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
l{L}aux=
| 1 | |
| 2 |
(A,A)+(f(\varphi),A).
The equation of motion for
A
A(\varphi)=-f(\varphi),
and the Lagrangian becomes
l{L}aux=-
| 1 | |
| 2 |
(f(\varphi),f(\varphi)).
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
l{L}0
\varphi
l{L}=l{L}0(\varphi)+l{L}aux=l{L}0(\varphi)-
| 1 | |
| 2 |
(f(\varphi),f(\varphi)).
Therefore, auxiliary fields can be employed to cancel quadratic terms in
\varphi
l{L}0
l{S}=\intl{L}dnx
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:
| infty | |
| \int | |
| -infty |
dA
| |||||
| e |
=
| ||||
| \sqrt{2\pi}e |
.