The Faddeev–Popov ghost, often simply called ghost fields or ghosts, are fictitious, non-physical fields introduced in the path integral formulation of quantum field theories, particularly in gauge theories such as Quantum Chromodynamics (QCD). They are an essential element for consistently quantizing gauge theories and for maintaining unitarity and causality when using the path integral method after fixing the gauge.
Context and Motivation
Gauge theories, like electromagnetism (QED) or the strong nuclear force (QCD), possess a symmetry known as gauge symmetry. This means that different configurations of the gauge fields (e.g., the photon field or gluon fields) can represent the exact same physical state. When attempting to quantize these theories using the path integral formulation, this redundancy leads to an "overcounting" problem: the integral includes an infinite volume associated with these unphysical gauge transformations.
To resolve this, a gauge-fixing condition must be imposed. However, simply adding a gauge-fixing term to the Lagrangian is not sufficient to produce a well-defined and unitary quantum theory. The method developed by Ludvig Faddeev and Victor Popov in 1967 addresses this issue.
The Faddeev–Popov Procedure
The Faddeev–Popov method involves inserting a specific form of unity into the path integral, which essentially restricts the integration to only one representative from each gauge orbit. This introduces a determinant factor (the Faddeev–Popov determinant) into the path integral measure. This determinant cannot be directly integrated out or absorbed into other terms in a simple way, especially in non-abelian gauge theories.
To handle this determinant, it is re-expressed as an integral over a pair of anticommuting (Grassmann) scalar fields. These auxiliary fields are the Faddeev–Popov ghosts ($c^a$) and anti-ghosts ($\bar{c}^a$). Their corresponding kinetic and interaction terms are added to the gauge-fixed Lagrangian:
$$ \mathcal{L}{\text{ghost}} = \partial^\mu \bar{c}^a D\mu^{ab} c^b $$
where $D_\mu^{ab}$ is the covariant derivative in the adjoint representation, incorporating interactions with the gauge bosons. For example, in QCD, the ghosts interact with gluons.
Properties of Faddeev–Popov Ghosts
- Statistics: Despite being scalar fields (spin 0), Faddeev–Popov ghosts are Grassmann fields, meaning they anticommute. This property implies they obey Fermi-Dirac statistics, which is crucial for them to cancel unphysical degrees of freedom that would otherwise contribute with "bosonic" signs.
- Physicality: Ghosts are purely mathematical constructs and are not physical particles. They do not appear as asymptotic states (incoming or outgoing particles) in scattering processes and are never detected experimentally. Their sole purpose is to maintain the consistency (unitarity and renormalizability) of the quantum theory when performing internal loop calculations in Feynman diagrams.
- Quantum Numbers: They carry the same gauge quantum numbers as the gauge bosons with which they interact (e.g., color charge for QCD ghosts), but unlike gauge bosons, they are scalars.
- BRST Symmetry: The introduction of ghost fields is deeply connected to BRST (Becchi–Rouet–Stora–Tyutin) symmetry, which is an extended gauge symmetry that provides a powerful framework for proving the renormalizability and unitarity of gauge theories.
Significance
Faddeev–Popov ghosts are indispensable for:
- Unitarity: They cancel the contributions of unphysical longitudinal and scalar polarization states of the gauge bosons that would otherwise propagate in loop diagrams, ensuring that the S-matrix remains unitary (i.e., probabilities are conserved).
- Renormalizability: Their presence ensures that non-abelian gauge theories remain renormalizable, allowing for a consistent quantum description without encountering infinities that cannot be removed.
The concept of Faddeev–Popov ghosts revolutionized the understanding and application of quantum field theory, particularly in the context of the Standard Model of particle physics, where non-abelian gauge theories (like the strong and weak forces) play a central role.