# 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):
${\displaystyle {\mathcal {L}}_{aux}={\frac {1}{2}}(A,A)+(f(\varphi ),A)}$.
The equation of motion for ${\displaystyle A}$ is: ${\displaystyle A(\varphi )=-f(\varphi )}$ and the Lagrangian becomes: ${\displaystyle {\mathcal {L}}_{aux}=-{\frac {1}{2}}(f(\varphi ),f(\varphi ))}$. 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 ${\displaystyle {\mathcal {L}}_{0}}$ describing a field ${\displaystyle \varphi }$ then the Lagrangian describing both fields is:
${\displaystyle {\mathcal {L}}={\mathcal {L}}_{0}(\varphi )+{\mathcal {L}}_{aux}={\mathcal {L}}_{0}(\varphi )-{\frac {1}{2}}(f(\varphi ),f(\varphi ))}$.
Therefore, auxiliary fields can be employed to cancel quadratic terms in ${\displaystyle \varphi }$ in ${\displaystyle {\mathcal {L}}_{0}}$ and linearize the action ${\displaystyle {\mathcal {S}}=\int {{\mathcal {L}}\,d^{n}x}.}$

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.

The quantum mechanical effect of adding an auxiliary field is the same as the classical, since the path integral over such a field is Gaussian. To wit:

${\displaystyle \int _{-\infty }^{\infty }\!dA\,e^{-{\frac {1}{2}}A^{2}+Af}={\sqrt {2\pi }}e^{\frac {f^{2}}{2}}}$.