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Dynamical gauge fields are assumed to be able to respond to sources.

What's the difference in the Lagrangians between a background field and a dynamical field?

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Typically, in the path integral formalism

$$ Z~=~\int \!{\cal D}\phi~ \exp\left(\frac{i}{\hbar}S[\phi]\right), $$

the dynamical$^1$ quantum field variables

$$\phi^{\alpha}(x)~=~B^{\alpha}(x)+\eta^{\alpha}(x)$$

are split into two parts:

  1. a classical background field configuration $B^{\alpha}(x)$, which satisfies the classical equations$^2$ of motion (with classical background sources$^3$ $J_{\alpha}(x)$),

  2. and a quantum fluctuation part $\eta^{\alpha}(x)$, which is (perturbatively) integrated over in the path integral $$ Z[B]~=~\int \!{\cal D}\eta~ \exp\left(\frac{i}{\hbar}S[B+\eta]\right), $$

See also e.g. this Wikipedia page.

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$^1$ The word dynamical will here for simplicity just mean that the field variable is an integration variable in the path integral. (Often the word dynamical is only used for propagating fields (as opposed to an non-propagating auxiliary field, e.g. Lagrange multipliers and ghosts)). In contrast, the word background is typically assigned to variables that are not integrated over in the path integral.

$^2$ We assume for simplicity that there (with the given boundary conditions and the given classical sources) is a unique classical solution for $B^{\alpha}(x)$, i.e. we ignore instantons here.

$^3$ The classical source configuration $J_{\alpha}(x)$ should typically satisfy certain consistency conditions, such as, e.g., a continuity equation.

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