One argument for the rationality of the action principle is its simplicity: by simply adding connections and then adding actions, one can obtain interaction equations without having to guess how matter, electromagnetic fields, and gravitational fields interact.
However, why the addition of actions is used, and whether the resulting interaction equations truly correspond to real phenomena, are not solved here …
Spinor fields, scalar fields, gauge fields, gravitational fields (Einstein-metric), or other fields are coupled in the following ways:
- Action summation
- gauge-connection
- metric-volume-form
- metric-connection
Varying with respect to 4 sets of variables gives 4 equations
We can set some fields to zero or fix certain fields to obtain partial coupling (for gravity, ==> flat-metric)
In the equation, the 4-current of the field becomes the source of the gauge potential (spinor field 4-current scalar field 4-current point particle 4-current)