minimal coupling (changes) in nLab

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Contents

Idea

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 GG-principal bundle where GG is the gauge group of the given gauge theory. (For instance G=U(1)×SU(2)×SU(3)G = U(1) \times SU(2) \times SU(3) in the standard model of particle physics).

The matter fields are sections ψ\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 ∇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

S gc(∇,ψ)=∫ Σ⟨ψ,D ∇ψ⟩. S_{gc}(\nabla, \psi) = \int_\Sigma \langle \psi, D_\nabla \psi\rangle \,.

Examples

All the couplings appearin in the standard model of particle physics are “minimal” in this sense.

• spinors in Yang-Mills theory

• Yukawa coupling

gauge field: models and components

physics	differential geometry	differential cohomology
gauge field	connection on a bundle	cocycle in differential cohomology
instanton/charge sector	principal bundle	cocycle in underlying cohomology
gauge potential	local connection differential form	local connection differential form
field strength	curvature	underlying cocycle in de Rham cohomology
gauge transformation	equivalence	coboundary
minimal coupling	covariant derivative	twisted cohomology
BRST complex	Lie algebroid of moduli stack	Lie algebroid of moduli stack
extended Lagrangian	universal Chern-Simons n-bundle	universal characteristic map