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):

ℒaux=12(A,A)+(f(φ),A).

The equation of motion for A is

A(φ)=−f(φ),

and the Lagrangian becomes

ℒaux=−12(f(φ),f(φ)).

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 ℒ0 describing a field φ, then the Lagrangian describing both fields is

ℒ=ℒ0(φ)+ℒaux=ℒ0(φ)−12(f(φ),f(φ)).

Therefore, auxiliary fields can be employed to cancel quadratic terms in φ in ℒ0 and linearize the action 𝒮=∫ℒdnx.

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:

∫−∞∞dAe−12A2+Af=2πef22.

## See also

* Bosonic field
* Fermionic field
* Composite Field

## References

1. ↑ Fujimori, Toshiaki; Nitta, Muneto; Yamada, Yusuke (2016-09-19). "Ghostbusters in higher derivative supersymmetric theories: who is afraid of propagating auxiliary fields?". Journal of High Energy Physics 2016 (9): 106. doi:10.1007/JHEP09(2016)106. Bibcode: 2016JHEP...09..106F. https://doi.org/10.1007/JHEP09(2016)106.
2. ↑ Antoniadis, I.; Dudas, E.; Ghilencea, D.M. (Mar 2008). "Supersymmetric models with higher dimensional operators". Journal of High Energy Physics 2008 (3): 45. doi:10.1088/1126-6708/2008/03/045. Bibcode: 2008JHEP...03..045A. https://dx.doi.org/10.1088/1126-6708/2008/03/045.

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Auxiliary field. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Physics:Auxiliary_field&oldid=371619"