The Ghost Fields: The Dr Ruth Galloway Mysteries 7

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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. A field with a negative ghost number (the number of ghosts excitations in the field) is called an anti-ghost. 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] 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 artifacts 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. [5] Bad ghosts [ edit ]

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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. 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. Griffiths, David J. (1987). Introduction to elementary particles. New York: Wiley. ISBN 0471603864. OCLC 19468842.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.

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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. 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. 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. 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.Hawking, Stephen W.; Hertog, Thomas (2002). "Living with Ghosts". Physical Review D. 65 (10): 103515. arXiv: hep-th/0107088. Bibcode: 2002PhRvD..65j3515H. doi: 10.1103/PhysRevD.65.103515. S2CID 2412236.