Locality in quantum field theory (QFT) mandates that the Lagrangian density ##\mathscr{L}## is a functional of fields and a finite number of their spatial and temporal derivatives evaluated at a single spacetime point ##x^{\mu}=(t,\mathbf{x})##. This requirement ensures that the physical state at a point depends solely on the immediate neighborhood, preventing direct interactions between fields at different spacetime points, thus avoiding action at a distance. The cluster decomposition property further supports this by ensuring that observables at spacelike separation commute, maintaining causality and consistency in predictions. Effective field theories, while often nonlocal, are truncated to local forms for tractability.
PREREQUISITESPhysicists, particularly those specializing in quantum field theory, theoretical physicists exploring the foundations of locality, and students seeking to understand the implications of locality and causality in QFT.
Would this also be true for a non-compact gauge group?vanhees71 said: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.