An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 4.1

Interacting Fields and Feynman Diagrams — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · interacting-fields-feynman-diagrams

Problem

Let us return to the problem of the creation of Klein-Gordon particles by a classical source. Recall from Chapter 2 that this process can be described by the Hamiltonian H=H0+∫d3x (−j(t,x)ϕ(x)),H = H_0 + \int d^3x\,(-j(t,\mathbf{x})\phi(x)), where H0H_0 is the free Klein-Gordon Hamiltonian, ϕ(x)\phi(x) is the Klein-Gordon field, and j(x)j(x) is a c-number scalar function. We found that, if the system is in the vacuum state before the source is turned on, the source will create a mean number of particles ⟨N⟩=∫d3p(2π)312Ep∣j~(p)∣2.\langle N\rangle = \int \frac{d^3p}{(2\pi)^3}\frac{1}{2E_\mathbf{p}}|\tilde{j}(p)|^2. In this problem we will verify that statement, and extract more detailed information, by using a perturbation expansion in the strength of the source. (a) Show that the probability that the source creates no particles is given by P(0)=∣⟨0∣ T{exp⁡[i∫d4x j(x)ϕI(x)]} ∣0⟩∣2.P(0) = \left|\langle 0|\,T\Big\{\exp\Big[i\int d^4x\,j(x)\phi_I(x)\Big]\Big\}\,|0\rangle\right|^2. (b) Evaluate the term in P(0)P(0) of order j2j^2, and show that P(0)=1−λ+O(j4)P(0) = 1 - \lambda + \mathcal{O}(j^4), where λ\lambda equals the expression given above for ⟨N⟩\langle N\rangle. (c) Represent the term computed in part (b) as a Feynman diagram. Now represent the whole pertubation series for P(0)P(0) in terms of Feynman diagrams. Show that this series exponentiates, so that it can be summed exactly: P(0)=exp⁡(−λ)P(0) = \exp(-\lambda). (d) Compute the probability that the source creates one particle of momentum kk. Perform this computation first to O(j)\mathcal{O}(j) and then to all orders, using the trick of part (c) to sum the series. (e) Show that the probability of producing nn particles is given by P(n)=(1/n!)λnexp⁡(−λ).P(n) = (1/n!)\lambda^n \exp(-\lambda). This is a Poisson distribution. (f) Prove the following facts about the Poisson distribution: ∑n=0∞P(n)=1;⟨N⟩=∑n=0∞n P(n)=λ.\sum_{n=0}^{\infty} P(n) = 1; \qquad \langle N\rangle = \sum_{n=0}^{\infty} n\,P(n) = \lambda. The first identity says that the P(n)P(n)'s are properly normalized probabilities, while the second confirms our proposal for ⟨N⟩\langle N\rangle. Compute the mean square fluctuation ⟨(N−⟨N⟩)2⟩\langle (N - \langle N\rangle)^2\rangle.

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