Is the vectorial representation of the Lorentzian Group unitary?

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Discussion Overview

The discussion centers on whether the vectorial representation of the Lorentzian Group is unitary. Participants explore theoretical aspects of representation theory, particularly in the context of the Lorentz group and its properties.

Discussion Character

  • Technical explanation, Debate/contested

Main Points Raised

  • One participant expresses a strong belief that the vectorial representation is not unitary but seeks confirmation from others.
  • Another participant claims that the Lorentz group has no unitary representation of finite dimension, except for the trivial representation, and notes that the vector representation has dimension 4.
  • A participant mentions performing a calculation involving the product of Lorentz matrices to seek validation of their reasoning.
  • Another participant attributes the non-unitarity to the non-compact nature of the Lorentz group, referencing a classical result found in a specific compendium.

Areas of Agreement / Disagreement

Participants do not reach a consensus, as there are differing views on the unitary nature of the vectorial representation and the implications of the Lorentz group's properties.

Contextual Notes

Some assumptions about the definitions of unitary representations and the implications of non-compactness are not fully explored, leaving certain mathematical steps unresolved.

IRobot
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I am 99% sure it is not, but I would like to hear that from someone else to be more serene.
 
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Thanks I already was thinking with that argument, I did the calculation of [itex]\Lambda \Lambda ^\dagger[/itex] for some random Lorentz Matrix. Was just looking for a confirmation.
 
Last edited:
It's because the Lorentz group (or its component connected to the identity) is non-compact. It's a classical result, a proof of which can be found in Cornwell's compendium.
 

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