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

We investigate the generally assumed inconsistency in light cone quantum field theory that the restriction of a massive, real scalar free field to the nullplane Σ = {x⁰ + x³ = 0} is independent of mass (Leutwyler, Klauder and Streit 1970 Nuovo Cimento A 66 536), but the restriction of the two-point function is mass dependent (see, e.g., Nakanishi and Yamawaki 1977 Nucl. Phys. B 122 15; Yamawaki K 1997 Proc. Int. Workshop New Nonperturbative Methods and Quantization on the Light Cone (Les Houches, France) Preprint hep-th/9707141). We resolve this inconsistency by showing that the two-point function has no canonical restriction to Σ in the sense of distribution theory. Only the so-called tame restriction of the two-point function, which we have introduced in (Ullrich P 2004 Uniqueness in the characteristic Cauchy problem of the Klein-Gordon equation and tame restrictions of generalized functions Preprint math-ph/0408022 (submitted)) exists. Furthermore, we show that this tame restriction is indeed independent of the mass. Hence the inconsistency is induced by the erroneous assumption that the two-point function has a (canonical) restriction to Σ.