Sidney Coleman — Quantum Field Theory Lectures · Problem 14.1

November 6 - Fourier transform of the new blob; A second interpretation of Feynman diagrams with lines off the mass shell; A third interpretation of the blob; Reformulation of scattering theory; S matrix elements without the turning on and off function; LSZ formula stated — Sidney Coleman

Physics — QFT · time-ordered-green-functions-interaction-picture

Problem

Convince yourself that, at least in the limit t±→±∞t_{\pm} \to \pm\infty when t+>t1>⋯>tn>t−t_+ > t_1 > \cdots > t_n > t_-, GF(n)(x1,…,xn)=lim⁡t±→±∞⟨0∣UI(t+,t1)ϕI(x1)UI(t1,t2)ϕI(x2)⋯ϕI(xn)UI(tn,t−)∣0⟩⟨0∣U(t+,t−)∣0⟩G_F^{(n)}(x_1, \ldots, x_n) = \lim_{t_\pm \to \pm\infty} \frac{\langle 0 | U_I(t_+, t_1) \phi_I(x_1) U_I(t_1, t_2) \phi_I(x_2) \cdots \phi_I(x_n) U_I(t_n, t_-) | 0 \rangle}{\langle 0 | U(t_+, t_-) | 0 \rangle} is equivalent to the time-ordered expression GF(n)(x1,…,xn)=lim⁡t±→±∞⟨0∣T[ϕI(x1)⋯ϕI(xn)e−i∫t−t+d4x HI]∣0⟩⟨0∣Te−i∫t−t+d4x HI∣0⟩.G_F^{(n)}(x_1, \ldots, x_n) = \lim_{t_\pm \to \pm\infty} \frac{\langle 0 | T\left[\phi_I(x_1) \cdots \phi_I(x_n) e^{-i \int_{t_-}^{t_+} d^4x\, \mathcal{H}_I}\right] | 0 \rangle}{\langle 0 | T e^{-i \int_{t_-}^{t_+} d^4x\, \mathcal{H}_I} | 0 \rangle}.

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