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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 A {\displaystyle A} contains an algebraic quadratic term and an arbitrary linear term, while it contains no kinetic terms :
The equation of motion for A {\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 L 0 {\displaystyle {\mathcal {L}}_{0}} describing a field φ {\displaystyle \varphi } , then the Lagrangian describing both fields is