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

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

The equation of motion for ${\displaystyle A}$ is

${\displaystyle A(\phi )=-f(\phi ),}$

and the Lagrangian becomes

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

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 ${\displaystyle {{ℒ}}_0}$ describing a field ${\displaystyle \phi }$, then the Lagrangian describing both fields is

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

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

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:

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

• Bosonic field
• Fermionic field
• Composite Field

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