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Minimal coupling
In physics, 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.
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 (q) times the four-potential \( (A_\mu) \) from the four-momentum \( (p_\mu) \) in the Lagrangian or Hamiltonian:
\( p_\mu := p_\mu - q \ A_\mu \)
Taken almost verbatim from Doughty's Lagrangian Interaction, pg. 456[1]
See the Hamiltonian mechanics article for a full derivation and examples.
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 field \varphi (called the inflaton in the context of inflation) is not coupled to the scalar curvature. Its only coupling to gravity is the coupling to the Lorentz invariant measure \( \sqrt{-g}\, d^4 x \) constructed from the metric (in Planck units):
\(S =\int d^4 x \, \sqrt{g} \, \left(-\frac{1}{2}R + \frac{1}{2}\nabla_\mu \varphi \nabla^\mu \varphi - V(\varphi)\right) \)
where \(g := \det g_{\mu\nu}. \)
See also
Gauge covariant derivative
Hamiltonian mechanics
Lagrangian mechanics
References
Doughty, Noel (1990). Lagrangian Interaction. Westview Press. ISBN 0-201-41625-5.
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