Discussing the mathematical formalism of generators (Lorentz Group)

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Discussion Overview

The discussion revolves around the mathematical formalism of the Lorentz group, specifically focusing on the generators of the group and their representations through matrix forms. Participants explore the implications of Lorentz transformations, including boosts and rotations, and the mathematical relationships governing them.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants describe the Lorentz group as encompassing rotations and boosts that satisfy the Lorentz condition, with representations being matrix embeddings of these transformations.
  • There is a discussion on the nature of boosts and rotations, noting that the composition of two boosts generally results in a boost combined with a rotation, which some participants refer to as the Wigner rotation.
  • One participant expresses uncertainty about the origin of specific equations related to infinitesimal transformations and seeks clarification on how they lead to the representation of Lorentz transformations.
  • Another participant explains that any Lorentz transformation can be expressed as an exponential of generators, referencing the Baker-Campbell-Hausdorff formula to support this claim.
  • Some participants challenge each other's understanding of the mathematical properties of the Lorentz group, particularly regarding commutation relations and their implications for the structure of the group.
  • There is an exploration of the commutator of two boosts, with one participant suggesting that it results in a rotation, leading to a discussion about the expected axis of rotation.

Areas of Agreement / Disagreement

Participants generally agree on the mathematical structure of the Lorentz group and the role of generators, but there are differing views on the implications of composing boosts and the specific mathematical details of certain transformations. The discussion remains unresolved regarding the derivation and understanding of specific equations.

Contextual Notes

Some participants express uncertainty about specific mathematical steps and the implications of the Baker-Campbell-Hausdorff formula. There are also references to the need for further exploration of commutation relations and their effects on the composition of transformations.

  • #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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