Uniqueness in the characteristic Cauchy problem of the Klein–Gordon equation and tame restrictions of generalized functions

Abstract

Abstract. We show that every tempered distribution, which is a solution of the (homogenous) Klein-Gordon equation, admits a "tame" restriction to the characteristic (hyper)surface {x 0 + x n = 0} in (1 + n)-dimensional Minkowski space and is uniquely determined by this restriction. The restriction belongs to the space S ′ ∂− (R n ) which we have introduced in [16]. Moreover, we show that every element of S ′ ∂− (R n ) appears as the "tame" restriction of a solution of the (homogeneous) Klein-Gordon equation.

On the problem of mass dependence of the two-point function of the real scalar free massive field on the light cone

“…The restriction problem, i.e., the problem that the real scalar free massive field has no canonical restriction to Σ = {x 0 + x 3 = 0}, manifests itself in the problem that the (positive-/ negative-frequency) Pauli-Jordan has no canonical restriction to Σ in the sense of distribution theory. By using the so-called tame restriction of a tempered distribution, which we have already introduced in 14 , we have seen that also the assumed inconsistency of the mass-dependence of the two-point function on Σ can be resolved. Thus the result of this paper contributes to the philosophy (introduced in 13 ) that S ∂ − (R 3 ) -instead of S(R 3 ) -is the right test function space when treating quantum fields on the null plane Σ.…”

“…Since the tame restriction of the free field to Σ is also independent of mass 13,15 no inconsistency appears if we take the tame restrictions (to Σ) on both sides of (V.2). First of all we have to recall the definition of the tame restriction of a generalized function to Σ -for details see 13,14 . Definition V.1.…”