Chen, W.F. (2008), "Quantum Field Theory and Differential Geometry", Int. J. Geom. Methods Mod. Phys., 10 (4): 1350003, arXiv: 0803.1340v2, doi: 10.1142/S0219887813500035, S2CID 16651244 If a given theory is self-consistent by the introduction of ghosts, these states are labeled "good". Good ghosts are virtual particles that are introduced for regularization, like Faddeev–Popov ghosts. Otherwise, "bad" ghosts admit undesired non-virtual states in a theory, like Pauli–Villars ghosts that introduce particles with negative kinetic energy. Griffiths, David J. (1987). Introduction to elementary particles. New York: Wiley. ISBN 0471603864. OCLC 19468842. Bad ghosts" represent another, more general meaning of the word "ghost" in theoretical physics: states of negative norm, [6] 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. [7]

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An example of the need of ghost fields is the photon, which is usually described by a four component vector potential A μ, even if light has only two allowed polarizations in the vacuum. To remove the unphysical degrees of freedom, it is necessary to enforce some restrictions; one way to do this reduction is to introduce some ghost field in the theory. While it is not always necessary to add ghosts to quantize the electromagnetic field, ghost fields are strictly needed when dealing with non-Abelian Yang–Mills theory extensions to the Standard Model. [1] [2] Daintith, John, ed. (2009). "Landau ghost". A Dictionary of Physics (6thed.). Oxford: Oxford University Press. ISBN 9780199233991. OCLC 244417456. Archived from the original on 2017-12-28 . Retrieved 2018-04-25.

Faddeev, Ludwig D.; Popov, Victor N. (1967). "Feynman diagrams for the Yang-Mills field". Physics LettersB. 25 (1): 29–30. Bibcode: 1967PhLB...25...29F. doi: 10.1016/0370-2693(67)90067-6. ISSN 0370-2693. A field with a negative ghost number (the number of ghosts excitations in the field) is called an anti-ghost. Becchi, Carlo Maria; Imbimbo, Camillo (2008-10-26). "Becchi-Rouet-Stora-Tyutin symmetry". Scholarpedia. 3 (10): 7135. Bibcode: 2008SchpJ...3.7135B. doi: 10.4249/scholarpedia.7135. ISSN 1941-6016.

Classical and quantum ghosts - arXiv.org

In physics, Faddeev–Popov ghosts (also called Faddeev–Popov gauge ghosts or Faddeev–Popov ghost fields) are extraneous 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] Faddeev, Ludwig D. (2009). "Faddeev-Popov ghosts". Scholarpedia. 4 (4): 7389. Bibcode: 2009SchpJ...4.7389F. doi: 10.4249/scholarpedia.7389. ISSN 1941-6016. The chilling discovery of a downed World War II plane with a body inside leads Ruth and DCI Nelson to uncover a wealthy family’s secrets in this Ruth Galloway mystery. The necessity for Faddeev–Popov ghosts follows from the requirement that quantum field theories yield unambiguous, non-singular solutions. This is not possible in the path integral formulation when a gauge symmetry is present since there is no procedure for selecting among physically equivalent solutions related by gauge transformation. 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.

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D [ A ] exp ⁡ i ∫ d 4 x ( − 1 4 F μ ν a F a μ ν ) . {\displaystyle \int {\mathcal {D}}[A]\exp i\int \mathrm {d}

Ghost (physics) - Wikipedia Ghost (physics) - Wikipedia

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. The ghost fields do not correspond to any real particles in external states: they appear as virtual particles in Feynman diagrams – or as the absence of some gauge configurations. However, they are a necessary computational tool to preserve unitarity. Unfortunately, this theory allows for superluminal propagation of information in some cases and has no lower bound on its energy. This model doesn't admit a Hamiltonian formulation (the Legendre transform is multi-valued because the momentum function isn't convex) because it is acausal. Quantizing this theory leads to problems.

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The exact form or formulation of ghosts is dependent on the particular gauge chosen, although the same physical results must be obtained with all gauges since the gauge one chooses to carry out calculations is an arbitrary choice. The Feynman–'t Hooft gauge is usually the simplest gauge for this purpose, and is assumed for the rest of this article.