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):
${\mathcal {L}}_{aux}={\frac {1}{2}}(A,A)+(f(\varphi ),A)$ .
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 ${\mathcal {S}}=\int {{\mathcal {L}}\,d^{n}x}.$

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:

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

References

Superspace, or One thousand and one lessons in supersymmetry arXiv:hep-th/0108200

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