Auxiliary field

 ( Quantum Field Theory ) 

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 ( of the field):

$\backslash mathcal\{L\}\_\backslash text\{aux\}\; =\; \backslash frac\{1\}\{2\}(A,\; A)\; +\; (f(\backslash varphi),\; A).$

The equation of motion for $A$ is

$A(\backslash varphi)\; =\; -f(\backslash varphi),$

and the Lagrangian becomes

$\backslash mathcal\{L\}\_\backslash text\{aux\}\; =\; -\backslash frac\{1\}\{2\}(f(\backslash varphi),\; f(\backslash varphi)).$

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 $\backslash mathcal\{L\}\_0$ describing a field $\backslash varphi$, then the Lagrangian describing both fields is

$\backslash mathcal\{L\}\; =\; \backslash mathcal\{L\}\_0(\backslash varphi)\; +\; \backslash mathcal\{L\}\_\backslash text\{aux\}\; =\; \backslash mathcal\{L\}\_0(\backslash varphi)\; -\; \backslash frac\{1\}\{2\}\backslash big(f(\backslash varphi),\; f(\backslash varphi)\backslash big).$

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

Examples of auxiliary fields are the complex scalar field F-term in a chiral superfield, the real scalar field D-term in a vector superfield, the scalar field B in BRST formalism 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:

$\backslash int\_\{-\backslash infty\}^\backslash infty\; dA\backslash ,\; e^\{-\backslash frac\{1\}\{2\}\; A^2\; +\; A\; f\}\; =\; \backslash sqrt\{2\backslash pi\}e^\{\backslash frac\{f^2\}\{2\}\}.$

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See also

• Bosonic field
• Fermionic field
• Composite field

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