7.7. Field theories with soliton solutions Consider the Lagrangian field theory in 1+1 dimensions

˜L

= 1 2

(∂μϕ)2 + m2

β2 [cos(βϕ)−1].

a. Expand in powers of β and show that this model corresponds to a Landau–

Ginzburg theory with an infinite number of couplings.

b. Write the equation of motion of the field ϕ(x, t).

c. Prove that the configurations

ϕ(±)(x,0) = ±4 arctan [exp(x−x0)],

(where x0 is an arbitrary point), are both classical solutions of the static version of the equation of motion.

d. Show that these configurations interpolate between two next-neighbour vacua.

These configurations correspond to topological excitations of the field, called soliton and anti-soliton.

e. Compute the stress-energy tensor and use the formula H = T00(x)dx to determine the energy of the solitons. Since they are static, their energy corresponds to their M. Prove that M = 8m β
.
Note that the coupling constant is at the denominator, so that this is a nonperturbative expression.