Invariant Tensors and Lorentz Transformation

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    Invariant Tensors
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SUMMARY

The discussion centers on the invariance of tensors under Lorentz transformations, specifically focusing on the Kronecker delta and the Levi-Civita epsilon as the only irreducible invariant tensors. Participants confirm the invariance of these tensors and express a need for a comprehensive proof that no additional invariant tensors exist. The conversation also touches on the definition of tensors and their behavior under linear transformations, suggesting that full tensors should maintain invariance across all such transformations.

PREREQUISITES
  • Understanding of Lorentz transformations
  • Familiarity with tensor algebra
  • Knowledge of Kronecker delta and Levi-Civita epsilon
  • Basic concepts of linear transformations
NEXT STEPS
  • Research the proof of invariance for the Kronecker delta and Levi-Civita epsilon under Lorentz transformations
  • Explore the general procedure for identifying invariant tensors in linear transformations
  • Study the properties of full tensors and their invariance across various transformations
  • Investigate advanced topics in tensor calculus related to irreducible representations
USEFUL FOR

This discussion is beneficial for physicists, mathematicians, and students studying theoretical physics, particularly those focused on tensor analysis and Lorentz transformations.

Heirot
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It is often stated that the Kronecker delta and the Levi-Civita epsilon are the only (irreducible) invariant tensors under the Lorentz transformation. While it is fairly easy to prove that the two tensors are indeed invariant wrt Lorentz transformation, I have not seen a proof that there aren't any more such tensors.

So, my question is, how to find all invariant tensors under some (linear) transformation? Is there a general procedure for this?

Thanks
 
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If an object is a full tensor, then it should be invariant under any linear transformation, shouldn't it?
 
Tensors as per definition are invariant objects, I guess the OP asked about tensor components in the canonical basis.
 

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