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 [math]\displaystyle{ A }[/math] 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 [math]\displaystyle{ A }[/math] is
and the Lagrangian becomes
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 [math]\displaystyle{ \mathcal{L}_0 }[/math] describing a field [math]\displaystyle{ \varphi }[/math], then the Lagrangian describing both fields is
Therefore, auxiliary fields can be employed to cancel quadratic terms in [math]\displaystyle{ \varphi }[/math] in [math]\displaystyle{ \mathcal{L}_0 }[/math] and linearize the action [math]\displaystyle{ \mathcal{S} = \int \mathcal{L} \,d^n x }[/math].
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:
Original source: https://en.wikipedia.org/wiki/Auxiliary field.
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