Interpolating conformal algebra in $(1+1)$ dimensions between the instant form and the light-front form of relativistic dynamics

TL;DR

This paper develops an interpolating conformal algebra in (1+1) dimensions bridging instant form and light-front dynamics, highlighting the utility of light-front form in simplifying quantum field theory calculations.

Contribution

It introduces a novel interpolating conformal algebra and matrix representations, clarifying the change in kinematic and dynamic generators between different forms of relativistic dynamics.

Findings

Maximizes kinematic generators in light-front dynamics

Minimizes dynamic generators in light-front dynamics

Simplifies solving (1+1) dimensional quantum field theories

Abstract

We present the interpolating conformal algebra between the instant form dynamics (IFD) and the light-front dynamics (LFD) in $(1+1)$ dimensions, along with a $4\times4$ interpolating projective spacetime matrix representation. While there are six generators in the $(1+1)$ dimensional conformal algebra, the number of kinematic and dynamic generators dramatically changes in LFD, maximizing (minimizing) the number of kinematic (dynamic) generators to four (two) with respect to two (four) kinematic (dynamic) generators in IFD, as well as in any other forms of dynamics between IFD and LFD. It confirms and signifies the utility of LFD, saving substantial dynamical efforts in solving the $(1+1)$ dimensional quantum field theories. We also present $2\times2$ Pauli matrix representation of $(1+0)$ and $(0+1)$ conformal groups, and creation/annihilation operators of quantum simple harmonic oscillator representations of $(1+0)$ dimensional conformal groups.