Confused About Lorentz Generators

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Discussion Overview

The discussion focuses on the generators of the Lorentz group, specifically the definitions and properties of the generators Ji and Mij, including the factor of (1/2) in their equations. The scope includes theoretical aspects of group generators in physics.

Discussion Character

  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant expresses confusion regarding the factor of (1/2) in the definition of Ji, questioning its necessity given the relationship between Ji and Mij.
  • Another participant explains that the factor of (1/2) is necessary to avoid double counting terms due to the antisymmetry of the indices in both the epsilon and M terms.
  • A participant seeks clarification by providing an example involving M23 and asks if the reasoning about the antisymmetry holds true in that case.
  • A later reply confirms the correctness of the example provided regarding the antisymmetry and additive nature of the terms.

Areas of Agreement / Disagreement

Participants appear to agree on the reasoning behind the factor of (1/2) and the antisymmetry of the terms, but the initial confusion expressed by the first participant indicates that some uncertainty remains regarding the definitions.

Contextual Notes

The discussion does not resolve the broader implications of the definitions or their applications in different contexts, and the initial confusion suggests that further clarification may be needed for those less familiar with the topic.

nigelscott
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I am looking at the generators of the Lorentz group. The literature commonly refers to the generators as
Mij, Ji and Ki and defines:

Ji = (1/2)∈ijkMjk

I am confused about the factor of (1/2) in this equation as I thought that Mij is essentially the same as Ji

This also shows up in

Λ= exp((1/2)ΩρσMρσ) ≡ 1 + (1/2)ΩρσMρσ

see http://www.phys.ufl.edu/~fry/6607/lorentz.pdf
 
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The 1/2 is needed not to count the same term of the sum twice, because the epsilon and M have the same index symmetry, i.e. they are both antisymmetric over the two indices summed over.
 
OK. Just to clarify. For M23 this would look like:

J1 = (1/2)[∈123M23 + ∈132M32]

Since M23 = -M32 and ∈123 = -∈132 the second term is additive.

Correct?
 
Last edited:

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