Faddeev–Popov determinant

Faddeev–Popov determinant
Name	Faddeev–Popov determinant
Field	Theoretical physics
Introduced	1967
Introduced by	Ludvig Faddeev, Victor Popov
Related	Gauge theory, Path integral, BRST symmetry

Faddeev–Popov determinant.

Introduction

The Faddeev–Popov determinant was introduced by Ludvig Faddeev and Victor Popov as a central tool in quantizing non-Abelian gauge theories such as those used in the Standard Model, and it plays a role in relating the path integral formulation employed by Richard Feynman and Julian Schwinger to canonical quantization approaches associated with Paul Dirac and Peter Higgs. It appears in treatments alongside concepts developed in work by Murray Gell-Mann, Sheldon Glashow, Abdus Salam, and Steven Weinberg, and it underpins perturbative computations used in experiments at CERN, Fermilab, and SLAC.

Gauge Fixing and Path Integrals

Gauge fixing appears in the path integral methods developed after contributions by Freeman Dyson and Paul Dirac to remove redundant configurations; the Faddeev–Popov determinant implements this removal in Yang–Mills path integrals connected to Yang and Mills’s original proposal and later applications to Quantum Chromodynamics studied by David Gross, Frank Wilczek, and David Politzer. Its insertion is related to fixing gauges such as Lorenz gauge used by Hendrik Lorentz, Coulomb gauge referenced in Marcel Grossmann’s contexts, and axial gauges applied in work by Gerard 't Hooft and Stanley Mandelstam, with implications for renormalization programs developed by Kenneth Wilson and Michael Peskin.

Derivation of the Faddeev–Popov Determinant

The derivation begins with the functional integral formalism built on earlier formal work of John von Neumann and Norbert Wiener, and employs gauge group integrals over groups like SU(2) and SU(3) appearing in the models of Werner Heisenberg and Murray Gell-Mann; one inserts a delta functional for a chosen gauge condition influenced by techniques in the path integral literature of Richard Feynman, then introduces the determinant of the gauge variation Jacobian, an idea formalized by Faddeev and Popov and used in subsequent studies by Gerard 't Hooft, Kenneth Wilson, and Steven Weinberg in renormalizable gauge theories. Evaluating this determinant often leverages spectral analysis methods related to John von Neumann’s operator theory and zeta-function regularization approaches pursued by Bernhard Riemann and Edward Witten in modern contexts.

Ghost Fields and BRST Symmetry

The Faddeev–Popov determinant can be represented as a functional integral over anticommuting Grassmann fields introduced in analogy with methods of Paul Dirac and Pascual Jordan; these ghost fields were incorporated into quantization schemes later unified with Becchi, Rouet, Stora, and Tyutin to produce BRST symmetry, which connects with symmetry treatments by Emmy Noether and Élie Cartan. BRST invariance plays a role in proofs of renormalizability by Gerard 't Hooft and Carlo Becchi and appears in approaches to anomaly cancellation investigated by Michael Green and John Schwarz in superstring theory contexts developed by Edward Witten and Joseph Polchinski.

Examples and Applications

In perturbative Quantum Chromodynamics computations relevant to experiments by the ATLAS Collaboration and the CMS Collaboration at CERN, the Faddeev–Popov determinant and associated ghost loops discovered in calculations by Gerard 't Hooft and Martinus Veltman contribute to beta function determinations by David Gross and Frank Wilczek; in electroweak theory work by Sheldon Glashow, Abdus Salam, and Steven Weinberg the determinant is used alongside Higgs sector analyses initiated by Peter Higgs and François Englert. In topological quantum field theory and studies by Edward Witten, as well as in lattice gauge theory implementations pioneered by Kenneth Wilson and Michael Creutz, the determinant and ghosts influence Monte Carlo simulations used at institutions such as SLAC and Brookhaven National Laboratory.

Mathematical Properties and Regularization

Mathematically, the Faddeev–Popov determinant is a functional determinant related to elliptic operators studied in spectral geometry by Atiyah and Singer and in index theory by Michael Atiyah and Isadore Singer; regularization methods applied to it include Pauli–Villars regularization developed by Wolfgang Pauli and Felix Villars, dimensional regularization advanced by Gerard ’t Hooft and William Bardeen, and zeta-function techniques investigated by Ray and Singer and employed in work by Edward Witten. Its analytic continuation and renormalization touch on methods used in perturbative expansions by Freeman Dyson and resummation techniques in studies by Stanley Coleman and Murray Gell-Mann, while rigorous treatments draw on functional analysis traditions established by John von Neumann and Laurent Schwartz.

Category:Quantum field theory