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Consider Maxwell-Chern-Simons theory in 2+1 dimension, with Lagrangian $$L = -(1/4)F_{\mu v}F^{\mu v} + (m^2/4) \epsilon_{\mu v \rho}A^\mu F^{v \rho},$$ when I make a gauge transformation $A_\mu \to A_\mu + d\lambda$, the lagrangian changes by a total derivative, which we can change it to surface integral. We assume the total derivative term, which can be converted to surface integral, vanishes at large distances. If we don't assume this does this change the symmetries or does it change the conserved quantities (i.e change the momentum, angular momentum operator etc.)

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It is a bit unclear what OP is actually asking, but perhaps it is worth stressing the following point:

  1. On one hand, pertinent boundary conditions (BCs) are assumed to derive Euler-Lagrange (EL) equations.

  2. On the other hand, BCs are not assumed in Noether's two theorems.

See also this Phys.SE post.

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