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):

:$\backslash mathcal\{L\}\_\backslash text\{aux\}\; =\; \backslash frac\{1\}\{2\}(A,\; A)\; +\; (f(\backslash varphi),\; A).$

The equation of motion for $A$ is

:$A(\backslash varphi)\; =\; -f(\backslash varphi),$

and the Lagrangian becomes

:$\backslash mathcal\{L\}\_\backslash text\{aux\}\; =\; -\backslash frac\{1\}\{2\}(f(\backslash varphi),\; f(\backslash varphi)).$

Auxiliary fields generally do not propagate,{{Cite journal |last1=Fujimori |first1=Toshiaki |last2=Nitta |first2=Muneto |last3=Yamada |first3=Yusuke |date=2016-09-19 |title=Ghostbusters in higher derivative supersymmetric theories: who is afraid of propagating auxiliary fields? |url=https://doi.org/10.1007/JHEP09(2016)106 |journal=Journal of High Energy Physics |volume=2016 |issue=9|page=106 |doi=10.1007/JHEP09(2016)106 |arxiv=1608.01843 |bibcode=2016JHEP...09..106F |s2cid=256040291 }} and hence the content of any theory can remain unchanged in many circumstances by adding such fields by hand.

If we have an initial Lagrangian $\backslash mathcal\{L\}\_0$ describing a field $\backslash varphi$, then the Lagrangian describing both fields is

:$\backslash mathcal\{L\}\; =\; \backslash mathcal\{L\}\_0(\backslash varphi)\; +\; \backslash mathcal\{L\}\_\backslash text\{aux\}\; =\; \backslash mathcal\{L\}\_0(\backslash varphi)\; -\; \backslash frac\{1\}\{2\}\backslash big(f(\backslash varphi),\; f(\backslash varphi)\backslash big).$

Therefore, auxiliary fields can be employed to cancel quadratic terms in $\backslash varphi$ in $\backslash mathcal\{L\}\_0$ and linearize the action $\backslash mathcal\{S\}\; =\; \backslash int\; \backslash mathcal\{L\}\; \backslash ,d^n\; x$.

Examples of auxiliary fields are the complex scalar field F in a chiral superfield,{{Cite journal |last1=Antoniadis |first1=I. |last2=Dudas |first2=E. |last3=Ghilencea |first3=D.M. |date=Mar 2008 |title=Supersymmetric models with higher dimensional operators |url=https://dx.doi.org/10.1088/1126-6708/2008/03/045 |journal=Journal of High Energy Physics |volume=2008 |issue=3 |pages=45|doi=10.1088/1126-6708/2008/03/045 |arxiv=0708.0383 |bibcode=2008JHEP...03..045A |s2cid=2491994 }} 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:

:$\backslash int\_\{-\backslash infty\}^\backslash infty\; dA\backslash ,\; e^\{-\backslash frac\{1\}\{2\}\; A^2\; +\; A\; f\}\; =\; \backslash sqrt\{2\backslash pi\}e^\{\backslash frac\{f^2\}\{2\}\}.$