16.4. Derrick theorem The aim of this exercise is to show that the static solitonic solution of finite energy can only exist for 1+1 dimensional theories. Consider, in the (d +1) dimensional Minkowski space, the Lagrangian L = 1 2

∂μφ∂μφ −U(φ), where U(φ) is a non-negative function, that vanishes at the vacua of the theory. The static energy E can be written as E =W1 +W2, where W1 = 1 2

 ddx(∇φ)2, W2 =  ddxU(φ).

Let φ(x) be a static solution of the equation of motion of the theory.

a. Determine the variation of W1 and W2 under the transformation φ(x)→φ(λx).

b. Using the condition that φ(x) is a solution of the equation of motion, show that the energy E[λ] is stationary for λ = 1

c. Since W1 ≥ 0 and W2 ≥ 0, show that only non-vanishing solutions exist for d ≤ 2.