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The different only shows up for field, not with particles. For fields, there are two types of changes that we can make that can keep the Lagrangian unchanged as mentioned at Physics from Symmetry by Jakob Schwichtenberg (2015) chapter "4.5.2 Noether's Theorem for Field Theories - Spacetime":
From defining properties of elementary particles:
From the spacetime theory alone, we can derive the Lagrangian for the free theories for each spin:
Then the internal symmetries are what add the interaction part of the Lagrangian, which then completes the Standard Model Lagrangian.


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