We show that the Lorentzian Snyder models, together with their Galilei and Carroll limiting cases, can be rigorously constructed through the projective geometry description of Lorentzian, Galilean and Carrollian spaces with nonvanishing constant curvature. The projective coordinates of such curved spaces take the role of momenta, while translation generators over the same spaces are identified with noncommutative spacetime coordinates. In this way, one obtains a deformed phase space algebra, which fully characterizes the Snyder model and is invariant under boosts and rotations of the relevant kinematical symmetries. While the momentum space of the Lorentzian Snyder models is given by certain projective coordinates on (Anti-)de Sitter spaces, we discover that the momentum space of the Galilean (Carrollian) Snyder models is given by certain projective coordinates on curved Carroll (Newton--Hooke) spaces. This exchange between the Galilei and Carroll limits emerging in the transition from the geometric picture to the phase space picture is traced back to an interchange of the role of coordinates and translation operators. As a physically relevant feature, we find that in Galilean Snyder spacetimes the time coordinate does not commute with space coordinates, in contrast with previous proposals for non-relativistic Snyder models, which assume that time and space decouple in the non-relativistic limit $c\to \infty$. This remnant mixing between space and time in the non-relativistic limit is a quite general Planck-scale effect found in several quantum spacetime models.

中文翻译：

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洛伦兹斯奈德时空和他们的伽利略和卡罗尔限制从射影几何

我们表明，洛伦兹斯奈德模型及其伽利略和卡罗尔极限情况可以通过洛伦兹、伽利略和卡罗尔空间的射影几何描述来严格构造，这些空间具有非零常曲率。这种弯曲空间的射影坐标扮演动量的角色，而相同空间上的平移生成器被标识为非对易时空坐标。这样，就得到了变形的相空间代数，它充分表征了 Snyder 模型，并且在相关运动学对称性的提升和旋转下是不变的。虽然洛伦兹斯奈德模型的动量空间由（反）德西特空间上的某些投影坐标给出，我们发现伽利略（卡罗尔）斯奈德模型的动量空间是由弯曲卡罗尔（牛顿-胡克）空间上的某些投影坐标给出的。在从几何图像到相空间图像的过渡中出现的伽利略和卡罗尔极限之间的这种交换可以追溯到坐标和平移算子角色的交换。作为一个物理相关的特征，我们发现在伽利略斯奈德时空中，时间坐标不与空间坐标交换，这与之前非相对论斯奈德模型的提议形成对比，后者假设时间和空间在非相对论极限 $c 中解耦\到\infty$。在非相对论极限中空间和时间之间的这种残余混合是在几个量子时空模型中发现的一种非常普遍的普朗克尺度效应。