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 contains an algebraic quadratic term and an arbitrary linear term, while it contains no kinetic terms (derivatives of the field):
The equation of motion for is
and the Lagrangian becomes
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 describing a field , then the Lagrangian describing both fields is
Therefore, auxiliary fields can be employed to cancel quadratic terms in in and linearize the action .
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:
See also
References
- ^ 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. arXiv:1608.01843. Bibcode:2016JHEP...09..106F. doi:10.1007/JHEP09(2016)106. S2CID 256040291.
- ^ Antoniadis, I.; Dudas, E.; Ghilencea, D.M. (Mar 2008). "Supersymmetric models with higher dimensional operators". Journal of High Energy Physics. 2008 (3): 45. arXiv:0708.0383. Bibcode:2008JHEP...03..045A. doi:10.1088/1126-6708/2008/03/045. S2CID 2491994.
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