Faddeev–Popov ghost
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In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, Faddeev–Popov ghosts (also called Faddeev–Popov gauge ghosts or Faddeev–Popov ghost fields) are extraneous
fields Fields may refer to: Music * Fields (band), an indie rock band formed in 2006 * Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song b ...
which are introduced into
gauge Gauge ( or ) may refer to: Measurement * Gauge (instrument), any of a variety of measuring instruments * Gauge (firearms) * Wire gauge, a measure of the size of a wire ** American wire gauge, a common measure of nonferrous wire diameter, ...
quantum field theories In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles ...
to maintain the consistency of the
path integral formulation The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional in ...
. They are named after
Ludvig Faddeev Ludvig Dmitrievich Faddeev (also ''Ludwig Dmitriyevich''; russian: Лю́двиг Дми́триевич Фадде́ев; 23 March 1934 – 26 February 2017) was a Soviet and Russian mathematical physicist. He is known for the discovery of the ...
and Victor Popov. A more general meaning of the word 'ghost' in
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
is discussed in Ghost (physics).


Overcounting in Feynman path integrals

The necessity for Faddeev–Popov ghosts follows from the requirement that
quantum field theories In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles ...
yield unambiguous, non-singular solutions. This is not possible in the
path integral formulation The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional in ...
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 Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...
of the path integrals contains a factor which does not allow obtaining various results directly from the
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
.


Faddeev–Popov procedure

It is possible, however, to modify the action, such that methods such as
Feynman diagram In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...
s 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 particle A virtual particle is a theoretical transient particle that exhibits some of the characteristics of an ordinary particle, while having its existence limited by the uncertainty principle. The concept of virtual particles arises in the perturbat ...
s in Feynman diagrams – or as the ''absence'' of gauge configurations. However, they are a necessary computational tool to preserve
unitarity In quantum physics, unitarity is the condition that the time evolution of a quantum state according to the Schrödinger equation is mathematically represented by a unitary operator. This is typically taken as an axiom or basic postulate of quant ...
. The exact form or formulation of ghosts is dependent on the particular
gauge Gauge ( or ) may refer to: Measurement * Gauge (instrument), any of a variety of measuring instruments * Gauge (firearms) * Wire gauge, a measure of the size of a wire ** American wire gauge, a common measure of nonferrous wire diameter, ...
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. Consider for example non-Abelian gauge theory with : \int \mathcal \exp i \int \mathrm d^4 x \left ( - \frac F^a_ F^ \right ). The integral needs to be constrained via gauge-fixing via G(A) = 0 to integrate only over physically distinct configurations. Following Faddeev and Popov, this constraint can be applied by inserting : 1 = \int \mathcal
alpha (x) Alpha (uppercase , lowercase ; grc, ἄλφα, ', or ell, άλφα, álfa) is the first Letter (alphabet), letter of the Greek alphabet. In the system of Greek numerals, it has a value of one. Alpha is derived from the Phoenician alphabet, P ...
\delta (G(A^)) \mathrm \frac into the integral. A^ denotes the gauge-fixed field.


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 In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type ...
) the ghosts are
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
scalar fields In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
( spin 0), but they anti-commute (like
fermions In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
). In general, anti-commuting ghosts are associated with
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...
ic symmetries, while
commuting Commuting is periodically recurring travel between one's place of residence and place of work or study, where the traveler, referred to as a commuter, leaves the boundary of their home community. By extension, it can sometimes be any regul ...
ghosts are associated with
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
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 mechanism In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other be ...
, 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 diagram In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduc ...
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 In physics, the ''S''-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT). More forma ...
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 c^a(x)\, in Yang–Mills theories (where a is an index in the adjoint representation of the
gauge group In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
) is given by : \mathcal_=\partial_\bar^\partial^c^+gf^\left(\partial^\bar^\right)A_^c^\;. The first term is a kinetic term like for regular complex scalar fields, and the second term describes the interaction with the
gauge field In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...
s as well as the
Higgs field The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the excited state, quantum excitation of the Higgs field, one of the field (physics), fields in particl ...
. Note that in ''abelian'' gauge theories (such as
quantum electrodynamics In particle physics, quantum electrodynamics (QED) 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 spec ...
) the ghosts do not have any effect since f^ = 0 and, consequently, the ghost particles do not interact with the gauge fields.


Footnotes


References


External links

* {{DEFAULTSORT:Faddeev-Popov ghost Gauge theories Quantum chromodynamics