Incoming links: Field (physics)
Static case of Maxwell's law for electricity only.
Is implied by Gauss' law if Maxwell's equations: physics.stackexchange.com/questions/44418/are-the-maxwells-equations-enough-to-derive-the-law-of-coulomb
The "static" part is important: if this law were true for moving charges, we would be able to transmit information instantly at infinite distances. This is basically where the idea of field comes in.
Quantum Field Theory lecture notes by David Tong (2007) puts it well:This is also mentioned e.g. at Video "The Quantum Experiment that ALMOST broke Locality by The Science Asylum (2019)".
In classical physics, the primary reason for introducing the concept of the field is to construct laws of Nature that are local. The old laws of Coulomb and Newton involve "action at a distance". This means that the force felt by an electron (or planet) changes immediately if a distant proton (or star) moves. This situation is philosophically unsatisfactory. More importantly, it is also experimentally wrong. The field theories of Maxwell and Einstein remedy the situation, with all interactions mediated in a local fashion by the field.
Unifies both special relativity and gravity.
Not compatible with the Standard Model, and the 2020 unification attempts are called theory of everything.
One of the main motivations for it was likely having forces not be instantaneous, but rather mediated by field to maintain the principle of locality, just like electromagnetism did earlier.
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":
- spacetime symmetry: act with the Poincaré group on the Four-vector spacetime inputs of the field itself, i.e. transforming into
- internal symmetry: act on the output of the field, i.e.:
From defining properties of elementary particles:
- spacetime:
- internal
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.