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.
For example in electrodynamics, in order to
maintain manifest Lorentz invariance, one
uses a four component vector potential
A
μ
(
x
)
{\displaystyle A_{\mu }(x)}
, 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.
== Good ghosts ==
=== Faddeev–Popov ghosts ===
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.Faddeev–Popov
ghosts are sometimes referred to as "good
ghosts".
=== Goldstone bosons ===
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.
== Bad ghosts ==
"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.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 ===
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
S
=
∫
d
4
x
[
a
X
2
−
b
X
]
{\displaystyle S=\int d^{4}x\left[aX^{2}-bX\right]}
where a and b are positive constants and
X
=
d
e
f
1
2
η
μ
ν
∂
μ
ϕ
∂
ν
ϕ
{\displaystyle X\ {\stackrel {\mathrm {def}
}{=}}\ {\frac {1}{2}}\eta ^{\mu \nu }\partial
_{\mu }\phi \partial _{\nu }\phi }
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.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 ===
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.
== See also ==
No-ghost theorem, related to bad ghosts
