Auxiliary field

In physics, and especially quantum field theory, an auxiliary field is one whose equations of motion admit a single solution. Therefore, the Lagrangian describing such a field ${\displaystyle A}$ contains an algebraic quadratic term and an arbitrary linear term, while it contains no kinetic terms (derivatives of the field):

The equation of motion for ${\displaystyle A}$ is

and the Lagrangian becomes

Auxiliary fields do not propagate, and hence the content of any theory remains unchanged by adding such fields by hand. If we have an initial Lagrangian ${\displaystyle {\mathcal {L}}_{0}}$ describing a field ${\displaystyle \varphi }$, then the Lagrangian describing both fields is

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