# Auxiliary field

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

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.

References

* Superspace, or One thousand and one lessons in supersymmetry [http://arxiv.org/abs/hep-th/0108200 arXiv:hep-th/0108200]

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