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 contains an algebraic quadratic term and an arbitrary linear term, while it contains no kinetic terms (derivatives of the field):
.
The equation of motion for is: and the Lagrangian becomes: . 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 describing a field then the Lagrangian describing both fields is:
.
Therefore, auxiliary fields can be employed to cancel quadratic terms in in and linearize the action

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:

.