The necessity for Faddeev–Popov ghosts follows from the requirement that quantum field theories yield unambiguous, non-singular solutions.
The path integrals overcount field configurations corresponding to the same physical state; the measure of the path integrals contains a factor which does not allow obtaining various results directly from the action.
It is possible, however, to modify the action, such that methods such as Feynman diagrams will be applicable by adding ghost fields which break the gauge symmetry.
Consider for example non-Abelian gauge theory with The integral needs to be constrained via gauge-fixing via
[3] The Faddeev–Popov ghosts violate the spin–statistics relation, which is another reason why they are often regarded as "non-physical" particles.
For example, in Yang–Mills theories (such as quantum chromodynamics) the ghosts are complex scalar fields (spin 0), but they anti-commute (like fermions).
[a] (A loop of gauge particles not wholly composed of 3-vertex couplings is not cancelled by ghosts.)
Note that in abelian gauge theories (such as quantum electrodynamics) the ghosts do not have any effect since the structure constants
Consequently, the ghost particles do not interact with abelian gauge fields.