Srednicki — Quantum Field Theory · Problem 92.1

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · derricks-theorem-solitons

Problem

Derrick's theorem says that, in a theory with scalar fields only, there are no solitons localized in more than one dimension. To prove this, consider a theory in DD space dimensions with a set of real scalar fields φi\varphi_i; any complex scalar fields are written as a pair of real ones. The lagrangian is L=−12∂μφi∂μφi−V(φi)\mathcal{L} = -\frac{1}{2}\partial^\mu\varphi_i\partial_\mu\varphi_i - V(\varphi_i), with V(φi)≥0V(\varphi_i) \geq 0. Suppose we have a soliton solution φi(x)\varphi_i(\mathbf{x}); its energy is E=T+UE = T + U, where T=12∫dDx (∇φi)2T = \frac{1}{2}\int d^Dx\,(\nabla\varphi_i)^2 and U=∫dDx V(φi)U = \int d^Dx\,V(\varphi_i). a) Now consider φi(x/α)\varphi_i(\mathbf{x}/\alpha), where α\alpha is a positive real number. Show that, for this field configuration, the energy is E(α)=αD−2T+αDUE(\alpha) = \alpha^{D-2}T + \alpha^D U. b) Argue that we must have E′(1)=0E'(1) = 0. c) Use this to prove the theorem.

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