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

TL;DR

The work addresses how conformal symmetry in $(1+1)$-D can be smoothly interpolated between instant-form and light-front dynamics by introducing an interpolation angle $\delta$ and constructing explicit $4\times4$ projective-spacetime representations. It derives six generators $P_{\hat{\pm}},D_{\hat{\pm}},\mathfrak{K}_{\hat{\pm}}$ that reproduce the IFD algebra at $\delta=0$ and the LFD algebra at $\delta=\pi/4$, and extends the structure to a Witt-type algebra with interpolating generators $l^{\hat{\pm}}_n$ whose commutators interpolate between the two limits. The analysis shows that LFD maximizes kinematic generators (yielding a direct sum $SO(2,1)\oplus SO(2,1)$) and provides $4\times4$ (and extendable to $6\times6$) projective-spacetime representations, clarifying how conformal transformations act on interpolating coordinates $x^{\hat{+}}$. This framework offers a concrete, algebraically controlled route to reduce dynamical complexity in $(1+1)$-D QFTs and lays groundwork for extending to $(3+1)$-D in future work.

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.