Gauge Fixing Term Research Articles

A study of the path-integral quantization of Abelian gauge theories when no explicit gauge-fixing term is included in the bilinear part of the gauge-field action

The mathematical problem of inverting the operator [Formula: see text] as it arises in the path-integral quantization of an Abelian gauge theory, such as quantum electrodynamics, when no gauge-fixing Lagrangian field density is included, is studied in this article.Making use of the fact that the Schwinger source functions, which are introduced for the purpose of generating Green's functions, are free of divergence, a result that follows from the conversion of the exponentiated action into a Gaussian form, the apparently noninvertible partial differential equation, [Formula: see text], can, by the addition and subsequent subtraction of terms containing the divergence of the source function, be cast into a form that does possess a Green's function solution. The gauge-field propagator is the same as that obtained by the conventional technique, which involves gauge fixing when the gauge parameter, α, is set equal to one.Such an analysis suggests also that, provided the effect of fictitious particles that propagate only in closed loops are included for the study of Green's functions in non-Abelian gauge theories in Landau-type gauges, then, in quantizing either Abelian gauge theories or non-Abelian gauge theories in this generic kind of gauge, it is not necessary to add an explicit gauge-fixing term to the bilinear part of the gauge-field action.

Stochastic diagrams and Feynman diagrams

We study the relationship between ordinary perturbation theory and perturbation theory obtained from stochastic quantization. We give a simple proof that, except in gauge theories, the several stochastic diagrams of a given topology are together equivalent to the corresponding Feynman diagram. Our analysis is presented in Minkowski space, but most of it may readily be adapted to euclidean space. The field propagator may be a non-diagonal matrix, such as is the case in real-time thermal field theory. We present a new version of the Langevin equation which directly reproduces the usual axial-gauge perturbation theory. Otherwise, we find that for gauge theories the relationship between ordinary and stochastic perturbation theory is not simple, and we present a recursive method of reconstructing Feynman diagrams from stochastic diagrams, without the need explicitly to introduce ghost fields. We consider both the original Parisi-Wu version of the Langevin equation, and Zwanziger's modified version with its stochastic gauge-fixing term.

Improved mean field calculations for U(1) lattice gauge theory in four dimensions

We perform a fully consistent mean-field calculation up to 4th order for the U(1) lattice gauge theory in four dimensions by treating the quadratic part of the gauge fixing term as an independent variable.

Infrared divergences and a non-local gauge for superspace Yang-Mills theory

The usual superspace approach to supersymmetric gauge theories suffers from problems with infrared divergences which greatly complicate multiloop calculations. We eliminate these divergences by introducing a non-local gauge-fixing term. In the background field method this term leads to unusual quantum-background interactions. Functional methods are presented for dealing with these interactions. As an example we compute the two-loop Yang-Mills β-function using the background field method in superspace. We also show how a non-local gauge can be used in ordinary, non-supersymmetric Yang-Mills theory.

Extended BRS symmetry in quantum gravity

The superspace formulation for BRS and anti-BRS symmetry, which has been successfully applied to ordinary and supersymmetric Yang-Mills theories, is extended to the case of quantum gravity. The action, which is constructed in the same way as in the previous cases, yields in component notation the known expressions for the gauge-fixing and Faddeev-Popov ghost terms.

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.

On Gauge Fixing in Quantum Gravity

AbstractIn the functional integral formalism of quantum field theory a gauge breaking term is added to the classical action in order to parametrize the orbits of the gauge group. In quantum gravity this gauge breaking term fixes the orientation of the Riemannian metric with respect to a fixed reference metric. We explain the notion of gauge invariance this procedure is based on. A special gauge fixing term is considered, which together with the Einstein‐Hilbert action leads to Rosen's bimetric theory with arbitrary background. Discussing some consequences of this gauging for quantum gravity, it is especially shown that there are some difficulties if back reaction is regarded.

Conformal QED

A conformally invariant quantum electrodynamics is constructed. The setting is realistic space-time (rather than Euclidean), and a complete Gupta–Bleuler quantization scheme is carried out. Conformal invariance of the quantum field theory (as opposed to either classical field theory or to a theory defined by its Feynman rules) requires a richer Gupta–Bleuler structure than has been considered previously. Yet the essential features of this structure are preserved. The requirement that the wave equation be of second order fixes a unique action that already contains the gauge-fixing terms that are required in any complete quantum field theory. The ‘‘Lorentz condition’’ turns out to be the transversality condition yαaα(y)=0 (in the manifestly covariant six-dimensional notation); this condition has to be treated in the same way as the Lorentz condition ∂μAμ(x)=0 (four-dimensional notation), as a boundary condition on the physical states.

Quantizing fourth-order gravity theories: The functional integral

The first part of a detailed analysis of the one-loop divergences of fourth-order gravity theories is presented. The functional-integral structure is determined and the three kinds of ghosts which appear are discussed. Field operators and gauge-fixing terms are derived and are found to be far more complicated than those of standard second-order theory, whose details are presented for comparison to the fourth-order case. We outline how, in future papers, we will approach the computation of the one-loop counterterms using pseudo-differential operator techniques and related methods. We also discuss briefly what other workers have done and are doing in studies of the problems inherent in fourth-order gravity and its extensions. Finally, we emphasize the need to investigate the full non-linear theory with coordinate-space methods and mention some of the problems brought about by linearization. Details of calculations and the multilevel-mass-spectrum theory are presented in several appendices.

Explicit two-loop calculation of decoupling in unbroken gauge theories with fermions

Unbroken gauge theories containing light as well as heavy fermions are considered in the limit of the mass of the heavy fermions tending to infinity. The effective coupling constant of the decoupled low-energy theory thus obtained, has been calculated up to two-loop level using the light particle irreducible vertex function and the background field formalism. In addition, to simplify the computation, a background field gauge-fixing term has been used, because in such a gauge the effective coupling constant can be calculated from a two-point function only. Our analysis reveals that in the non-abelian theory, the simple algorithm proposed by Ovrut and Schnitzer for computing the effective coupling constant up to the two-loop level is valid only in the α = 0 or α = −3 background field gauge. A general procedure correct for all values of α is described.

Quantization of gauge fields in gauges involving "extra ghosts"

Nielsen's recently proposed quantization rules, with gauge-fixing terms of the form $\frac{{F}_{a}{\ensuremath{\gamma}}^{\mathrm{ab}}{F}_{b}}{2\ensuremath{\alpha}}$, where ${\ensuremath{\gamma}}^{\mathrm{ab}}$ is field dependent, are shown to lead to an $S$ matrix which is both unitary and gauge independent. The demonstration relies on the canonical form of the path integral.

Superfield formalism for gauge theories with ghosts and gauge fixing terms from dynamics on superspace

The geometrical condition of soul-flatness needed in the superfield approach to quantized gauge theories and extended BRS transformations, is derived from a lagrangian theory on a superspace S 4.2. Invariance of the superlagrangian and the lagrangian of quantized gauge theories under Sp(2, R) and hermitian conjugation is discussed in different gauges.

Extended BRS symmetry in supersymmetric Yang-Mills theory

The superspace formulation of pure Yang-Mills theory is extended to N=1 supersymmetric Yang-Mills. BRS and anti-BRS symmetry are incorporated in a natural way, and a compact expression for the lagrangian, including the gauge-fixing and Faddeev-Popov ghost terms, is easily derived.

Loop expansion in massless three-dimensional QED

It is shown how the loop expansion in massless three-dimensional QED can be made finite, up to three loops, by absorbing the infrared divergences in a gauge-fixing term. The same method removes leading and first subleading singularities to all orders of perturbation theory, and all singularities of the fermion self-energy to four loops.

Local BRST symmetry and the geometry of gauge-fixing

Extended BRST transformations are constructed for arbitrary gauge theories by consideration of their algebra alone. Dynamics are introduced by writing the most general lagrangian which is globally extended-BRST invariant and provides a gauge-fixing term of the fork 1 2 F αγ αβF β . It is shown that all F α ≡ γ αβ F β must be variational derivatives of one universal scalar function F proportional to the squares of the classical gauge fields, while γ αβ, in principle still arbitrary, is interpreted as the generalized Killing form. A superspace formulation of Yang-Mills theory is used to promote the BRST symmetry to a local symmetry.

Quantised gauge theory, dimensional reduction and OSp(4+NM/2) supersymmetry

The Kelin-Kaluza version of the combined Einstein-Yang-Mills system is based on a larger-dimension manifold, admitting an O(4+N) symmetry, which undergoes compactification. The theory provides a unified treatment of all gauge fields and its action consists of a single invariant Lagrangian and, separately, a single gauge-fixing term. The authors show that both Lagrangians can be combined within a larger framework, admitting an OSp(4+N/2) supersymmetry, which undergoes appropriate dimensional reduction; here the extra graded dimensions are associated with the fictitious fields needed for consistent quantisation and unitarity.

Generalized BRST quantization of gauge field theory

Gauge fixing terms of the form 1 2 F αγ αβF β are considered, where γ αβ is field-dependent. It is shown in general that Nielsen's recently proposed quantization rules are consistent with the usual Faddeev-Popov rules and that the S-matrix is both unitary and independent of the choice of F α and γ αβ . A specific example in scalar chromodynamics is given and one-loop unitarity is demonstrated explicitly.

Symmetry restoration and the background field method in gauge theories

Spontaneously broken gauge theories in a constant external electromagnetic field are shown to exhibit a first-order phase transition to a restored symmetry phase when the external field exceeds a certain critical value. The effects of fields characterized by various values of the two Lorentz invariants F 1 = 1 2 (B 2 − E 2) and F 2 = E · B are discussed. In a simple SU(2) model the critical field strength is found to be g R 2( F 1) crit = 0.057 m w 4, m w being the vector boson mass. A number of theoretical developments in the background field formalism are presented. A new gauge-fixing term, the background field R gauge, is introduced. The configuration space heat kernel method for evaluating functional determinants, extended to allow the use of dimensional regularization, is employed, and it is shown how to perform background field calculations in a gauge specified by an arbitrary parameter α. Further applications of these methods are discussed.

Covariant gauge formalism for photons in friedmann space-time

When quantization is performed, the photon Lagrangian needs a gauge-fixing term which breaks conformal (Weyl) invariance. This is shown to imply in a Friedmann (conformally flat) space-time the creation of unphysical, unobservable particles lying in zero-norm states; no transverse photons are created. The case of a massive vector field for which also physical particles are produced is finally discussed and in both cases we show the structure of the space of states in the Kugo-Ojima formalism.

Superfield formulation of extended BRS symmetry

In the light of a recently discovered new BRS-like symmetry in gauge theories, we reformulate a superfield treatment of lagrangian gauge theories supplemented by gauge-fixing and Faddeev-Popov terms.