From Special Relativity to Feynman Diagrams.pdf

2 p 2 1 cos θ 4 p 2 sin 2 θ 2 1229 u 2 p 2 1 cos θ

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2 | p | 2 ( 1 cos θ) = − 4 | p | 2 sin 2 θ 2 , (12.29) u = − 2 | p | 2 ( 1 + cos (θ)) + ( E 1 E 2 ) 2 c 2 = − 4 | p | 2 cos 2 θ 2 + ( E 1 E 2 ) 2 c 2 , (12.30) where θ is the angle between p and q . The variable t represents the norm of the momentum p = p q transferred during the process. 12.3 Dynamics of Interaction Processes In the description we have given in last section of decay and scattering processes, we have encoded the dynamics of the event, namely the details of the interaction, in
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444 12 Fields in Interaction the Lorentz-invariant functions ˆ and , separating it from the kinematics, which is captured by the phase-space element d ( n ) . In this section we are going to express these quantities in terms of the interaction Hamiltonian H I . 12.3.1 Interaction Representation As anticipated in the introduction, in perturbation theory the Hamiltonian of the interacting system is computed on free fields, namely on fields evolving according to H 0 . Inthe Schrödingerpicture ,seeSect. 9.3.2 ,operators,includingtheHamiltonian,are constant while states | ψ( t ) S evolve in time according to the Schrödinger equation: i t | ψ( t ) S = ˆ H | ψ( t ) S = ( ˆ H 0 + ˆ H I ) | ψ( t ) S . (12.31) Both ˆ H 0 and ˆ H I can be expressed in terms of Hamiltonian-density operators ˆ H 0 = d 3 x H 0 , ˆ H I = d 3 x H I , (12.32) H 0 and H I being functions of free-field operators and their derivatives computed at some fixed time t = 0, and thus not evolving H 0 = H 0 ( ˆ φ 0 ( 0 , x ), ∂ μ ˆ φ 0 ( 0 , x )), H I = H I ( ˆ φ 0 ( 0 , x ), ∂ μ ˆ φ 0 ( 0 , x )). (12.33) Clearly, the Schrödinger picture does not provide a relativistically-covariant descrip- tion of the interaction since the free-field operators are all computed at t = 0 . The time evolution of states was described, in Sect.9.3.2 of Chap.9 , in terms of a time-evolution operator U ( t , t 0 ) , defined by property ( 9.73 ): | ψ ; t S = U ( t , t 0 ) | ψ ; t 0 S . (12.34) Let us recall the main properties of U ( t , t 0 ) discussed in Sect.9.3.2 . The inverse of U ( t , t 0 ) is the operator which maps the state at t back to t 0 : U ( t , t 0 ) 1 = U ( t 0 , t ). Substituting ( 12.34 ) into ( 12.31 ) we find that U ( t , t 0 ) is solution to the following equation: i d dt U ( t , t 0 ) = ˆ HU ( t , t 0 ), (12.35) with the initial condition U ( t 0 , t 0 ) = 1 . From hermiticity of ˆ H it follows that U ( t , t 0 ) is unitary. Indeed let us first show that U ( t , t 0 ) U ( t , t 0 ) is constant: d dt ( U ( t , t 0 ) U ( t , t 0 )) = d dt U ( t , t 0 ) U ( t , t 0 ) + U ( t , t 0 ) d dt U ( t , t 0 ) = i U ( t , t 0 ) ˆ H U ( t , t 0 ) + U ( t , t 0 ) i ˆ HU ( t , t 0 ) = iU ( t , t 0 ) ˆ HU ( t , t 0 ) iU ( t , t 0 ) ˆ HU ( t , t 0 ) = 0 . (12.36)
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12.3 Dynamics of Interaction Processes 445 Being constant this operator should be equal to its value at t = t 0 , namely U ( t , t 0 ) U ( t , t 0 ) = 1 , and thus U ( t , t 0 ) is unitary. In the Heisenberg picture states | ψ H are constant while operators evolve in time. The relation between states in the Heisenberg and in the Schrödinger pictures is defined by the time-evolution operator: | ψ H = U ( t , t 0 ) | ψ( t ) S . Similarly the operators O ( t ) H and O in the two representations are related to one another in such a way that their mean values on the states is the same: O ( t ) H = U ( t , t 0 ) O U ( t , t 0 ).
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