Quantum field that enables consistent quantization
In the terminology of
quantum field theory
, a
ghost
,
ghost field, ghost particle
, 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
.
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.
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 to consistently and rigorously quantize non-Abelian
Yang?Mills theory
, such as done with
BRST quantization
.
[1]
[2]
A field with a negative ghost number (the number of ghosts excitations in the field) is called an
anti-ghost
.
Good ghosts
[
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]
Faddeev?Popov ghosts
[
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]
Faddeev?Popov ghosts are extraneous
anticommuting
fields
which are introduced to maintain the consistency of the
path integral formulation
. They are named after
Ludvig Faddeev
and
Victor Popov
.
[3]
[4]
Goldstone bosons
[
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]
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
[
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]
"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]
Ghost condensate
[
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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
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.
[8]
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
[
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]
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.
[9]
See also
[
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]
References
[
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]
- ^
Faddeev, Ludwig D.
(2009).
"Faddeev-Popov ghosts"
.
Scholarpedia
.
4
(4): 7389.
Bibcode
:
2009SchpJ...4.7389F
.
doi
:
10.4249/scholarpedia.7389
.
ISSN
1941-6016
.
- ^
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
.
- ^
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
.
- ^
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
- ^
Griffiths, David J.
(1987).
Introduction to elementary particles
. New York: Wiley.
ISBN
0471603864
.
OCLC
19468842
.
- ^
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
.
- ^
Itzhak Bars, John Terning (2010).
Extra Dimensions in Space and Time
. p. 70.
Bibcode
:
2010edst.book.....B
.
- ^
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
(5): 074.
arXiv
:
hep-th/0312099
.
Bibcode
:
2004JHEP...05..074H
.
doi
:
10.1088/1126-6708/2004/05/074
.
ISSN
1029-8479
.
S2CID
16844964
.
- ^
Daintith, John, ed. (2009). "Landau ghost".
A Dictionary of Physics
(6th ed.). Oxford: Oxford University Press.
ISBN
9780199233991
.
OCLC
244417456
. Archived from
the original
on 2017-12-28
. Retrieved
2018-04-25
.
External links
[
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]