The passage discusses the contraction of Lie groups and their representations, focusing on the transition from relativistic to classical mechanics. It introduces the concept of contraction, where a Lie group is transformed into another group by singular transformations of its infinitesimal elements. The contraction process involves reducing the number of parameters and commuting certain infinitesimal operators, leading to a limit where the original group contracts to an infinitesimally small neighborhood of a subgroup. The authors provide examples of contractions, such as the contraction of the inhomogeneous Lorentz group to the Galilei group and the contraction of the three-dimensional rotation group to the Euclidean group. They also explore the representation theory of these contracted groups, showing how representations of the original groups can be transformed or approximated by sequences of representations to yield representations of the contracted groups. In the context of the Lorentz groups, the authors specifically examine the contraction of the special theory of relativity (the inhomogeneous Lorentz group) to the Galilei group. They analyze the irreducible representations of the Lorentz group and show that while some representations cannot be contracted, others can be approximated up to a factor by adjusting the parameters and operators. This generalization of the contraction concept allows for the contraction of representations with positive rest mass, though the behavior of representations with zero rest mass is not discussed in detail.The passage discusses the contraction of Lie groups and their representations, focusing on the transition from relativistic to classical mechanics. It introduces the concept of contraction, where a Lie group is transformed into another group by singular transformations of its infinitesimal elements. The contraction process involves reducing the number of parameters and commuting certain infinitesimal operators, leading to a limit where the original group contracts to an infinitesimally small neighborhood of a subgroup. The authors provide examples of contractions, such as the contraction of the inhomogeneous Lorentz group to the Galilei group and the contraction of the three-dimensional rotation group to the Euclidean group. They also explore the representation theory of these contracted groups, showing how representations of the original groups can be transformed or approximated by sequences of representations to yield representations of the contracted groups. In the context of the Lorentz groups, the authors specifically examine the contraction of the special theory of relativity (the inhomogeneous Lorentz group) to the Galilei group. They analyze the irreducible representations of the Lorentz group and show that while some representations cannot be contracted, others can be approximated up to a factor by adjusting the parameters and operators. This generalization of the contraction concept allows for the contraction of representations with positive rest mass, though the behavior of representations with zero rest mass is not discussed in detail.