# What is the fundamental difference between ghost and auxiliary fields?

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I am somehow confused by the notion of auxiliary fields, such as for example the fields F and D which appear in supersymmetry, and the notion of ghost fields which appear for example in the BRST formalism. Both kinds of fields are deemed to be not physical.

Can somebody once and for all in a clear cut way explain to me what the fundamental differences (and similarities) between ghost and auxiliary fields are?

asked Apr 14, 2013
Related: physics.stackexchange.com/q/41738/2451

This post imported from StackExchange Physics at 2014-03-09 16:25 (UCT), posted by SE-user Qmechanic

## 1 Answer

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There is a good reference that can help provide a good understanding of the relationship of auxiliary fields used in supersymmetry and their relationship to ghost fields.  It is called Superspace or One Thousand and One Lessons in Supersymmetry [1].

Auxiliary fields are defined as having non-derivative kinetic terms, or rather, it has terms that are NOT derivatives of the field.

A further advantage of superfields is that they automatically include, in addition
to the dynamical degrees of freedom, certain unphysical fields: (1) auxiliary fields (fields
with nonderivative kinetic terms), needed classically for the off-shell closure of the supersymmetry algebra, and (2) compensating fields (fields that consist entirely of gauge
degrees of freedom), which are used to enlarge the usual gauge transformations to an
entire multiplet of transformations forming a representation of supersymmetry; together
with the auxiliary fields, they allow the algebra to be field independent. The compensators are particularly important for quantization, since they permit the use of supersymmetric gauges, ghosts, Feynman graphs, and supersymmetric power-counting.

The auxiliary fields are most commonly used to cancel unwanted quadratic terms in a Lagrangian.  They do not propagate, or rather they do not change with time.

Ghost fields are somewhat different.  Ghost fields have virtual particles associated with them and not physical particles like ordinary fields.  They were originally introduced in order to maintain unitarity in guage theories.  In quantum field theories, they are included on internal lines of Feynman diagrams but not external lines.  As such there are creation and annihilation operators associated with Ghost fields, but they are entirely "fictitious" and follow "fictitious rules" [2].

The important concept in the above long quote is that the combination of auxiliary fields and compensating fields:

allow the algebra to be field independent

Which is important because it brings the complete underlying algebra to the forefront and be the key component of study.

[1] Superspace Or One Thousand and One Lessons in Supersymmetry
S.J. Gates (MIT, LNS), Marcus T. Grisaru (Brandeis U.), M. Rocek (SUNY, Stony Brook), W. Siegel (UC, Berkeley). 1983. 548 pp.
YITP-SB-01-53

[2] Veltman, M. (1994). Diagrammatica: The path to Feynman rules. Cambridge: Cambridge University Press.

answered Nov 15, 2014 by (140 points)
edited Nov 15, 2014 by Username
Wow, thanks for this nice lucid answer, very helpful!

I like the comment "allow the algebra to be field independent".This is off-topic but this is a general and very very important comment that I like to use when I need to quote the Coleman-Mandula theorem which is, first of all, a Lie algebra result and properly proved. Thus, the next time an experimentalist or some crazy person tells (who like for example E8 exceptional theories) you that Coleman-Mandula does not have to be true say exactly the same statement. "the Lie algebra is independent of the fields and particles".

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