Auxiliary field

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

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

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

A ( φ ) = − f ( φ ) , {\displaystyle A(\varphi )=-f(\varphi ),} {\displaystyle A(\varphi )=-f(\varphi ),}

and the Lagrangian becomes

L aux = − 1 2 ( f ( φ ) , f ( φ ) ) . {\displaystyle {\mathcal {L}}_{\text{aux}}=-{\frac {1}{2}}(f(\varphi ),f(\varphi )).} {\displaystyle {\mathcal {L}}_{\text{aux}}=-{\frac {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 0 {\displaystyle {\mathcal {L}}_{0}} {\displaystyle {\mathcal {L}}_{0}} describing a field φ {\displaystyle \varphi } {\displaystyle \varphi }, then the Lagrangian describing both fields is

L = L 0 ( φ ) + L aux = L 0 ( φ ) − 1 2 ( f ( φ ) , f ( φ ) ) . {\displaystyle {\mathcal {L}}={\mathcal {L}}_{0}(\varphi )+{\mathcal {L}}_{\text{aux}}={\mathcal {L}}_{0}(\varphi )-{\frac {1}{2}}{\big (}f(\varphi ),f(\varphi ){\big )}.} {\displaystyle {\mathcal {L}}={\mathcal {L}}_{0}(\varphi )+{\mathcal {L}}_{\text{aux}}={\mathcal {L}}_{0}(\varphi )-{\frac {1}{2}}{\big (}f(\varphi ),f(\varphi ){\big )}.}

Therefore, auxiliary fields can be employed to cancel quadratic terms in φ {\displaystyle \varphi } {\displaystyle \varphi } in L 0 {\displaystyle {\mathcal {L}}_{0}} {\displaystyle {\mathcal {L}}_{0}} and linearize the action S = ∫ L d n x {\displaystyle {\mathcal {S}}=\int {\mathcal {L}}\,d^{n}x} {\displaystyle {\mathcal {S}}=\int {\mathcal {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:

∫ − ∞ ∞ d A e − 1 2 A 2 + A f = 2 π e f 2 2 . {\displaystyle \int _{-\infty }^{\infty }dA\,e^{-{\frac {1}{2}}A^{2}+Af}={\sqrt {2\pi }}e^{\frac {f^{2}}{2}}.} {\displaystyle \int _{-\infty }^{\infty }dA\,e^{-{\frac {1}{2}}A^{2}+Af}={\sqrt {2\pi }}e^{\frac {f^{2}}{2}}.}

See also

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. arXiv:1608.01843. Bibcode:2016JHEP...09..106F. doi:10.1007/JHEP09(2016)106. S2CID 256040291.
  2. 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.