ) = − i δ 4 ( x μ y μ ) and corresponds to L 1 together with the associated boundary conditions. The solution of the differential equation( 11.135 ) is then given by the formula f ( x μ ) = f 0 ( x μ ) + i d 4 yG ( x μ , y μ ) g ( y μ ). (11.136) where f 0 ( x μ ) is the general solution of the associated homogeneous equation. This is easily verified applying the operator L to both sides of ( 11.136 ). Let us then consider the Klein–Gordon equation describing the interaction of a classical field φ( x ) with an external source J ( x ) : x + m 2 c 2 2 φ( x ) = J ( x ). (11.137) Identifying L ( x μ ) with the operator x + m 2 c 2 2 and g ( x μ ) with J ( x μ ), the general solution of ( 11.137 ) can be written as φ( x ) = φ 0 ( x ) + i d 4 yD ( x , y ) J ( y ) (11.138) where φ 0 ( x ) is the general solution to the homogeneous part of the Klein–Gordon equation x + m 2 c 2 2 φ 0 ( x ) = 0 while the last term is a particular solution of the inhomogeneous equation. Acting with the Klein–Gordon operator on ( 11.133 ) we find
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11.4 Invariant Commutation Rules and Causality 393 Fig.11.5 Prescription for the retarded Green function +
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