Faddeev-Popov Fields Research Articles

Erratum: One-loop corrections to the string tension of the vortex in the Abelian Higgs model [Phys. Rev. D78, 085008 (2008)

We present an exact numerical computation of the one-loop correction of the string tension for the Nielsen-Olesen vortex in the Abelian Higgs model. The computations proceeds via the computation of the Euclidean Green's function for the gauge, Higgs and Faddeev-Popov fields using mode functions, and taking the appropriate trace. Renormalization is an essential part of this computation, it is done by removing leading order contributions from the numerical results so as to make these finite, and to add the divergent parts back, after suitable regularization and renormalization. We encounter and solve some problems which are specific to gauge theories and topological solutions. The corrections to the energy are found to be sizeable, but still smaller than the classical energy as long as g**2 is smaller than unity.

One-loop corrections to the string tension of the vortex in the Abelian Higgs model

We present an exact numerical computation of the one-loop correction of the string tension for the Nielsen-Olesen vortex in the Abelian Higgs model. The computation proceeds via the computation of the Euclidean Green's function for the gauge, Higgs, and Faddeev-Popov fields using mode functions, and taking the appropriate trace. Renormalization is an essential part of this computation. It is done by removing leading order contributions from the numerical results so as to make these finite, and to add the divergent parts back, after suitable regularization and renormalization. We encounter and solve some problems which are specific to gauge theories and topological solutions. The corrections to the energy are found to be sizable, but still smaller than the classical energy as long as g{sup 2} is smaller than unity.

Generally covariant geometro-stochastic quantum gravity

A geometro-stochastic formulation of quantum gravity is presented which is covariant with respect to Poincare gauge transformations as well as to internal gauges related to local coordinate changes in the base manifold of quantum-gravitational fibre bundles. In this formulation the graviton is a massless geometro-stochastic exciton of spin 2 displaying a stringlike mean stochastic extension of the order of the Planck length. A quantum pregeometry is obtained by constructing from the states of such gravitons Fock-type fibres over the general linear frame bundleGL(M) of a base differential manifoldM. These fibres are then soldered toM by means of soldering forms supplied by various pseudo-Riemannian/Lorentzian geometries in suitable regions of that manifold. Graviton propagation is governed by a second-quantized connection∇ invariant under external as well as internal gauge transformations, which in combination give rise to a quantum-gravitational gauge group. The parallel transport of related gauge frames involves geometrized versions of Faddeev-Popov fields derived from Maurer-Cartan structural equations for that gauge group in a supergauge context. The quantum-gravitational connection∇ gives rise to a geometro-stochastic propagation of quantum-gravitational states which is interrelated to the evolution of various Lorentzian geometries in the base manifoldM. For each set of suitable initial conditions this mutual interaction reduces the quantum pregeometry overGL(M) to a quantum space-time represented by gravitational quantum (super)fibre bundles over supermanifolds based on mean Lorentzian geometries (M,ḡ) inM.

Canonical operator theory of non-Abelian gauge fields

Non-Abelian gauge theory in a manifestly covariant gauge is formulated as a theory of canonical field operators and embedded in an indefinite metric space. A gauge-fixing field is included and every field component has a nonvanishing adjoint momentum with which it has canonical commutation (or anticommutation) relations. Faddeev-Popov fields are represented as scalar fermion fields with ghost particle excitations. Feynman rules are derived from the canonical formulation. A discussion is given of the relation between the existence of a subsidiary condition and the existence of « pure gauge » states that dynamically detach from observable states.

Superfiber bundle structure of gauge theories with Faddeev–Popov fields

In this work the mathematical structure of principal superfiber bundle Ps is used to give a geometrical description of gauge theories. The base space of Ps is an S4,2-supermanifold Xs (four commuting and two anticommuting variables), and the structure group an Sn,0 supergroup Gs, where n is the dimension of the gauge group for the classical theory. The body of Ps is the usual principal fiber bundle P of gauge theories. Gauge and Faddeev–Popov fields arise as superfields, components of the connections in Ps, in a local coordinate system. BRS (Becchi, Rouet, and Stora) and anti-BRS transformations are gauge transformations, in Ps, of parameters the ghost and antighost superfields, respectively. In the case of soul-flat connections, which are connections in Ps coming from connections in P, the BRS and anti-BRS transformations are finite translations along the anticommuting directions of Xs, and generate an S0,2-supergroup.

The supercurrent in N = 1 supersymmetric Yang-Mills theories : (II). Renormalization

We show to all orders of perturbation theory that in a general N = 1 supersymmetric Yang-Mills theory maintaining parity there exists a supercurrent. Since all Faddeev-Popov fields and gauge contributions are taken into account, the relation between supercurrent anomalies and the Callan-Symanzik equation can be established. The connection with the axial anomaly coefficient is somewhat obscured by the fact that the supercurrent cannot be improved.

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.

Geometrical structure of Faddeev–Popov fields and invariance properties of gauge theories

Let 𝒫n be the trivial principal bundle with structural group G and base space 𝒫n−1, 𝒫1 being the usual fiber bundle of gauge theories. In order to give a geometrical interpretation to the Faddeev–Popov fields, as well as to the Becchi, Rouet, and Stora transformations, we need to use the fiber bundle 𝒫3. The gauge fields and the Faddeev–Popov ghost and antighost fields appear as part of certain 1-forms defined on the base space 𝒫2. The anticommuting character of the ghost and antighost fields is essentially due to their identification with 1-forms. The Becchi, Rouet, and Stora transformations are identified with generalized infinitesimal gauge transformations on 𝒫3 of parameters related to the ghost fields. We obtain a further invariance of the action given by a similar generalized infinitesimal gauge transformation on 𝒫3 related to the antighost fields.

On a Superfield Theoretical Treatment of the Higgs-Kibble Mechanism

The superfield theoretical aspect of the Riggs-Kibble mechanism is discussed by realiz­ ing the supersymmetry of Becchi, Rouet and Stora (BRS) as translation and dilatation operations on the real elements of the Grassmann algebra. The Slavnov-Taylor identity becomes a statement on each connected Green's function separately in this formulation, and some of the pathological cases discussed in the literature can be avoided by imposing the manifest BRS covariance in terms of the superfields. This is applied to the discussion of non-linear (quadratic) gauge conditions. The manifestly BRS covariant treatment of non­ linear gauge conditions requires a quartic self-coupling of Faddeev-Popov fields to ensure the multiplicative renormalization, although the physical S-matrix is independent of this ghost coupling. The proof of unitarity (i.e., the ghost cancellation) follows the same mechanism as the divergence cancellation in the ordinary superfield theory. The gauge independence is proved by a simple classification of operators according to their BRS trans­ formation properties. We also briefly comment on the canonical treatment of superfields. The renormalization of the Yang-Mills fieldsn and the Riggs-Kibble mecha­ nism2> has been extensively discussed in the literature.s>.•> The Faddeev-Popov Lagrangian 5> provides a starting point for these discussions of the renormalizability and unitarity. In connection with the Faddeev-Popov Lagrangian, Becchi, Rcuet and Stora (BRS) a> introduced a supersymmetry pseudo-algebra which gives rise to a very neat way to derive the Slavnov-Taylor identityn (i.e., the Ward-Takahashi identity associated with the BRS symmetry). It was also recognized that a detailed study of the BRS symmetry is sufficient to investigate the unitarity of the S-matrix elements. 8> Recently, Kugo and Ojima9> further clarified the contents of the BRS symmetry and the structure of the physical Hilbert space. They made the following important observations: (i) The auxiliary Lagrangian multiplier field makes the BRS symmetry manifest, and (ii) the ghost number of the Faddeev-Popov fields should be properly associated with the scale transformation of the ghost fields. Although the latter point is not crucial in the perturbative treatmeneo> of the Yang-Mills fields, the structure of the physical space is much simplified by this interpretation. It is also nicer to keep the Lagrangian manifestly Hermitian at all the stages of the calculation. In this paper, we discuss the superfield theoryll)~J•> associated with the BRS symmetry. Some of the initial attempts 15> toward this direction have been made,