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arXiv · 2504.13306

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
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Abstract

We present a comprehensive study on $SIM(2)$ and $ISIM(2)$ groups, their representations and algebraic aspects. These groups, together with $HOM(2)$, arise as the symmetry groups of Very Special Relativity (VSR), where full Lorentz invariance is reduced while retaining many relativistic consequences. After obtaining $SIM(2)$ through the In\"on\"u-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.

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J. E. Rodrigues, J. M. B. Matzenbacher, G. M. Caires da Rocha, J. M. Hoff da Silva. 2025-04-17. From Lorentz to $SIM(2)$: contraction, four-dimensional algebraic relations and projective representations. https://doi.org/10.1142/s021988782650177x

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