Faddeevpopov GhostEdit
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Faddeev–Popov ghost
The Faddeev–Popov ghost, often simply called a ghost in the literature on quantum field theory, is a mathematical device used in the quantization of gauge theories to handle gauge redundancy in the path integral formalism. Named for Ludvig D. Faddeev and Victor Popov, who introduced the method in 1967, the construction ensures that gauge fixing does not spoil gauge invariance or unitarity after quantization. The ghost fields are anticommuting scalar fields (Grassmann-valued) that, despite not corresponding to physical particles in the observed spectrum, are essential for the consistency of calculations in non-Abelian gauge theories.
In short, the Faddeev–Popov mechanism allows one to gauge-fix a theory with redundant gauge degrees of freedom and to represent the resulting gauge-fixing determinant as an integral over auxiliary ghost fields. This makes it possible to perform perturbative calculations in a way that preserves the underlying symmetries of the theory and yields finite, renormalizable results in many cases.
Origins and purpose - The method was developed to address the redundancy that arises when integrating over gauge-equivalent configurations in the path integral for non-Abelian gauge theories such as Yang–Mills theory and the Standard Model of particle physics. The introduction of the Faddeev–Popov determinant corrects for overcounting due to gauge symmetry. - The procedure generalizes the earlier idea of gauge fixing and provides a practical route to compute quantum corrections in theories where gauge fields interact nontrivially. In Abelian theories like QED, the determinant is field-independent, and ghosts decouple from the dynamics; in non-Abelian theories, ghosts play an active role in loop corrections.
Mathematical construction - Gauge fixing is implemented by imposing a condition G[A] = 0 on the gauge fields A, and one introduces the Faddeev–Popov determinant det[M[A]] to compensate for changes of variables under gauge transformations. Here M[A] is the operator that results from the variation of the gauge-fixing condition with respect to the gauge parameter. - The determinant det[M[A]] can be represented as a functional integral over ghost fields c and c̄ (the ghost and anti-ghost), which are Grassmann-valued scalars: det[M[A]] = ∫ D[c, c̄] exp(i ∫ d^dx c̄^a M^{ab}[A] c^b). The resulting ghost action S_ghost typically takes the form ∫ d^dx c̄^a M^{ab}[A] c^b, with M[A] involving the gauge field A and the structure constants of the gauge group. - In covariant gauges, such as the Landau gauge or the Feynman gauge, M[A] reduces to expressions built from the covariant derivative D_μ and the gauge fields. In Abelian theories like QED, the operator M[A] simplifies so that ghost fields cancel and do not affect physical observables; in non-Abelian theories, they contribute nontrivially to loop corrections.
Role in gauge theories - Ghosts appear as internal lines in Feynman diagrams for non-Abelian gauge theories and are essential for preserving unitarity and gauge invariance after gauge fixing. They ensure that unphysical polarizations of gauge bosons do not contribute to physical amplitudes. - The presence of ghost fields is closely tied to the renormalizability of non-Abelian gauge theories. Ghosts, together with gauge fields, participate in loop corrections in a way that maintains the consistency of perturbation theory, enabling precise predictions in Quantum Chromodynamics (QCD) and the electroweak sector of the Standard Model. - The formalism is often discussed alongside BRST symmetry, a global fermionic symmetry that combines gauge and ghost fields. BRST invariance provides a rigorous framework for identifying physical states as elements of a cohomology class, ensuring that unphysical gauge artifacts do not contaminate the physical spectrum.
BRST symmetry and gauge fixing - The Faddeev–Popov construction can be embedded in a BRST-symmetric framework, where the gauge-fixing term and the ghost term form a BRST-exact combination. This symmetry plays a central role in proving renormalizability and in organizing perturbative expansions. - BRST cohomology defines the physical Hilbert space as states that are BRST-closed but not BRST-exact, offering a clean criterion for selecting observable states in gauge theories.
Non-perturbative issues and debates - A well-known non-perturbative complication is the Gribov ambiguity: in non-Abelian gauge theories, the gauge-fixing condition does not always intersect each gauge orbit exactly once. Multiple gauge copies (Gribov copies) can satisfy the same gauge condition, signaling a limitation of the standard Faddeev–Popov approach beyond perturbation theory. - This issue leads to refinements such as the Gribov–Zwanziger framework, which modifies the gauge-fixed action to account for copies inside a restricted region of configuration space. The discussion remains active in lattice studies and continuum approaches, with ongoing debates about how best to implement nonperturbative gauge fixing and what implications this has for confinement and the infrared behavior of propagators. - Ghost propagators and their infrared properties have been subjects of investigation using lattice gauge theory and functional methods. While perturbation theory treats ghosts as internal devices, some nonperturbative analyses explore how ghost dynamics might influence long-range phenomena, though the consensus remains that ghosts are not physical particles in asymptotic states.
Applications and impact - In the Standard Model, ghost fields contribute to quantum corrections in processes governed by non-Abelian gauge symmetries, such as those in QCD and the electroweak sector. They are indispensable for maintaining gauge-invariant, renormalizable calculations across multiple orders in perturbation theory. - The formalism extends to a wide range of gauge theories and remains a cornerstone of modern quantum field theory. It underpins precision calculations that test the Standard Model and guide explorations of physics beyond it, including grand unified theories and various approaches to quantum gravity that employ gauge-like redundancies. - The interplay between gauge fixing, ghost fields, and BRST symmetry is routinely discussed in the context of renormalization and the unitarity of the S-matrix, ensuring that physical predictions do not depend on the arbitrary choices made during gauge fixing.
See also - Ludvig Faddeev - Victor Popov - gauge theory - path integral - ghost field - Faddeev–Popov determinant - Yang–Mills theory - QCD - QED - BRST symmetry - Gribov ambiguity - Gribov–Zwanziger action - Feynman diagram - Renormalization - Lattice gauge theory