Faddeev-Popov ghost
Encyclopedia
In physics
, Faddeev–Popov ghosts (also called ghost fields) are additional field
s 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
.
There is also a more general meaning of the word "ghost" in theoretical physics
, which is discussed below (see general ghosts in theoretical physics).
, quantum field theories
should yield unambiguous, non-singular solutions. This is not possible when a gauge symmetry is present since there is no procedure for selecting any one solution from a range of physically equivalent solutions, all related by a gauge transformation. The problem stems from the path integral
s overcounting field configurations related by gauge symmetries since those correspond to the same physical state; the measure
of the path integrals contains a factor, which does not allow obtaining various results directly from the original action
using the regular methods (e.g., Feynman diagram
s). It is possible, however, to modify the action, such that the regular methods will be applicable by adding some additional fields, which break the gauge symmetry, which are called the ghost fields. This technique is called the Faddeev–Popov procedure (see also BRST quantization
). The ghost fields are a computational tool in that they do not correspond to any real particles in external states: they only appear as virtual particle
s in Feynman diagram
s - or as the absence of some gauge configurations. However they are necessary to preserve unitarity
.
The exact form or formulation of ghosts is dependent on the particular gauge
chosen, although the same physical results are obtained with all the gauges. The Feynman-'t Hooft gauge
is usually the simplest gauge for this purpose, and is assumed for the rest of this article.
For example, in Yang–Mills theories (such as quantum chromodynamics
) the ghosts are complex
scalar field
s (spin
0), but they anti-commute
(like fermions).
In general, anti-commuting
ghosts are associated with boson
ic symmetries, while commuting
ghosts are associated with fermion
ic symmetries.
, the associated ghost field acquires the same mass (in the Feynman-'t Hooft gauge only, not true for other gauges).
s the ghosts appear as closed loops wholly composed of 3-vertices, attached to the rest of the diagram via a gauge particle at each 3-vertex. Their contribution to the S-matrix
is exactly cancelled (in the Feynman-'t Hooft gauge) by a contribution from a similar loop of gauge particles with only 3-vertex couplings or gauge attachments to the rest of the diagram. (A loop of gauge particles not wholly composed of 3-vertex couplings is not cancelled by ghosts.) The opposite sign of the contribution of the ghost and gauge loops is due to them having opposite fermionic/bosonic natures. (Closed fermion loops have an extra −1 associated with them; bosonic loops don't.)
in Yang–Mills theories (where 
is an index in the adjoint representation of the gauge group) is given by
The first term is a kinetic term like for regular complex scalar fields, and the second term describes the interaction with the gauge fields. Note that in abelian gauge theories (such as quantum electrodynamics
) the ghosts do not have any effect since
and, consequently, the ghost particles do not interact with the gauge fields.
: 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
.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, Faddeev–Popov ghosts (also called ghost fields) are additional field
Field (physics)
In physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...
s which are introduced into gauge quantum field theories
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
to maintain the consistency of the path integral formulation
Path integral formulation
The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics...
. They are named after Ludvig Faddeev
Ludvig Faddeev
-References:...
and Victor Popov
Victor Popov
Victor Nikolaevich Popov was a Russian theoretical physicist known for his contribution to the quantization of non-abelian gauge fields. His work with Ludvig Faddeev on that subject introduced the fundamental objects now known as Faddeev–Popov ghosts....
.
There is also a more general meaning of the word "ghost" in theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, which is discussed below (see general ghosts in theoretical physics).
Overcounting in Feynman path integrals
The necessity for Faddeev–Popov ghosts follows from the requirement that in the path integral formulationPath integral formulation
The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics...
, quantum field theories
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
should yield unambiguous, non-singular solutions. This is not possible when a gauge symmetry is present since there is no procedure for selecting any one solution from a range of physically equivalent solutions, all related by a gauge transformation. The problem stems from the path integral
Path integral formulation
The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics...
s overcounting field configurations related by gauge symmetries since those correspond to the same physical state; the measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
of the path integrals contains a factor, which does not allow obtaining various results directly from the original action
Action (physics)
In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
using the regular methods (e.g., Feynman diagram
Feynman diagram
Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...
s). It is possible, however, to modify the action, such that the regular methods will be applicable by adding some additional fields, which break the gauge symmetry, which are called the ghost fields. This technique is called the Faddeev–Popov procedure (see also BRST quantization
BRST quantization
In theoretical physics, BRST quantization is a relatively rigorous mathematical approach to quantizing a field theory with a gauge symmetry...
). The ghost fields are a computational tool in that they do not correspond to any real particles in external states: they only appear as virtual particle
Virtual particle
In physics, a virtual particle is a particle that exists for a limited time and space. The energy and momentum of a virtual particle are uncertain according to the uncertainty principle...
s in Feynman diagram
Feynman diagram
Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...
s - or as the absence of some gauge configurations. However they are necessary to preserve unitarity
Unitarity (physics)
In quantum physics, unitarity is a restriction on the allowed evolution of quantum systems that insures the sum of probabilities of all possible outcomes of any event is always 1....
.
The exact form or formulation of ghosts is dependent on the particular gauge
Gauge fixing
In the physics of gauge theories, gauge fixing denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailed local field...
chosen, although the same physical results are obtained with all the gauges. The Feynman-'t Hooft gauge
Gauge fixing
In the physics of gauge theories, gauge fixing denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailed local field...
is usually the simplest gauge for this purpose, and is assumed for the rest of this article.
Spin-statistics relation violated
The Faddeev–Popov ghosts violate the spin-statistics relation, which is another reason why they are often regarded as "non-physical" particles.For example, in Yang–Mills theories (such as quantum chromodynamics
Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
) the ghosts are complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
scalar field
Scalar field theory
In theoretical physics, scalar field theory can refer to a classical or quantum theory of scalar fields. A field which is invariant under any Lorentz transformation is called a "scalar", in contrast to a vector or tensor field...
s (spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
0), but they anti-commute
Anticommutativity
In mathematics, anticommutativity is the property of an operation that swapping the position of any two arguments negates the result. Anticommutative operations are widely used in algebra, geometry, mathematical analysis and, as a consequence, in physics: they are often called antisymmetric...
(like fermions).
In general, anti-commuting
Anticommutativity
In mathematics, anticommutativity is the property of an operation that swapping the position of any two arguments negates the result. Anticommutative operations are widely used in algebra, geometry, mathematical analysis and, as a consequence, in physics: they are often called antisymmetric...
ghosts are associated with boson
Boson
In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....
ic symmetries, while commuting
Commutativity
In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...
ghosts are associated with fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
ic symmetries.
Gauge fields and associated ghost fields
Every gauge field has an associated ghost, and where the gauge field acquires a mass via the Higgs mechanismHiggs mechanism
In particle physics, the Higgs mechanism is the process in which gauge bosons in a gauge theory can acquire non-vanishing masses through absorption of Nambu-Goldstone bosons arising in spontaneous symmetry breaking....
, the associated ghost field acquires the same mass (in the Feynman-'t Hooft gauge only, not true for other gauges).
Appearance in Feynman diagrams
In Feynman diagramFeynman diagram
Feynman diagrams are a pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, first developed by the Nobel Prize-winning American physicist Richard Feynman, and first introduced in 1948...
s the ghosts appear as closed loops wholly composed of 3-vertices, attached to the rest of the diagram via a gauge particle at each 3-vertex. Their contribution to the S-matrix
S matrix
In physics, the scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process...
is exactly cancelled (in the Feynman-'t Hooft gauge) by a contribution from a similar loop of gauge particles with only 3-vertex couplings or gauge attachments to the rest of the diagram. (A loop of gauge particles not wholly composed of 3-vertex couplings is not cancelled by ghosts.) The opposite sign of the contribution of the ghost and gauge loops is due to them having opposite fermionic/bosonic natures. (Closed fermion loops have an extra −1 associated with them; bosonic loops don't.)
Ghost field Lagrangian
The Lagrangian for the ghost fields in Yang–Mills theories (where is an index in the adjoint representation of the gauge group) is given byThe first term is a kinetic term like for regular complex scalar fields, and the second term describes the interaction with the gauge fields. Note that in abelian gauge theories (such as quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...
) the ghosts do not have any effect since
and, consequently, the ghost particles do not interact with the gauge fields.General ghosts in theoretical physics
The Faddeev–Popov ghosts are sometimes referred to as "good ghosts". The "bad ghosts" represent another, more general meaning of the word "ghost" in theoretical physicsTheoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
: 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
Negative probability
In 1942, Paul Dirac wrote a paper "The Physical Interpretation of Quantum Mechanics" where he introduced the concept of negative energies and negative probabilities:...
thus violating unitarity
Unitarity (physics)
In quantum physics, unitarity is a restriction on the allowed evolution of quantum systems that insures the sum of probabilities of all possible outcomes of any event is always 1....
.