Gamma5 x d^2 Lorentz invariant?

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Discussion Overview

The discussion centers on the Lorentz invariance of the term \(\overline{\Psi} \gamma^5 \partial^2 \Psi\), where \(\Psi\) is a Dirac field. Participants explore how this term transforms under Lorentz transformations and the implications of parity transformations.

Discussion Character

  • Technical explanation, Debate/contested

Main Points Raised

  • One participant asks whether the term \(\overline{\Psi} \gamma^5 \partial^2 \Psi\) is Lorentz invariant and how it transforms under Lorentz transformations.
  • Another participant asserts that it transforms as a pseudoscalar.
  • A subsequent reply suggests that it behaves as a scalar under Lorentz transformations but changes sign under parity, questioning if this implies Lorentz invariance.
  • Another participant clarifies that since parity is a Lorentz transformation, the term should be considered invariant under Lorentz transformations that do not include spatial inversion.

Areas of Agreement / Disagreement

Participants express differing views on the implications of parity transformations on Lorentz invariance, indicating that the discussion remains unresolved regarding the overall invariance of the term.

Contextual Notes

The discussion does not clarify the specific conditions under which the term is considered invariant or the implications of different types of Lorentz transformations.

cbetanco
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is [itex]\overline{\Psi}[/itex] γ5 [itex]\partial[/itex]2 [itex]\Psi[/itex] Lorentz Invariant? How does this term transform under Lorentz transformations? Here [itex]\Psi[/itex] is a Dirac field.

Thanks
 
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It transforms as a pseudoscalar.
 
So it changes as a scalar with respect to Lorentz transformations, but changes sign under parity? This means it is Lorentz invariant?
 
The parity transformation is a Lorentz transformation, so you would have to say it is invariant under Lorentz transformations which do not include r --> -r.
 

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