= Internal and spacetime symmetries
https://physics.stackexchange.com/questions/106392/internal-and-spacetime-symmetries
The different only shows up for <field (physics)>, 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 $L(\Phi(x), \partial \Phi(x), dx)$ into $L(\Phi'(x'), \partial \Phi'(x'), x')$
* <internal symmetry>: act on the output of the field, i.e.: $L(\Phi(x) + \delta \Phi(x), \partial (\Phi(x) + \delta \Phi(x)), x)$
From <defining properties of elementary particles>:
* spacetime:
* <mass>
* <spin (physics)>
* internal
* <electric charge>
* <Weak charge>
* <color charge>
From the spacetime theory alone, we can derive the <Lagrangian> for the free theories for each <spin (physics)>:
* <spin 0>: <Klein-Gordon equation>
* <spin half>: <Dirac equation>
* <spin 1>: <Proca equation>
Then the internal symmetries are what add the interaction part of the <Lagrangian>, which then completes the <Standard Model Lagrangian>.