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Interpolating Helicity Spinors Between the Instant Form and the Light-front Form

We discuss the helicity spinors interpolating between the instant form dynamics (IFD) and the front form dynamics, or the light-front dynamics (LFD), and present the interpolating helicity amplitudes as well as their squares for the scattering of two fermions, and the annihilation of fermion and anti-fermion. We parametrize the interpolation between the two dynamics, IFD and LFD, by an interpolation angle and derive not only the generalized helicity spinors in the $(0,J)\oplus(J,0)$ chiral representation that links naturally the two typical IFD vs. LFD helicity spinors but also the generalized Melosh transformation that relates these generalized helicity spinors to the usual Dirac spinors. Analyzing the directions of the particle momentum and spin with the variation of the interpolation angle, we inspect the whole landscape of the generalized helicity intermediating between the usual Jacob-Wick helicity in the IFD and the light-front helicity in the LFD. Our analysis clarifies the characteristic difference of the helicity amplitudes between the IFD and the LFD. In particular, we find that the behavior of the angle between the momentum direction and the spin direction bifurcates at a critical interpolation angle and the IFD and the LFD separately belong to the two different branches bifurcated at this critical interpolation angle. This finding further clarifies any conceivable confusion in the prevailing notion of the equivalence between the infinite momentum frame and the LFD. The existence of the universal J-curve found in our previous works of scalar field theory and the sQED theory is confirmed in the present work of interpolating helicity amplitudes for the fermion scattering and annihilation processes. In conjunction with the bifurcation of branches, the two boundaries appear in the interpolating helicity amplitudes and interestingly the J-curve persists within these two boundaries.