BRS Invariance Research Articles

General theory of renormalization of gauge theories in nonlinear gauges

We discuss the general theory of renormalization of unbroken gauge theories in the nonlinear gauges in which the gauge-fixing term is of the form\( - \frac{1}{2}\sum\limits_\alpha {f_\alpha ^2 [A] = - \frac{1}{2}} \sum\limits_\alpha {\frac{1}{{\eta _\alpha }}(\partial ^\mu A_\mu ^\alpha + \zeta _{\beta \gamma }^\alpha A_\mu ^\beta A^{\gamma \mu } )^2 } \) We show that higher loop renormalization modifiesfα [A] to contain ghost terms of the form\(f_\alpha [A,c,\bar c] = \eta _\alpha ^{ - \frac{1}{2}} (\partial ^\mu A_\mu ^\alpha + \zeta _{\beta \gamma }^\alpha A_\mu ^\beta A^{\gamma \mu } + \tau _\gamma ^{\alpha \beta } \bar c_\beta c_\gamma )\) and show how the corresponding ghost terms are deduced fromfα [A, c, c] uniquely. We show that the theory can be renormalized while preserving a modified form of BRS invariance by multiplicative and independent renormalizations onA, c, g, η, ζ, τ. We briefly discuss the independence of the renormalized S-matrix from η,ζ, τ.

Quantisation in a nonlinear gauge and Feynman rules

Certain aspects of the use of a non-linear gauge fixing condition in SU(N) gauge field theory are studied. The Faddeev-Popov ghost Lagrangian is obtained systematically using two different but related methods. Feynman rules are explicitly derived from the generating functional for the full Lagrangian in non-linear gauge. The tree-level amplitude for gauge field-gauge field scattering is shown to be independent of the parameter signalling the non-linearity of the gauge fixing condition and hence the same as the one in the linear covariant (Lorentz) gauge. The full Lagrangian is shown to have BRS invariance and the explicit form for BRS transformations is given. Using this, the corresponding Slavnov-Taylor identities are derived. The Lagrange multiplier formalism is used to derive the ghost Lagrangian independently and shown to be the same as that obtained by the gauge variation of the gauge fixing condition. The corresponding situation in Abelian gauge theories is reviewed. The coupling of the ghosts with gauge fields and the Gribov ambiguity are discussed.

Renormalization of a gauge theory in a nonlinear gauge

We discuss renormalization of an O(3) gauge model with the gauge fixing term given by ℒg.f.=-1/ζ|(∂μ-igA3μ)W+μ|2-(1/2α)(∂A3)2. We utilize earlier results on the general theory of renormalization of gauge theories in quadratic gauges to prove multiplicative renormalizability of the theory together with a subtractive renormalization of gauge fixing and ghost terms. We show that this model has a double BRS invariance and that it is preserved under renormalization.

Gravity as a gauge theory with Cartan connection

A gauge formulation of gravity, based on the notion of the connection of Cartan, is given. The Cartan connection involves two principal fiber bundles P and P′ with groups G and G′, respectively; G′ is a subgroup of G and can be regarded as a symmetry group to which G is broken. When the differential form defining the connection in P gives absolute parallelism in P′, one speaks about the Cartan connection. General geometric framework is specialized to the case in which G is the de Sitter group SO(4,1) and G′ is the Lorentz group SO(3,1). The action of the Yang–Mills type is similar to, but not identical with, the action derived earlier by Townsend. The field equations in P are translated into a system of coupled equations for curvature and torsion in P′. Under contraction of SO(4,1) to the Poincaré group ISO(3,1), and for vanishing torsion, the equations become Yang’s equation and Einstein’s equation in vacuum. The BRS invariance of the theory, supplemented by Faddeev–Popov and gauge fixing terms, is analyzed in some detail.

Hidden BRS invariance in classical mechanics

We give in this paper a path integral formulation of classical mechanics. We do so by writing down the associated classical-generating functional. This functional exhibits an unexpected BRS-like and antiBRS-like invariance. This invariance allows for a simple expression, in term of superfields, of this generating functional. Associated to the BRS and antiBRS charges there is also a ghost charge whose conservation turns out to be nothing else than the wel-known Liouville theorem of classical mechanics.

BRS Invariance of String Vertex on General Ghost Vacuum

BRS Invariance of String Vertex on General Ghost Vacuum

Nonperturbative BRS invariance and the Gribov problem

The requirement of lattice BRS invariance is argued to successfully replace local gauge invariance to any order in a perturbative expansion. In a nonperturbative setting, however, one encounters an obstacle which, on a formal level, exists also in the continuum formulation. The origin of the problem and possible ways to avoid it are discussed.

Stochastic regularization of non-abelian gauge theories

We study stochastic regularization as applied to Yang-Mills theory. A perturbative computation to one loop shows that either the regularization is incomplete and some divergences remain or gauge invariance and BRS invariance are destroyed. These two situations arise for the gauge invariant and gauge fixed Langevin equations respectively.

BRS Hamiltonian quantization of a spinless relativistic particle in relativistic gauges

The mechanics of a spinless relativistic particle is quantized by using the covariant hamiltonian path integral of Balatin, Fradkin, and Vilkovisky (BFV). The local noninternal symmetry of he classical theory is fixed by using a “relativistic gauge” via BRS symmetry, a single derivation of the BFV path integral based on the underlying hamiltonian theory is given, and the boundary conditions on the path integral are determined by the BRS invariance requirement. The resulting path integral is explicitly evaluated to yield the Klein-Gordon propagator.

Conformal quantum Yang–Mills

Conformally covariant quantization of non-Abelian gauge theory is presented, and the invariant propagators needed for perturbative calculations are found. The vector potential acquires a richer gauge structure displayed in the larger Gupta–Bleuler triplet whose center is occupied by conformal QED. Path integral formulation and BRS invariance are shown on a formal level in one covariant gauge.

Renormalization and stochastic quantization

A set of problems which includes for example the Langevin equation, a spin system in a random magnetic field or the quantization of gauge theories, share a common algebraic structure: the field from which correlation functions are calculated is given by a local equation in terms of a second field (called generically in this article the noise) for which a probability distribution is provided. For all these problems one can construct an effective action which is automatically BRS invariant (in special cases it is even supersymmetric). This BRS invariance is responsible for the structural stability of the effective action after renormalization. These properties are illustrated on the Langevin equation. A Langevin equation associated with two-dimensional field theories on manifolds is presented.

Nonperturbative BRS invariance

In the context of lattice gauge theories the standard requirement of local gauge invariance is replaced by BRS invariance. The most general BRS invariant lagrangian is shown to contain as many parameters as the gauge invariant one. This is done by explicitly solving the lattice BRS cohomology problem.

A new improved temporal gauge: The soft temporal gauge

A new gauge called the “soft temporal gauge” is considered, which automatically leads to a “softening” of the gauge singularities at k 0 = 0, while preserving the nice features of the traditional temporal gauge. In the physical sector, Landshoff's α-prescription is obtained. Using the Slavnov-Taylor identities (derived from BRS invariance), a simple proof is given that, for non-anomalous Yang-Mills theories, Gauss' law exactly holds.

Gravitational counterterms in an axial gauge.

All the counterterms of quantum Einstein gravity are calculated up to bilinear terms and one-loop order in an axial gauge as a sum of gauge-invariant and the Becchi-Rouet-Stora- (BRS) invariant terms. Contrary to the de Donder gauge condition, 10 out of 33 coefficients for counterterms remain undetermined in this gauge. Some relations among the counterterms, which satisfy the BRS invariance, are implicitly obtained in the course of the calculations.

On background gauge-invariant regulators

Background gauge invariant regulators are studied. It is found that BRS invariance is needed only for the proper treatment of subdivergences. Gauge-invariant Pauli-Villars regularization is used as an example of such a regulator.

Canonical Quantum Theory of Gravitational Field with Higher Derivatives. III: A Formulation with an Additional BRS Invariance

Canonical Quantum Theory of Gravitational Field with Higher Derivatives. III: A Formulation with an Additional BRS Invariance

Covariant quantization of string based on BRS invariance

Covariant canonical quantization of the bosonic string is performed, based on the BRS invariance principle. Defining a physical state by a modified form of Kugo-Ojima's subsidiary condition Q B|phys〉 = 0 ( Q B = BRS charge), we prove a no-ghost theorem in a simple manner. There an interesting relation between critical dimension and BRS symmetry is noticed; namely, the conditions D = 26 and α 0 = 1 for the space-time dimension D and the zero-intercept α 0 of leading trajectory are required by the nilpotency Q B 2 = 0 of the BRS charge. In addition, ghost number fractionization is observed in this system.

Transition amplitude in the quantum theory of the relativistic membrane

Abstract The transition amplitude between two configurations of the n-dimensional relativistic membrane is written explicitly. Its expression contains coupling terms of order 2n in the ghost fields, despite the fact that the algebra of the gauge transformations is closed in the covariant formalism. The inclusion of these additional coupling terms ensures the BRS invariance of the effective action.

General procedure of gauge fixing based on BRS invariance principle

We propose a simple gauge-fixing procedure based on the BRS invariance principle. It does not refer to the path integral at all and is applicable to the cases for which the standard path-integral method of Faddeev and Popov does not work.

Local OSp(4/2) supersymmetry and extended BRS transformations for gravity

When the gravitational field and its fictitious partners are grouped into anOSp(4/2) supermultiplet, an extended BRS invariance emerges. The action and BRS transformations (ordinary and dual) are written in supersymmetric form, and the extended gauge identities are deduced, in parallel with recent work on Yang-Mills theory.