# Ghost (physics)

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In the terminology of quantum field theory, a **ghost**, **ghost field**, or **gauge ghost** is an unphysical state in a gauge theory. Ghosts are necessary to keep gauge invariance in theories where the local fields exceed a number of physical degrees of freedom.^{[1]}

For example in electrodynamics, in order to maintain manifest Lorentz invariance, one uses a four component vector potential , whereas the photon has only two polarizations. Thus, one needs a suitable mechanism in order to get rid of the unphysical degrees of freedom. Introducing fictitious fields, the ghosts, is one way of achieving this goal.

## Contents

## Good ghosts[edit]

### Faddeev–Popov ghosts[edit]

Faddeev–Popov ghosts are extraneous fields which are introduced to maintain the consistency of the path integral formulation. They are named after Ludvig Faddeev and Victor Popov.^{[2]}^{[3]}

Faddeev–Popov ghosts are sometimes referred to as "good ghosts".

### Goldstone bosons[edit]

Goldstone bosons are sometimes referred to as ghosts. Mainly, when speaking about the vanishing bosons of the spontaneous symmetry breaking of the electroweak symmetry through the Higgs mechanism. These *good* ghosts are artefacts of gauge fixing. The longitudinal polarization components of the W and Z bosons correspond to the Goldstone bosons of the spontaneously broken part of the electroweak symmetry SU(2)⊗U(1), which, however, are not observable. Because this symmetry is gauged, the three would-be Goldstone bosons, or ghosts, are "eaten" by the three gauge bosons (*W ^{±}* and

*Z*) corresponding to the three broken generators; this gives these three gauge bosons a mass, and the associated necessary third polarization degree of freedom.

^{[4]}

## Bad ghosts[edit]

"Bad ghosts" represent another, more general meaning of the word "ghost" in theoretical physics: states of negative norm,^{[5]} 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.^{[6]}

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.

### Ghost condensate[edit]

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A **ghost condensate** is a speculative proposal in which a ghost, an excitation of a field with a wrong sign of the kinetic term, acquires a vacuum expectation value. This phenomenon breaks Lorentz invariance spontaneously. Around the new vacuum state, all excitations have a positive norm, and therefore the probabilities are positive definite.

We have a real scalar field φ with the following action

where *a* and *b* are positive constants and

using the sign convention in the (+, −, −, −) metric signature.

The theories of ghost condensate predict specific non-Gaussianities of the cosmic microwave background. These theories have been proposed by Nima Arkani-Hamed, Markus Luty, and others.^{[7]}

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.

### Landau ghost[edit]

The Landau pole is sometimes referred as the **Landau ghost**. Named after Lev Landau, this ghost is an inconsistency in the renormalization procedure in which there is no asymptotic freedom at large energy scales.^{[8]}

## See also[edit]

- No-ghost theorem, related to bad ghosts

## References[edit]

**^**Faddeev, Ludwig D. (2009). "Faddeev-Popov ghosts".*Scholarpedia*.**4**(4): 7389. Bibcode:2009SchpJ...4.7389F. doi:10.4249/scholarpedia.7389. ISSN 1941-6016.**^**Faddeev, Ludwig D.; Popov, Victor N. (1967). "Feynman diagrams for the Yang-Mills field".*Physics Letters B*.**25**(1): 29–30. Bibcode:1967PhLB...25...29F. doi:10.1016/0370-2693(67)90067-6. ISSN 0370-2693.**^**W. F. Chen. (2008), "Quantum Field Theory and Differential Geometry",*Int. J. Geom. Methods Mod. Phys.*,**10**(4): 1350003, arXiv:0803.1340v2, doi:10.1142/S0219887813500035**^**Griffiths, David J. (1987).*Introduction to elementary particles*. New York: Wiley. ISBN 0471603864. OCLC 19468842.**^**Hawking, Stephen W.; Hertog, Thomas (2001). "Living with Ghosts".*Physical Review D*.**65**(10). arXiv:hep-th/0107088. Bibcode:2002PhRvD..65j3515H. doi:10.1103/PhysRevD.65.103515.**^**Itzhak Bars, John Terning.*Extra Dimensions in Space and Time*. p. 70.**^**Arkani-Hamed, Nima; Cheng, Hsin-Chia; Luty, Markus A.; Mukohyama, Shinji (2004-05-29). "Ghost Condensation and a Consistent Infrared Modification of Gravity".*Journal of High Energy Physics*.**2004**(05): 074–074. arXiv:hep-th/0312099. Bibcode:2004JHEP...05..074H. doi:10.1088/1126-6708/2004/05/074. ISSN 1029-8479.**^**Daintith, John, ed. (2009).*A Dictionary of Physics*.*Landau Ghost*entry (6th ed.). Oxford: Oxford University Press. ISBN 9780199233991. OCLC 244417456.

## External links[edit]

- Copeland, Ed; Padilla, Antonio (26 October 2011). "Ghost Particles".
*Sixty Symbols*. Brady Haran for the University of Nottingham.