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In quantum field theory, the term minimal coupling refers to the kind of interaction betweem fermionic particles and force gauge fields.
The force gauge fields are modeled by principal connections on a $G$-principal bundle where $G$ is the gauge group of the given gauge theory. (For instance $G = U(1) \times SU(2) \times SU(3)$ in the standard model of particle physics).
The matter fields are sections $\psi$ of a spinor bundle associated to this principal bundle. Therefore there is an induced connection on a vector bundle $\nabla$ on this spinor bundle.
Let $D_\nabla$ be the Dirac operator of the given Riemannian metric and this conneciton $\nabla$. The minimal coupling term in the action functional on the space of these sections is
All the couplings appearin in the standard model of particle physics are “minimal” in this sense.
gauge field: models and components