note-math

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

$${\int }_{𝑀}𝑑\text{Vol}\left(𝑔\right)\left(\text{spin }+\text{ scalar }+\text{ gauge }+\text{ gravity}\right)$$

Varying with respect to 4 sets of variables gives 4 equations

$$\begin{array}{c}\text{spin }\\ \text{ scalar }\\ \text{ gauge }\\ \text{ gravity }\\ \end{array}$$

We can set some fields to zero or fix certain fields to obtain partial coupling (for gravity, $\mathrm{\Gamma }=0$ ==> flat-metric)

In the equation, the 4-current of the field becomes the source of the gauge potential (#link(<current-gauge-spinor>)[spinor field 4-current] #link(<current-gauge-KG>)[scalar field 4-current] #link(<current-gauge-particle>)[point particle 4-current])