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):

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

The equation of motion for ${\displaystyle A}$ is

${\displaystyle A(\varphi )=-f(\varphi ),}$

and the Lagrangian becomes

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

Auxiliary fields generally do not propagate,[1] 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 ${\displaystyle {\mathcal {L}}_{0}}$ describing a field ${\displaystyle \varphi }$, then the Lagrangian describing both fields is

${\displaystyle {\mathcal {L}}={\mathcal {L}}_{0}(\varphi )+{\mathcal {L}}_{\text{aux}}={\mathcal {L}}_{0}(\varphi )-{\frac {1}{2}}{\big (}f(\varphi ),f(\varphi ){\big )}.}$

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,[2] 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:

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