nLab interaction

Idea

Given a local Lagrangian density $L$ defining the action functional of a local field theory, typically it splits as a sum

$$L = L_{free} + L_{int}$$

of a term $L_{free}$ that defines a free field theory (kinetic energy) and a remainder $L_{int}$ defining the interactions of a corresponding interacting field theory. That remainder then describes the interactions that the otherwise free fields undergo.

The interaction picture of quantum physics serves to decompose the quantization of local field theories such that the interaction is seen as a perturbation of the quantization of the free field theory. This underlies most constructions of perturbative quantum field theory, including the mathematically rigorous formulation via causal perturbation theory.

Examples

• phi^n interaction

• electron-photon interaction

• The holonomy of a circle bundle with connection gives the gauge interaction term for a particle sigma-model charged under an electromagnetic field given by a line bundle with connection.

• More generally the volume holonomy of a circle n-bundle with connection gives the higher gauge interaction of an $(n-1)$-brane sigma model, for instance the string sigma-model charged under the B-field or the membrane sigma-model charged under the supergravity C-field.

Related concepts

• renormalizable interaction

• interacting field theory

• interacting field algebra

• perturbative quantum field theory

• scattering, scattering matrix

• hidden sector

action functional	kinetic action	interaction	path integral measure
$\exp(-S(\phi)) \cdot \mu =$	$\exp(-(\phi, Q \phi)) \cdot$	$\exp(I(\phi)) \cdot$	$\mu$
BV differential	elliptic complex +	antibracket with interaction +	BV-Laplacian
$d_q =$	$Q$ +	$\{I,-\}$ +	$\hbar \Delta$