Why Are Two Indices Used for the Generators of Lorentz and Poincare Groups?

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SUMMARY

The discussion clarifies that the generators of the Lorentz and Poincaré groups are labeled with two indices due to the nature of transformations they represent. Specifically, just as a 3D rotation is defined by angles between two axes, the use of two indices for Lorentz transformations serves to specify relationships between different dimensions. While one index could suffice, using two provides a clearer representation of the transformation's complexity.

PREREQUISITES
  • Understanding of Lorentz transformations
  • Familiarity with Poincaré group concepts
  • Basic knowledge of matrix representations
  • Comprehension of 3D rotations and their mathematical implications
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  • Research the mathematical structure of Lorentz transformations
  • Explore the properties of the Poincaré group in theoretical physics
  • Study the role of indices in tensor notation
  • Learn about the implications of multi-index notation in advanced mathematics
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The discussion is beneficial for physicists, mathematicians, and students studying theoretical physics, particularly those focusing on group theory and its applications in relativity.

Bobhawke
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Just a quick question here: I was going through my notes and I noticed that the generators of both these groups are labeled two indices. I was wondering if there is any particular reason for this, since it seems to me that one index would work perfectly well.

Thanks
 
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There is no more reason to attach 2 indices to a Lorentz transformation than to an ordinary 3D rotation, or a matrix in general. A rotation will be specified by angles between one axis (first index) and another axis (second index). If it pleases you, you can store them in a vector, but that is clumsy.
 

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