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 [math]\displaystyle{ A }[/math] contains an algebraic quadratic term and an arbitrary linear term, while it contains no kinetic terms (derivatives of the field):

[math]\displaystyle{ \mathcal{L}_\text{aux} = \frac{1}{2}(A, A) + (f(\varphi), A). }[/math]

The equation of motion for [math]\displaystyle{ A }[/math] is

[math]\displaystyle{ A(\varphi) = -f(\varphi), }[/math]

and the Lagrangian becomes

[math]\displaystyle{ \mathcal{L}_\text{aux} = -\frac{1}{2}(f(\varphi), f(\varphi)). }[/math]

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

[math]\displaystyle{ \mathcal{L} = \mathcal{L}_0(\varphi) + \mathcal{L}_\text{aux} = \mathcal{L}_0(\varphi) - \frac{1}{2}\big(f(\varphi), f(\varphi)\big). }[/math]

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:

[math]\displaystyle{ \int_{-\infty}^\infty dA\, e^{-\frac{1}{2} A^2 + A f} = \sqrt{2\pi}e^{\frac{f^2}{2}}. }[/math]

See also

• Bosonic field
• Fermionic field
• Composite Field

References

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