Why do we require locality in quantum field theory?

[DrDu]

I think what Schwartz discusses there is the realization of an Abelian massive vector field as a gauge field. This is a remarkable model, because it shows that in the Abelian case, i.e., gauge group U(1), you can formulate a gauge-symmetric renormalizable model with massive gauge bosons without the Higgs mechanism. This construct does not work for the non-Abelian case. There you need the Higgs mechanism to consistently describe massive gauge bosons and/or fermions for chiral gauge groups as in the electroweak sector of the Standard Model.

Would this also be true for a non-compact gauge group?

 

[vanhees71]

Non-compact gauge groups are pretty evil. The prime example for that is the notorious trouble the quantization of gravity (general relativity) provides. For the mathematical reasons, why for the local gauge symmetry we have to assume compact semisimple gauge groups (or direct products of such groups and U(1)'s as in the Standard Model), see Weinberg, Quantum Theory of Fields, vol. 2, Sect. 15.2.