Faddeev–Popov ghost

In physics, Faddeev–Popov ghosts (also called Faddeev–Popov gauge ghosts or Faddeev–Popov ghost fields) are extraneous fields 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.[1][2]

A more general meaning of the word ghost in theoretical physics is discussed in Ghost (physics).

Overcounting in Feynman path integrals

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.

Faddeev–Popov procedure

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 gauge configurations. However, they are a necessary computational tool to preserve unitarity.

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.

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) the ghosts are complex scalar fields (spin 0), but they anti-commute (like fermions).

In general, anti-commuting ghosts are associated with fermionic symmetries, while commuting ghosts are associated with bosonic 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, 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 diagrams 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.[3] (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 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 as well as the Higgs field. 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.


  1. ^ Faddeev, L. D.; Popov, V. (1967), "Feynman diagrams for the Yang-Mills field", Phys. Lett. B, 25 (1): 29, Bibcode:1967PhLB...25...29F, doi:10.1016/0370-2693(67)90067-6
  2. ^ W. F. Chen. (2008), "Quantum Field Theory and Differential Geometry", Int. J. Geom. Methods Mod. Phys., 10 (4): 1350003, arXiv:0803.1340v2, doi:10.1142/S0219887813500035
  3. ^ R. P. Feynman, (1963). "Quantum Theory of Gravitation", Acta Physica Polonica 24, 697−722. Feynman discovered empirically that "boxing" and simply dismissing these diagrams restored unitarity. "Because, unfortunately, I also discovered in the process that the trouble is present in the Yang−Mills theory; and, secondly, I have incidentally discovered a tree−ring connection which is of very great interest and importance in the meson theories and so on. And so I'm stuck to have to continue this investigation, and of course you appreciate that this is the secret reason for doing any work, no matter how absurd and irrational and academic it looks: we all realize that no matter how small a thing is, if it has physical interest and is thought about carefully enough, you're bound to think of something that's good for something else."

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BRST quantization

In theoretical physics, the BRST formalism, or BRST quantization (where the BRST refers to Becchi, Rouet, Stora and Tyutin) denotes a relatively rigorous mathematical approach to quantizing a field theory with a gauge symmetry. Quantization rules in earlier quantum field theory (QFT) frameworks resembled "prescriptions" or "heuristics" more than proofs, especially in non-abelian QFT, where the use of "ghost fields" with superficially bizarre properties is almost unavoidable for technical reasons related to renormalization and anomaly cancellation.

The BRST global supersymmetry introduced in the mid-1970s was quickly understood to rationalize the introduction of these Faddeev–Popov ghosts and their exclusion from "physical" asymptotic states when performing QFT calculations. Crucially, this symmetry of the path integral is preserved in loop order, and thus prevents introduction of counterterms which might spoil renormalizability of gauge theories. Work by other authors a few years later related the BRST operator to the existence of a rigorous alternative to path integrals when quantizing a gauge theory.

Only in the late 1980s, when QFT was reformulated in fiber bundle language for application to problems in the topology of low-dimensional manifolds (topological quantum field theory), did it become apparent that the BRST "transformation" is fundamentally geometrical in character. In this light, "BRST quantization" becomes more than an alternate way to arrive at anomaly-cancelling ghosts. It is a different perspective on what the ghost fields represent, why the Faddeev–Popov method works, and how it is related to the use of Hamiltonian mechanics to construct a perturbative framework. The relationship between gauge invariance and "BRST invariance" forces the choice of a Hamiltonian system whose states are composed of "particles" according to the rules familiar from the canonical quantization formalism. This esoteric consistency condition therefore comes quite close to explaining how quanta and fermions arise in physics to begin with.

In certain cases, notably gravity and supergravity, BRST must be superseded by a more general formalism, the Batalin–Vilkovisky formalism.

Deaths in February 2017

The following is a list of notable deaths in February 2017.

Entries for each day are listed alphabetically by surname. A typical entry lists information in the following sequence:

Name, age, country of citizenship at birth, subsequent country of citizenship (if applicable), reason for notability, cause of death (if known), and reference.

Functional determinant

In functional analysis, a branch of mathematics, it is sometimes possible to generalize the notion of the determinant of a square matrix of finite order (representing a linear transformation from a finite-dimensional vector space to itself) to the infinite-dimensional case of a linear operator S mapping a function space V to itself. The corresponding quantity det(S) is called the functional determinant of S.

There are several formulas for the functional determinant. They are all based on the fact that the determinant of a finite matrix is equal to the product of the eigenvalues of the matrix. A mathematically rigorous definition is via the zeta function of the operator,

where tr stands for the functional trace: the determinant is then defined by

where the zeta function in the point s = 0 is defined by analytic continuation. Another possible generalization, often used by physicists when using the Feynman path integral formalism in quantum field theory (QFT), uses a functional integration:

This path integral is only well defined up to some divergent multiplicative constant. To give it a rigorous meaning it must be divided by another functional determinant, thus effectively cancelling the problematic 'constants'.

These are now, ostensibly, two different definitions for the functional determinant, one coming from quantum field theory and one coming from spectral theory. Each involves some kind of regularization: in the definition popular in physics, two determinants can only be compared with one another; in mathematics, the zeta function was used. Osgood, Phillips & Sarnak (1988) have shown that the results obtained by comparing two functional determinants in the QFT formalism agree with the results obtained by the zeta functional determinant.

Gauge fixing

In the physics of gauge theories, gauge fixing (also called choosing a gauge) 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 configurations. Any two detailed configurations in the same equivalence class are related by a gauge transformation, equivalent to a shear along unphysical axes in configuration space. Most of the quantitative physical predictions of a gauge theory can only be obtained under a coherent prescription for suppressing or ignoring these unphysical degrees of freedom.

Although the unphysical axes in the space of detailed configurations are a fundamental property of the physical model, there is no special set of directions "perpendicular" to them. Hence there is an enormous amount of freedom involved in taking a "cross section" representing each physical configuration by a particular detailed configuration (or even a weighted distribution of them). Judicious gauge fixing can simplify calculations immensely, but becomes progressively harder as the physical model becomes more realistic; its application to quantum field theory is fraught with complications related to renormalization, especially when the computation is continued to higher orders. Historically, the search for logically consistent and computationally tractable gauge fixing procedures, and efforts to demonstrate their equivalence in the face of a bewildering variety of technical difficulties, has been a major driver of mathematical physics from the late nineteenth century to the present.

Index of physics articles (F)

The index of physics articles is split into multiple pages due to its size.

To navigate by individual letter use the table of contents below.

Popov (disambiguation)

Popov is a common Russian, Bulgarian and Serbian last name.

Popov may also refer to:

Popov (crater), lunar crater

Popov (vodka), American brand of vodka

Faddeev–Popov ghost, an object in theoretical physics

Popov Island

Popov Castle


For the game, see Lexicant, a variant of ghost (game).In a supersymmetric quantum field theory, a superghost is a fermionic Faddeev–Popov ghost, which is used in the gauge fixing of a fermionic symmetry generator.

Yang–Mills theory

Yang–Mills theory is a gauge theory based on the SU(N) group, or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using these non-abelian Lie groups and is at the core of the unification of the electromagnetic force and weak forces (i.e. U(1) × SU(2)) as well as quantum chromodynamics, the theory of the strong force (based on SU(3)). Thus it forms the basis of our understanding of the Standard Model of particle physics.

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