Noether's first theorem and the energy-momentum tensor ambiguity problem

Understand

Noether's theorems are widely praised as some of the most beautiful and useful results in physics.

• However, if one reads the majority of standard texts and literature on the application of Noether's first theorem to field theory, one immediately finds that the ``canonical Noether energy-momentum tensor" derived from the 4-parameter translation of the Poincar\'e group does not correspond to what's widely accepted as the ``physical'' energy-momentum tensor for central theories such as electrodynamics.
• This gives the impression that Noether's first theorem is in some sense not working.
• In recognition of this issue, common practice is to ``improve" the canonical Noether energy-momentum tensor by adding suitable ad-hoc ``improvement" terms that will convert the canonical expression into the desired result.