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Andre' Quanta
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Which are the discret subgroups of O(1,3)?
Shyan said:The biggest subgroup of the Lorentz group is the proper orthochronous Lorentz group, which is called the restricted Lorentz group. Its the group of Lorentz transformations with unit determinant that preserve a past(future)-directedness of time-like vectors.
If you decompose the Lorentz group as the union of restricted Lorentz group and another set, the other set won't be a group, its just a set of transformations. You still can decompose it further into a union of three more sets. The set of proper antichronous Lorentz transformations, the set of improper orthochronous Lorentz transformations and the set of improper antichronous Lorentz transformations.
Yeah...sorry! I read the question and it reminded me of that decomposition. Then looks like I got completely distracted from the main question. Sorry!micromass said:How does that answer his question?
A discret subgroup of O(1,3) is a subset of the special orthogonal group in four dimensions that is closed under multiplication and has a discrete topology. It is commonly used in the study of hyperbolic geometry and the theory of relativity.
Discret subgroups of O(1,3) are closely related to Lorentz transformations, which are transformations that preserve the spacetime interval in special relativity. In fact, the elements of a discret subgroup of O(1,3) can be thought of as discrete versions of Lorentz transformations.
Studying discret subgroups of O(1,3) is important for understanding the geometry of spacetime in special relativity. They can also be used to construct models of hyperbolic geometry, which has applications in fields such as physics, mathematics, and computer graphics.
Discret subgroups of O(1,3) can be classified based on their properties, such as the number of generators, type of generators, and the values of their parameters. This classification system helps to identify and differentiate between different types of discret subgroups.
Yes, there are several real-world applications of discret subgroups of O(1,3). They are used in the study of relativistic systems, such as black holes and particle collisions. They also have applications in computer graphics for creating 3D models with hyperbolic geometry, and in cryptography for creating secure encryption algorithms.