Interpolating Quantum Electrodynamics between Instant and Front Forms

TL;DR

This paper develops a formal interpolation between instant and front forms of relativistic dynamics in QED, highlighting unique fermion constraint effects and their impact on scattering amplitudes across different frames.

Contribution

It introduces a formal derivation of interpolating QED connecting instant and front forms, emphasizing the fermion constraint and helicity differences, with explicit calculations of scattering amplitudes.

Findings

Fermion propagator differs significantly between IFD and LFD.

Helicity in LFD depends on the reference frame, unlike in IFD.

Scattering amplitudes exhibit frame and angle dependence across the interpolation.

Abstract

The instant form and the front form of relativistic dynamics proposed by Dirac in 1949 can be linked by an interpolation angle parameter $\delta$ spanning between the instant form dynamics (IFD) at $\delta =0$ and the front form dynamics which is now known as the light-front dynamics (LFD) at $\delta =\pi/4$. We present the formal derivation of the interpolating quantum electrodynamics (QED) in the canonical field theory approach and discuss the constraint fermion degree of freedom which appears uniquely in the LFD. The constraint component of the fermion degrees of freedom in LFD results in the instantaneous contribution to the fermion propagator, which is genuinely distinguished from the ordinary equal-time forward and backward propagation of relativistic fermion degrees of freedom. As discussed in our previous work, the helicity of the on-mass-shell fermion spinors in LFD is also distinguished from the ordinary Jacob-Wick helicity in the IFD with respect to whether the helicity depends on the reference frame or not. To exemplify the characteristic difference of the fermion propagator between IFD and LFD, we compute the helicity amplitudes of typical QED processes such as $e^+ e^- \to \gamma \gamma$ and $e \gamma \to e \gamma$ and present the whole landscape of the scattering amplitudes in terms of the frame dependence or the scattering angle dependence with respect to the interpolating angle dependence. Our analysis clarifies any conceivable confusion in the prevailing notion of the equivalence between the infinite momentum frame approach and the LFD.