From Special Relativity to Feynman Diagrams.pdf

I k t t dt ˆ h i t k 1250 and the series 1243 is

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i k t t 0 dt ˆ H I ( t ) k , (12.50) and the series ( 12.43 ) is easily summed to an exponential: U I ( t , t 0 ) = exp i t t 0 dt ˆ H I ( t ) . (12.51) In general, however, [ ˆ H I ( t ), ˆ H I ( t ) ] = 0 and the correct solution to ( 12.42 ) is not given by the above exponential but rather by the following formal expansion: U I ( t , t 0 ) = T exp i t t 0 dt ˆ H I ( t ) = k = 0 1 k ! i k t t 0 dt 1 t t 0 dt 2 . . . t t 0 dt k T [ ˆ H I ( t 1 ) ˆ H I ( t 2 ) . . . ˆ H I ( t k ) ] , (12.52) where the symbol T [ exp (. . .) ] represents the prescription that the integrand in each multiple integral originating from the expansion of the exponential should be time- ordered. This means that, when acting by means of U I ( t , t 0 ) on a state | ψ( t 0 ) I , the values of the operator ˆ H I ( t ) at earlier times should be applied to it before those computed at later times. Let | ψ( t ) I describe the state of the system at the time t . Long before the interac- tion, the system consists of free-particles described by the state | ψ in (we shall often omit the subscript “I” of the interaction representation), so that: | ψ in = lim t →−∞ | ψ( t ) I = | ψ( −∞ ) I . (12.53) At a time t , the state | ψ( t ) I can be formally expressed in terms of | ψ in using the time-evolution operator U I :
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12.3 Dynamics of Interaction Processes 449 | ψ( t ) I = U I ( t , −∞ ) | ψ in . (12.54) Long after interaction the system is described by the free-particle state | ψ( +∞ ) I , related to the initial state as follows: | ψ( +∞ ) I = U I ( +∞ , −∞ ) | ψ in = S | ψ in , (12.55) where we have defined the scattering matrix S ( S -matrix) as S U I ( +∞ , −∞ ) = T exp i +∞ −∞ dt ˆ H I ( t ) = +∞ n = 0 i n 1 n ! +∞ −∞ dt 1 . . . +∞ −∞ dt n T ˆ H I ( t 1 ) . . . ˆ H I ( t n ) . (12.56) If we now use ( 12.32 ) in the interaction representation, we can express S in a Lorentz invariant fashion, in terms of the interaction Hamiltonian density H I ( t , x ) = H I ( x ) : S T exp i c +∞ −∞ d 4 x H I ( x ) = +∞ n = 0 i c n 1 n ! +∞ −∞ d 4 x 1 . . . +∞ −∞ d 4 x n T H I ( x 1 ) . . . H I ( x n ) . (12.57) Just as U I , S is a unitary operator acting on the Fock space of free-particle states: SS = 1 , (12.58) and it encodes the information about the interaction. In general one is interested in the probability of finding the system, after the interaction, in a free-particle state | ψ out . If the particles are initially prepared in a state | ψ in , this probability P ( in ; out ) reads P ( in ; out ) = | ψ out | ψ( +∞ ) | 2 ψ out | ψ out ψ( +∞ ) | ψ( +∞ ) = | ψ out | S | ψ in | 2 ψ out | ψ out ψ in | ψ in , (12.59) where we have used ( 12.55 ). The transition amplitude A ( in ; out ) (also called, for scattering processes, scattering amplitude ) is then given by the matrix element of S between the initial and final states. Let us define an operator T so that S = 1 + i T . The identity operator only contributes to the transition amplitude when the initial and final states coincide, namely when there is no interaction. Excluding this case we can write A ( in ; out ) ψ out | S | ψ in = i ψ out | T | ψ in . (12.60)
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450 12 Fields in Interaction Since the system is isolated, the total four-momentum is conserved. If we think of the
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