From Special Relativity to Feynman Diagrams.pdf

Inthefollowingforthesakeofsimplicityweshalllimitourselvestotwoincoming

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Inthefollowing,forthesakeofsimplicity,weshalllimitourselvestotwoincoming spin0particles,describedbycomplexfields φ i ( x ) ,althoughthefinalrelationbetween the transition probability, the differential cross section and the S -matrix elements, straightforwardly extends to spin 1/2 and 1 particles. Being φ i ( x ) positive energy solutions, ρ i > 0 can be consistently regarded as single-particle densities. We also observe that, substituting ( 12.63 ) inside ( 12.65 ) we find: j i μ ( x ) = i c d p 2 E p V i d p 2 E p V i 2 × f i ( p ) f i ( p )( i ) p i μ e i ( p p ) · x i f i ( p ) f i ( p ) p i μ e i ( p p ) · x 2 c 2 ¯ p i μ | φ( x ) | 2 , (12.67) where we have used the property that f i ( p ) are narrowly peaked about average values ¯ p i and thus we have approximated, in the integral, p i μ , p i μ with ¯ p i μ . 8 Using the 5 The final relations we are going to derive apply to the photon field as well. 6 In Sects.12.2.1 and 12.2.2 , in contrast to the present section, we were considering collections of particles decaying or interacting (i.e. colliding beams) and thus the densities ρ were referred to multi-particle systems. 7 Recall our choice of normalization for the single-particle momentum eigenstates: p , r | p , r = ( 2 π ) 3 V δ 3 ( p p rr . 8 For fermionic fields, in the same approximation, we would find j i μ( x ) ¯ p i μ mc ψ i ( x i ( x ).
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452 12 Fields in Interaction same approximation, and restricting to the time component ρ i = j 0 i of the currents, we can set p 0 p 0 E ¯ p / c , so that the energy factors arising from the time- derivatives cancel against the factor 1 / E p E p , and we can write the densities ρ i in the following form: ρ i ( x ) 1 V i d p d p f i ( p ) f i ( p ) e i ( p p ) · x , (12.68) Let | ψ out describe a system of free outgoing wave packets whose momenta are also narrowly distributed about some average values, the average final total momentum being ¯ P out μ . Let us now write the matrix element of T in plane waves components ψ out | T | ψ in = d p 1 d p 2 f 1 ( p 1 ) f 2 ( p 2 ) ψ out | T | p 1 | p 2 = d p 1 d p 2 f 1 ( p 1 ) f 2 ( p 2 )( 2 π ) 4 δ 4 ( ¯ P out μ p 1 μ p 2 μ ) × ψ out | T | p 1 | p 2 . (12.69) To evaluate the transition probability, we need to compute the squared modulus of the above amplitude: | ψ out | T | ψ in | 2 = d p 1 d p 2 d p 1 d p 2 f 1 ( p 1 ) f 1 ( p 1 ) f 2 ( p 2 ) f 2 ( p 2 ) × ( 2 π ) 8 δ 4 ( ¯ P out μ p 1 μ p 2 μ 4 ( ¯ P out μ p 1 μ p 2 μ ) × ψ out | T | p 1 | p 2 ψ out | T | p 1 | p 2 . (12.70) Let us now approximate the matrix elements ψ out | T | p 1 | p 2 , ψ out | T | p 1 | p 2 with the corresponding value computed on ¯ p i , ψ out | T p 1 p 2 . Writing δ 4 ( ¯ P out μ p 1 μ p 2 μ 4 ( ¯ P out μ p 1 μ p 2 μ ) as δ 4 ( ¯ P out μ p 1 μ p 2 μ 4 ( p 1 μ + p 2 μ p 1 μ p 2 μ ) and expressing the second delta function as follows: ( 2 π ) 4 δ 4 ( p 1 μ + p 2 μ p 1 μ p 2 μ ) = d 4 xe i ( p 1 + p 2 p 1 p 2 ) · x , (12.71) the expression ( 12.70 ) can be recast in the form | ψ out | T | ψ in | 2 = d 4 x d p 1 d p 2 d p 1 d p 2 f 1 ( p 1 ) f 1 ( p 1 ) × f 2 ( p 2 ) f 2 ( p 2 ) e i ( p 1 + p 2 p
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