Discussing the mathematical formalism of generators (Lorentz Group)

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SUMMARY

The discussion centers on the mathematical formalism of the Lorentz group, specifically its generators and representations. Key equations include the infinitesimal transformations represented by equations (10.11) and (10.12), and the expression of any Lorentz transformation as an exponential of generators in equation (10.16). The participants explore the implications of closed commutation relations and the Baker-Campbell-Hausdorff formula, which facilitate the representation of Lorentz transformations through matrix exponentials. The concept of Wigner rotation is also introduced, emphasizing the relationship between boosts and rotations.

PREREQUISITES
  • Understanding of Lorentz transformations and the Lorentz group
  • Familiarity with matrix representations and operators
  • Knowledge of the Baker-Campbell-Hausdorff formula
  • Basic concepts of group theory and commutation relations
NEXT STEPS
  • Study the derivation of infinitesimal Lorentz transformations in detail
  • Learn about the Baker-Campbell-Hausdorff formula and its applications in physics
  • Explore the concept of Wigner rotation and its implications in special relativity
  • Investigate the mathematical properties of the generators of the Lorentz group
USEFUL FOR

Physicists, mathematicians, and students studying quantum field theory, particularly those interested in the mathematical foundations of the Lorentz group and its applications in theoretical physics.

  • #31
samalkhaiat said:
Rotations or non-rotations, by the object (M_{\alpha \beta})^{\mu}{}_{\nu}, we always mean the (\mu\nu) matrix elements of the matrix M_{\alpha \beta}.

Alright, so the upper index refers to rows and the lower to columns always. Thanks for the clarification.
 
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  • #32
PeroK said:
[...] the Baker-Campbell-Hausdorf formula [...]

I've found a reliable source about the Baker-Campbell-Hausdorf formula: problem 3 here. The solution is here :smile: .
 
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Likes   Reactions: vanhees71 and PeroK

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