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):

${{ℒ}}_{\text{aux}}=\frac{1}{2}(A,A)+(f(\phi ),A).$

The equation of motion for $A$ is

$A(\phi )=-f(\phi ).$

and the Lagrangian becomes

${{ℒ}}_{\text{aux}}=-\frac{1}{2}(f(\phi ),f(\phi )).$

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 $\phi$, then the Lagrangian describing both fields is

${ℒ}={{ℒ}}_0(\phi )+{{ℒ}}_{\text{aux}}={{ℒ}}_0(\phi )-\frac{1}{2}(f(\phi ),f(\phi )).$

Therefore, auxiliary fields can be employed to cancel quadratic terms in $\phi$ in ${{ℒ}}_0$ and linearize the action ${𝒮}=\int {ℒ}{d}^{n}x$.

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:

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}dA{e}^{-\frac{1}{2}{A}^2+Af}=\sqrt{2\pi }{e}^{\frac{f^2}{2}}.$

References

• Superspace, or One thousand and one lessons in supersymmetry arXiv:hep-th/0108200

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