The sum is ∑ ∑ i = 0 1 n i exp ⁡ ⁡ ( β β α α i ϕ ϕ ) = exp ⁡ ⁡ ( β β ϕ ϕ ) + exp ⁡ ⁡ ( − − β β ϕ ϕ ) . Then if β β is purely imaginary, β β = i b with b real and, without loss of generality, positive, then this is 2 cos ⁡ ⁡ ( b ϕ ϕ ) . The Lagrangian is then L = 1 2 ∂ ∂ μ μ ϕ ϕ ∂ ∂ μ μ ϕ ϕ + 2 m 2 b 2 cos ⁡ ⁡ ( b ϕ ϕ ) , which is the sine-Gordon Lagrangian.