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):
L a u x = 1 2 ( A , A ) + ( f ( φ ) , A ) .
The equation of motion for A is: A ( φ ) = − f ( φ ) and the Lagrangian becomes: L a u x = − 1 2 ( f ( φ ) , f ( φ ) ) . 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 L 0 describing a field φ then the Lagrangian describing both fields is:
L = L 0 ( φ ) + L a u x = L 0 ( φ ) − 1 2 ( f ( φ ) , f ( φ ) ) .
Therefore, auxiliary fields can be employed to cancel quadratic terms in φ in L 0 and linearize the action S = ∫ 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: