Gauge mechanical model inN dimensions

The Christ-Lee mechanical model is generalized toN spatial dimensions. Its quantization as a gauge system is carried out, emphasizing the relationship between gauge-fixing and curvilinear coordinates in configuration space.     

Supersymmetric quantization of linearized supergravity

We give a manifestly supersymmetric quantization scheme for linearized supergravity, motivated by the desire to develop a background field method for the full non-linear theory. Supersymmetric gauge-fixing constraints are constructed and the corresponding ghost action is discussed. It is found that the Faddeev-Popov action itself possesses invariances, requiring “secondary” gauge fixing, which in turn leads to “secondary” ghost fields, the latter having normal statistics. The gauge-fixing constraints are used to construct gauge-fixing terms in the action, with a total of four gauge-fixing parameters. The superpropagators are found and may be greatly simplified by certain choices of these parameters.     

Generalized Slavnov-Taylor,BRST and covariance identities from the geometry of the gauge surface

We establish the equivalence between the extended BRST invariances, and the conventional Slavnov-Taylor transformations together with a new “dual” analogue. However, the latter (a non-local gauge transformation, generating anA-dependent translation of the gauge-fixing surface) isnot an invariance of the Faddeev-Popov determinant, contrary to the published claim.     

How to fix the gauge in internal and external gauge theories

A recently presented method, that permits one to calculate gauge-fixing conditions from a given gauge-breaking term, is applied to internal as well as external Yang-Mills theories. As to the internal case, the known gauge-fixing conditions can easily be reproduced in a unified way. For the external case, i.e., the Poincaré gauge theory of gravitation, new gauge-fixing conditions are obtained, in particular the full nonlinear generalization of the Coulomb and axial gauge. They prove to be simultaneously valid for theories with or without torsion.     

Stochastic quantization and gauge-fixing of the linearized gravitational field

Due to the indefiniteness of the euclidean gravitational action the Parisi-Wu stochastic quantization scheme fails in the case of the gavitational field. Therefore we apply a recently proposed modification of stochastic quantization that works in Minkowski space and preserves all the advantages of the original Parisi-Wu method; in particular no gauge-fixing is required. Additionally stochastic gauge-fixing may be introduced and is also studied in detail. The graviton propagators obtained with and without stochastic gauge-fixing all exhibit a noncausal contribution, but apart from this effect the gauge-invariant quantities are the same as those of standard quantization.     

Gauge independence of the effective potential revisited

We apply the formalism of extended BRS symmetry to the investigation of the gauge dependence of the effective potential in a spontaneously symmetry broken gauge theory. This formalism, which includes a set of Grassmann parameters defined as the BRS variations of the gauge-fixing parameters, allows us to derive in a quick and unambiguous way the related Nielsen identities, which express the physical gauge independence, in a class of generalized 't Hooft gauges, of the effective potential. We show in particular that the validity of the Nielsen identities does not require any constraint on the gauge-fixing parameters, contrary to some claims found in the literature. We use the method of algebraic renormalization, which leads to results independent of the particular renormalization scheme used.     

Faddeev–Jackiw quantization of the higher derivative Chern–Simons gauge theory in the first-order formalism

We analyse the physical constraints of the higher derivative Chern–Simons gauge model by means of Faddeev–Jackiw symplectic approach in the first-order formalism. Within such framework, we systematically determine the zero-mode structure of the corresponding symplectic matrix. In addition, we calculate the Faddeev–Jackiw quantum brackets by choosing appropriate gauge-fixing conditions and evaluate the determinant of the non-singular symplectic matrix as well as the transition-amplitude. Finally, we present a detailed Hamiltonian analysis using Dirac–Bergmann algorithm method and show that the Dirac brackets coincide with the FJ brackets when all the second-class constraints are treated as zero equations.     

Gauge-fixing conditions and natural decompositions of Yang-Mills fields

A general method of constructing canonical decompositions of Yang-Mills fields is presented. The gauge dependence is concentrated to a minimal set of variables which parametrize the group space of the gauge group. The decompositions fall into different classes for different boundary conditions and are characterized by a specific connection to gauge-fixing conditions which are consistent with these boundary conditions. All classes connected to gauge-fixing conditions on the field strength E_aⁱ yield local decompositions generalizing a similar decomposition given by Goldstone and Jackiw for SU(2) Yang-Mills. All classes connected to gauge-fixing conditions. which also involve A_aⁱ yield non-local decompositions in space. Explicit canonical decompositions are given for those classes which are connected to the Coulomb and axial gauges as well as the gauges of the form B_a^b_kE_b^k = 0, where B_a^b_k are constants.     

A BRST gauge-fixing procedure for Yang–Mills theory on sphere

A gauge-fixing procedure for the Yang–Mills theory on an n  -dimensional sphere (or a hypersphere) is discussed in a systematic manner. We claim that Adler's gauge-fixing condition used in massless Euclidean QED on a hypersphere is not conventional because of the presence of an extra free index, and hence is unfavorable for the gauge-fixing procedure based on the BRST invariance principle (or simply BRST gauge-fixing procedure). Choosing a suitable gauge condition, which is proved to be equivalent to a generalization of Adler's condition, we apply the BRST gauge-fixing procedure to the Yang–Mills theory on a hypersphere to obtain consistent results. Field equations for the Yang–Mills field and associated fields are derived in manifestly O(n+1)$\mathrm{O}(n+1)$ covariant or invariant forms. In the large radius limit, these equations reproduce the corresponding field equations defined on the n-dimensional flat space.     

String field theory in the Siegel gauge

We specialize the gauge-fixing procedure for the Witten action of the open bosonic string, given in a preceding paper, choosing the Siegel gauge. We find that the BRST-invariant gauge-fixed action is the gauge invariant one with ghost number unrestricted plus a gauge-fixing term. The BRST invariance of the measure in the functional integral is briefly discussed. As a technical tool the Hodge dual of a string functional is defined.     

Conformal invariance in Weyl gravity

A conformal-invariant model of Weyl gravity, based on a nondecomposable representation of the conformal group, allows one to have a conformal-invariant propagator in an arbitrary gauge, as well as a conformal-invariant gauge-fixing term in the Lagrangian approach. It is shown that in the gauge-invariant sector this theory coincides with ordinary Weyl gravity (with conformal-noninvariant gauge fixing). The corresponding BRST transformations are found and are used for derivation of the Slavnov-Taylor identities.     

Infrared behavior and gauge artifacts in de Sitter spacetime: the photon field

We study the infrared (long-distance) behavior of the free photon field in de Sitter spacetime. Using a two-parameter family of gauge-fixing terms, we show that the infrared (IR) behavior of the two-point function is highly gauge-dependent and ranges from vanishing to growing. This situation is in disagreement with its counterpart in flat spacetime, where the two-point function vanishes in the IR region for any choice of the gauge-fixing parameters. A criterion to isolate the physical part of the two-point function is given and is shown to lead to a well-behaved two-point function in the IR region.     

Gauge and Mass Parameter Dependence¶of Renormalized Green's Functions

Using the framework of algebraic renormalization we discuss the dependence of the renormalization group flow on gauge-fixing and mass parameters. We demonstrate that the freedom of finite renormalizations can be used to remove this dependence from the coefficients of the renormalization group equation. Received: 18 May 2000 / Accepted: 28 May 2000     

The extended loop group: An infinite dimensional manifold associated with the loop space

A set of coordinates in the non-parametric loop-space is introduced. We show that these coordinates transform under infinite dimensional linear representations of the diffeomorphism group. An extension of the group of loops in terms of these objects is proposed. The enlarged group behaves locally as an infinite dimensional Lie group. Ordinary loops form a subgroup of this group. The algebraic properties of this new mathematical structure are analyzed in detail. Applications of the formalism to field theory, quantum gravity and knot theory are considered.     

Solving general gauge theories on inner product spaces

By means of a generalized quartet mechanism we show in a model independent way that a BRST quantization on an inner product space leads to physical states of the form ph〉 = exp [Q, ψ]ph〉₀ where Q is the nilpotent BRST operator, ψ a hermitian fermionic gauge-fixing operator, and ph〉_o BRST invariant states determined by a hermitian set of BRST doublets in involution. ph〉₀ does not belong to an inner product space although ph〉 does. Since the BRST quartets are split into two sets of hermitian BRST doublets there are two choices for ph〉₀ and the corresponding ψ. When applied to general, both irreducible and reducible, gauge theories of arbitrary rank within the BFV formulation we find that ph〉₀ are trivial BRST invariant states which only depend on the matter variables for one set of solutions, and for the other set ph〉₀ are solutions of a Dirac quantization. This generalizes previous Lie group solutions obtained by means of a bigrading.     

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.     

A stagnant gauge for QCD

It is shown that the gauge-fixing parameter α of standard covariant gauges may legitimately be replaced by an operator α(□). In particular, α may be chosen so that the gluon propagator has a “stagnant” tensor structure proportional to g_ην for all momenta. This choice of gauge simplifies explicit calculations and leads to renormalization group equations with no $\text{?}\text{?α}$ term.     

Finite BRST Mapping in Higher-Derivative Models

We continue the study of finite field-dependent BRST (FFBRST) symmetry in the quantum theory of gauge fields. An expression for the Jacobian of path integral measure is presented, depending on a finite field-dependent parameter, and the FFBRST symmetry is then applied to a number of well-established quantum gauge theories in a form which incudes higher-derivative terms. Specifically, we examine the corresponding versions of the Maxwell theory, non-Abelian vector field theory, and gravitation theory. We present a systematic mapping between different forms of gauge-fixing, including those with higher-derivative terms, for which these theories have better renormalization properties. In doing so, we also provide the independence of the S-matrix from a particular gauge-fixing with higher derivatives. Following this method, a higher-derivative quantum action can be constructed for any gauge theory in the FFBRST framework.     

Admissible coordinate systems for static,spherically symmetric stellar models

For static, spherically symmetric stellar models it is shown that imposing the condition that the determinant of the metrical coefficients takes on its flat space-time value everywhere is sufficient to ensure that the coordinates are admissible in the sense of Lichnerowicz. The general method of solution is illustrated by integrating the equations for a star of constant, uniform density.     

Renormalization of non-abelian gauge theories in curved space-time

We use indirect, renormalization group arguments to calculate the gravitational counterterms needed to renormalize an interacting non-abelian gauge theory in curved space-time. This method makes it straightforward to calculate terms in the trace anomaly which first appear at high order in the coupling constant, some of which would need a 4-loop calculation to find directly. The role of gauge invariance in the theory is considered, and we discuss briefly the effect of using coordinate-dependent gauge-fixing terms. We conclude by suggesting possible applications of this work to models of the very early universe.