Interpolating scattering amplitudes between the instant form and the front form of relativistic dynamics

Abstract

Among the three forms of relativistic Hamiltonian dynamics proposed by Dirac in 1949, the instant form and the front form can be interpolated by introducing an interpolation angle between the ordinary time $t$ and the light-front time $(t+z/c)/\sqrt{2}$. Using this method, we introduce the interpolating scattering amplitude that links the corresponding time-ordered amplitudes between the two forms of dynamics and provide the physical meaning of the kinematic transformations as they allow the invariance of each individual time-ordered amplitude for an arbitrary interpolation angle. In particular, it exhibits that the longitudinal boost is kinematical only in the front form dynamics, or the light-front dynamics (LFD), but not in any other interpolation angle dynamics. It also shows that the disappearance of the connected contributions to the current arising from the vacuum occurs when the interpolation angle is taken to yield the LFD. Since it doesn't require the infinite momentum frame (IMF) to show this disappearance and the proof is independent of reference frames, it resolves the confusion between the LFD and the IMF. The well-known utility of IMF usually discussed in the instant form dynamics is now also extended to any other interpolation angle dynamics using our interpolating scattering amplitudes.