nLab
minimal coupling

Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

\infty-Chern-Weil theory

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)= Σψ,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.

gauge field: models and components

physicsdifferential geometrydifferential cohomology
gauge fieldconnection on a bundlecocycle in differential cohomology
instanton/charge sectorprincipal bundlecocycle in underlying cohomology
gauge potentiallocal connection differential formlocal connection differential form
field strengthcurvatureunderlying cocycle in de Rham cohomology
gauge transformationequivalencecoboundary
minimal couplingcovariant derivativetwisted cohomology
BRST complexLie algebroid of moduli stackLie algebroid of moduli stack
extended Lagrangianuniversal Chern-Simons n-bundleuniversal characteristic map
Revised on January 7, 2013 19:19:44 by Urs Schreiber (89.204.154.29)