Concept

# Auxiliary field

Summary
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 generally do not propagate, 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 \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}{
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