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

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.