What are the Discret Subgroups of O(1,3)?

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Discussion Overview

The discussion revolves around identifying the discrete subgroups of the Lorentz group O(1,3). It includes theoretical considerations and decompositions of the Lorentz group into various subsets.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants describe the proper orthochronous Lorentz group as the largest subgroup of the Lorentz group, emphasizing its properties related to unit determinant and preservation of time-like vector direction.
  • There is a proposal to decompose the Lorentz group into the restricted Lorentz group and other sets, with the latter not forming a group but rather a collection of transformations.
  • Further decomposition into four sets is mentioned, including proper antichronous Lorentz transformations and both proper and improper orthochronous and antichronous transformations.
  • One participant expresses confusion about how the provided information addresses the original question regarding discrete subgroups.
  • Another participant acknowledges distraction from the main question and apologizes for the deviation.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the original question regarding discrete subgroups, and there is a recognition of distraction from the main topic.

Contextual Notes

The discussion lacks clarity on the specific nature of discrete subgroups and does not resolve the relationship between the mentioned transformations and the original inquiry.

Andre' Quanta
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Which are the discret subgroups of O(1,3)?
 
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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.
 
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.

How does that answer his question?
 
micromass said:
How does that answer his question?
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!
 

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