Auxiliary field

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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 contains an algebraic quadratic term and an arbitrary linear term, while it contains no kinetic terms (derivatives of the field):
\mathcal{L}_{aux}=\frac{1}{2}(A,A)+(f(\varphi),A).
The equation of motion for A is: A(\varphi)=-f(\varphi) and the Lagrangian becomes: \mathcal{L}_{aux}=-\frac{1}{2}(f(\varphi),f(\varphi)). 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 \mathcal{L}_{0} describing a field \varphi then the Lagrangian describing both fields is:
\mathcal{L}=\mathcal{L}_{0}(\varphi)+\mathcal{L}_{aux}=\mathcal{L}_{0}(\varphi)-\frac{1}{2}(f(\varphi),f(\varphi)).
Therefore, auxiliary fields can be employed to cancel quadratic terms in \varphi in \mathcal{L}_{0} and linearize the action \mathcal{S} = \int{\mathcal{L}\,d^nx}.

Examples of auxiliary fields are the complex scalar field F in a chiral superfield, the real scalar field D in a vector superfield, the scalar field B in BRST and the field in the Hubbard-Stratonovich transformation.

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