# Minimal coupling

In analytical mechanics and quantum field theory, minimal coupling refers to a coupling between fields which involves only the charge distribution and not higher multipole moments of the charge distribution. This minimal coupling is in contrast to, for example, Pauli coupling, which includes the magnetic moment of an electron directly in the Lagrangian.

## Electrodynamics

In electrodynamics, minimal coupling is adequate to account for all electromagnetic interactions. Higher moments of particles are consequences of minimal coupling and non-zero spin.

Mathematically, minimal coupling is achieved by subtracting the charge (${\displaystyle q}$) times the four-potential (${\displaystyle A_{\mu }}$) from the four-momentum (${\displaystyle p_{\mu }}$) in the Lagrangian or Hamiltonian:

${\displaystyle p_{\mu }\mapsto p_{\mu }-q\ A_{\mu }}$

See the Hamiltonian mechanics article for a full derivation and examples. (Taken almost verbatim from Doughty's Lagrangian Interaction, pg. 456)[1]

## Inflation

In studies of cosmological inflation, minimal coupling of a scalar field usually refers to minimal coupling to gravity. This means that the action for the inflaton field ${\displaystyle \varphi }$ is not coupled to the scalar curvature. Its only coupling to gravity is the coupling to the Lorentz invariant measure ${\displaystyle {\sqrt {g}}\,d^{4}x}$ constructed from the metric (in Planck units):

${\displaystyle S=\int d^{4}x\,{\sqrt {g}}\,\left(-{\frac {1}{2}}R+{\frac {1}{2}}\nabla _{\mu }\varphi \nabla ^{\mu }\varphi -V(\varphi )\right)}$

where ${\displaystyle g:=\det g_{\mu \nu }}$, and utilizing the Gauge covariant derivative.

## References

1. ^ Doughty, Noel (1990). Lagrangian Interaction. Westview Press. ISBN 0-201-41625-5.