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):

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

The equation of motion for $A$ is

$A(\varphi )=-f(\varphi ),$

and the Lagrangian becomes

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

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

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

Therefore, auxiliary fields can be employed to cancel quadratic terms in $\varphi$ 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