17.8. Interpretation of the two-dimensional S-matrix.

The non-relativistic S-matrix of a particle of mass m = 1 relative to the potential V(x) =

−2aπδ(x) is given by

˜S

(k) = k+iπa k−iπa

.

If we would like to generalize this result to the relativistic case, we must use the rapidity variable θ. Note that for small values of the momentum, θ k. Substituting in the expression of S, we have

˜S

(θ) = θ +iπa

θ −iπa

.

This expression, however, does not fulfill the important property S(θ) = S(θ ±2πi) of the relativistic S-matrix.

a. Discuss how it can be iteratively implemented the periodicity of the relativistic S-matrix starting from ˜S (θ).

b. Use the infinite product representation of the hyperbolic function sinhx sinhx = x ∞

k=1 1+ x kπ

2 , to show that the final result can be expressed as S(θ) = sinh 12 (θ +iπa)
sinh 12 (θ −iπa)
= sa(θ).