Faddeev-Popov Procedure Research Articles

Compact QED in the Landau gauge: A lattice-gauge-fixing case study.

We derive different representations of compact QED fixed to Landau gauge by the lattice Faddeev-Popov procedure. Our analysis finds that (A)Nielsen-Olesen vortices arising from the compactness of the gauge-fixing action are {\it quenched\/}, that is, the Faddeev-Popov determinant cancels them out and they do not influence correlation functions such as the photon propagator; (B)Dirac strings are responsible for the nonzero mass pole of the photon propagator. Since in $D=3+1$ the photon mass undergoes a rapid drop to zero at $\beta_c$, the deconfinement point, this result predicts that Dirac strings must be sufficiently dilute at $\beta > \beta_c$. Indeed, numerical simulations reveal that the string density undergoes a rapid drop to near zero at $\beta\sim \beta_c$.

Regularization of chiral gauge theories

In this paper we present an explicit two-loop calculation which shows that, as a consequence of the breaking of chiral gauge symmetry induced by the regularization, non-gauge invariant counterterms are present in the ghost sector of a chiral gauge theory. This occurrence prevents one from inverting the Faddeev-Popov procedure and removing the gauge fixing in numerical simulations.

Faddeev-Popov method for anomalous quasigroups

A generalization of the Faddeev-Popov procedure to deal with either anomalous and non-anomalous gauge theories with closed, irreducible gauge algebra is presented. An expression for the Wess-Zumino action is obtained.

A solvable four-dimensional model with gauge symmetry

Recently, interest has been focused on quantum field theories in which a Lagrange multiplier field occurs in the classical Lagrangian. This has the effect of restricting the sum over classical paths in the path integral to solutions of the classical field equation. Examples of such theories are the dynamical theory of two-dimensional gravity proposed by Jackiw and Teitelboim, the Chern–Simons formulation of gravity in 2 + 1 dimensions by Witten, and the path-integral formulation of classical systems by Gozzi. We examine, in this paper, a gauge theory in which a vector field [Formula: see text] acts as a Lagrange multiplier in the classical Lagrangian, ensuring that a vector field [Formula: see text] satisfies the Yang–Mills equations of motion. Quantization can be carried out either using BRST quantization or by using the Faddeev–Popov procedure. Either by explicitly integrating over the field [Formula: see text] and its associated ghost fields, or by directly examining the Feynman perturbation theory, it can be established that all diagrams beyond one-loop order vanish, allowing one to compute the one-particle irreducible generating functional exactly. Background-field quantization is introduced to simplify the renormalization program. The β function is computed in closed form. In an appendix we show how our interaction can be derived from Yang–Mills theory based on a group G or by considering the Yang–Mills theory for a group IG. This can be extended to deal with four interacting gauge fields. A second appendix deals with a scalar model that superficially resembles our vector model.

The Faddeev-Popov procedure and application to bosonic strings: An infinite dimensional point of view

A generalisation of the finite dimensional presentation of the Faddeev-Popov perocedure is derived, in an infinite dimensional framework for gauge theories with finite dimensional moduli space using heat-kernel regularised determinants. It is shown that the infinite dimensional Faddeev-Popov determinant is-up to a finite dimensional determinant determined by a choice of a slice-canonically determined by the geometrical data defining the gauge theory, namely a fibre bundleP→P/G with structure groupG and the invariance group of a metric structure given on the total spaceP. The case of (closed) bosonic string theory is discussed.

Gauge invariance and current algebra in two-dimensional non-Abelian chiral theories

The author studies two-dimensional massless chiral QCD in a gauge invariant formulation. The Faddeev-Popov procedure is used to obtain a gauge invariant effective action after bosonisation, which is then quantised canonically using Dirac's prescription for constrained systems. The gauge current algebra is deduced and covariant current conservation is shown by means of the equations of motion of the bosonic matter fields.

Anti-symmetric tensor gauge theories

Anti-symmetric tensor gauge theories are quantized via the Faddeev-Popov procedure leading to a BRS invariant theory without the need for second order ghosts. The quantum mechanical equivalence of these theories to non-linear sigma models is established using the spontaneously broken global symmetry currents. Upon gauging these symmetries the Higgs mechanism is displayed, while for anomalous symmetries an analog of the Wess-Zumino action is constructed in terms of the anti-symmetric tensor gauge fields.

Correct Fujikawa regularization prescription in the chiral Schwinger model and the Wess-Zumino term.

In the solutions for the chiral Schwinger model found in the literature, where path-integral methods were applied, the problem of Hermiticity of the regulator operator has been forgotten. We use two Hermitian prescriptions to solve this model exactly. We also include the Wess-Zumino term through the Faddeev-Popov procedure and study its behavior under both Hermitian prescriptions.

On Gribov ambiguity in finite temperature Abelian gauge theory

It is shown that there is Gribov ambiguity in finite temperature Abelian gauge theory if the gauge theory is defined on the full gauge orbit space. This is demonstrated geometrically and by an explicit construction of solutions to the Gribov equation. Its effects on the evaluation of the temperature-dependent partition function are noted using the Faddeev-Popov procedure.

On the Gribov ambiguity in the Polyakov string

The global aspects of the gauge fixing in the Polyakov path integral for the bosonic string are considered within the Ebin–Fischer–Mareden approach to the geometry of spaces of Riemannian metrics and conformal structures. It is shown that for surfaces of higher genus, the existence of local conformal gauges is sufficient to derive the globally defined integral over the Teichmüller space. The generalized Faddeev–Popov procedure for incomplete gauges is formulated and used to derive the global expression for the Polyakov path integral in the cases of torus and sphere. The Gribov ambiguity in the functional integral over surfaces without boundary can be successfully overcome for arbitrary genus.

The integration ofG-invariant functions and the geometry of the Faddeev-Popov procedure

Various versions of the Fubini theorem on the principal fibre bundle are derived. The formal generalizations of these theorems are used as a basic tool for the investigation of the geometrical setting of the Faddeev-Popov procedure in the cases of pure Yang-Mills theories, spinless particle and Polyakov's bosonic string.

On the Gribov ambiguity in the Faddeev-Popov procedure

The standard Faddeev-Popov procedure for quantization of a gauge theory is modified so as to be valid even when the Gribov problem is present. The hamiltonian is employed and a definite expression for the path integral is obtained for a wide class of gauges.

The supercurrent in N = 1 supersymmetric Yang-Mills theories : (I). The classical case

We construct, in the tree approximation, the complete off-shell supercurrent, including Faddeev-Popov fields and gauge contributions for a general supersymmetric Yang-Mills theory maintaining parity. Due to the conformally non-covariant character of the Faddeev-Popov procedure this supercurrent cannot be improved. The other currents of the superconformal group can therefore no longer be obtained by simply forming moments.

Faddeev-Popov procedure and Gribov copies

In simple examples, we find that gauges with Gribov copying are valid gauges. One need only compute the Faddeev-Popov determinant Δ exactly and procede. Δ provides the correct measure for the copies. A cut-off procedure keeping only the first copy is seen to work as well. In quantum chromodynamics, we point out a class of gauges, the field-strength gauges, for which Δ is easily computed.