Faddeev–Popov ghost

In physics, Faddeev–Popov ghosts (also called gauge ghosts or ghost fields) are additional fields which are introduced into gauge quantum field theories to maintain the consistency of the path integral formulation. They are named after Ludvig Faddeev and Victor Popov.[1][2]

There is also a more general meaning of the word "ghost" in theoretical physics, which is discussed below (see general ghosts in theoretical physics).

Overcounting in Feynman path integrals

The necessity for Faddeev–Popov ghosts follows from the requirement that in the path integral formulation, quantum field theories should yield unambiguous, non-singular solutions. This is not possible when a gauge symmetry is present since there is no procedure for selecting any one solution from a range of physically equivalent solutions, all related by a gauge transformation. The problem stems from the path integrals overcounting field configurations related by gauge symmetries, since those correspond to the same physical state; the measure of the path integrals contains a factor which does not allow obtaining various results directly from the original action using the regular methods (e.g., Feynman diagrams). It is possible, however, to modify the action, such that the regular methods will be applicable by adding some additional fields, which break the gauge symmetry, which are called the ghost fields. This technique is called the "Faddeev–Popov procedure" (see also BRST quantization). The ghost fields are a computational tool in that they do not correspond to any real particles in external states: they only appear as virtual particles in Feynman diagrams – or as the absence of some gauge configurations. However they are necessary to preserve unitarity.

The exact form or formulation of ghosts is dependent on the particular gauge chosen, although the same physical results are obtained with all the gauges. The Feynman-'t Hooft gauge is usually the simplest gauge for this purpose, and is assumed for the rest of this article.

Spin-statistics relation violated

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).

In general, anti-commuting ghosts are associated with fermionic symmetries, while commuting ghosts are associated with bosonic symmetries.

Gauge fields and associated ghost fields

Every gauge field has an associated ghost, and where the gauge field acquires a mass via the Higgs mechanism, the associated ghost field acquires the same mass (in the Feynman-'t Hooft gauge only, not true for other gauges).

Appearance in Feynman diagrams

In Feynman diagrams the ghosts appear as closed loops wholly composed of 3-vertices, attached to the rest of the diagram via a gauge particle at each 3-vertex. Their contribution to the S-matrix is exactly cancelled (in the Feynman-'t Hooft gauge) by a contribution from a similar loop of gauge particles with only 3-vertex couplings or gauge attachments to the rest of the diagram.[3] (A loop of gauge particles not wholly composed of 3-vertex couplings is not cancelled by ghosts.) The opposite sign of the contribution of the ghost and gauge loops is due to them having opposite fermionic/bosonic natures. (Closed fermion loops have an extra −1 associated with them; bosonic loops don't.)

Ghost field Lagrangian

The Lagrangian for the ghost fields c^a(x)\, in Yang–Mills theories (where a is an index in the adjoint representation of the gauge group) is given by

\mathcal{L}_{\text{ghost}}=\partial_{\mu}\bar{c}^{a}\partial^{\mu}c^{a}+gf^{abc}\left(\partial^{\mu}\bar{c}^{a}\right)A_{\mu}^{b}c^{c}\;.

The first term is a kinetic term like for regular complex scalar fields, and the second term describes the interaction with the gauge fields. Note that in abelian gauge theories (such as quantum electrodynamics) the ghosts do not have any effect since f^{abc} = 0 and, consequently, the ghost particles do not interact with the gauge fields.

General ghosts in theoretical physics

The Faddeev–Popov ghosts are sometimes referred to as "good ghosts". The "bad ghosts" represent another, more general meaning of the word "ghost" in theoretical physics: states of negative norm—or fields with the wrong sign of the kinetic term, such as Pauli–Villars ghosts—whose existence allows the probabilities to be negative thus violating unitarity.

Changing the symmetry

Ghost particles could obtain the symmetry or break it in gauge fields. The "good ghost" particles actually obtain the symmetry by unchanging the "gauge fixing lagrangian" in a gauge transformation, while bad ghost particles break the symmetry by bringing in the non-abelian G-matrix which does change the symmetry, and this was the main reason to introduce the Gauge covariant and contravariant derivatives.

References

  1. ↑ Faddeev, L. D.; Popov, V. (1967), "Feynman diagrams for the Yang-Mills field", Phys. Lett. B 25 (1): 29, doi:10.1016/0370-2693(67)90067-6
  2. ↑ W. F. Chen. (2008), "Quantum Field Theory and Differential Geometry", Int. J. Geom. Meth. Mod. Phys. 10 (4): 1350003, arXiv:0803.1340v2, doi:10.1142/S0219887813500035
  3. ↑ R. P. Feynman, (1963). "Quantum Theory of Gravitation", Acta Physica Polonica 24, 697−722. Feynman discovered empirically that "boxing" and simply dismissing these diagrams restored unitarity. "Because, unfortunately, I also discovered in the process that the trouble is present in the Yang−Mills theory; and, secondly, I have incidentally discovered a tree−ring connection which is of very great interest and importance in the meson theories and so on. And so I'm stuck to have to continue this investigation, and of course you appreciate that this is the secret reason for doing any work, no matter how absurd and irrational and academic it looks: we all realize that no matter how small a thing is, if it has physical interest and is thought about carefully enough, you're bound to think of something that's good for something else."


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