From Lorentz to $SIM(2)$: contraction, four-dimensional algebraic relations and projective representations
J. E. Rodrigues, J. M. B. Matzenbacher, G. M. Caires da Rocha, J. M. Hoff da Silva
TL;DR
This work analyzes the $SIM(2)$ and $ISIM(2)$ groups as Lorentz subgroups with a privileged spatial direction, deriving a complete four-dimensional algebraic framework for $\mathfrak{sim(2)}$ and $\mathfrak{isim(2)}$ via an Inönü–Wigner contraction from the Lorentz group and constructing explicit 4×4 matrix representations for both homogeneous and inhomogeneous algebras. Using Bargmann’s theory, it tracks the sources of local phase factors in their projective representations, showing that most phases are removable but that a nontrivial central charge persists (and the second group cohomology is isomorphic to $\mathbb{Z}$). The analysis provides a systematic method to diagnose local phases in non-abelian settings through the $R$-set construction and demonstrates how a decoupled abelian extension arises under contraction. Overall, the paper supplies a rigorous algebraic and representation-theoretic foundation for $SIM(2)$-type symmetries and clarifies their potential physical implications in contexts where a privileged spacetime direction plays a role.
Abstract
We present a comprehensive study on $SIM(2)$ and $ISIM(2)$ groups, their representations and algebraic aspects. After obtaining $SIM(2)$ through the Inönü-Wigner contraction procedure, a complete four-dimensional algebraic representation is shown for $\mathfrak{sim(2)}$ and $\mathfrak{isim(2)}$. Besides that, we apply Bargmann's formalism to investigate the (projective) representations for both cases, keeping track of the source of phase factors. We complete the study by presenting a particularly simple analysis to probe the existence of local phase factors, which is useful when dealing with non-abelian groups.