Gauge Fixing Condition Research Articles

FADDEEV–JACKIW FORMALISM FOR CONSTRAINED SYSTEMS WITH GRASSMANN DYNAMICAL FIELD VARIABLES

The Faddeev–Jackiw canonical quantization formalism for constrained systems with Grassmann dynamical variables within the framework of the field theory is reviewed. First, by means of a iterative process, the symplectic supermatrix is constructed and their associated constraints are found. Next, by taking into account the phase space of the system, the constraint structure is considered. It is found that, if there are no auxiliary dynamical field variables, the supermatrix whose elements are the Bose–Fermi brackets between the constraints associated with the independent dynamical field variables coincides with the symplectic supermatrix corresponding to these independent variables. An alternative procedure to obtain the first-class constraints is given. It is shown that for systems with gauge symmetries, by means of suitable gauge-fixing conditions, a nonsingular final symplectic supermatrix can be found. Then, two possible ways of calculating the Faddeev–Jackiw brackets are pointed out. The relation between the Faddeev–Jackiw and Dirac brackets is discussed. Throughout the previous developments, the Faddeev–Jackiw and Dirac algorithms are compared. Finally, the Faddeev–Jackiw canonical quantization method is applied to a simple model and the obtained results are compared with the ones corresponding to the use of the Dirac procedure on this model.

SU(3) gauge family symmetry and prediction for the lepton-flavor mixing and neutrino masses with maximal spontaneous CP violation

A model for the lepton-flavor mixing and CP violation is proposed based on the SUF(3) gauge family symmetry and the Majorana feature of neutrinos. A consistent prediction for the lepton-flavor mixing and masses is shown to be resulted from the appropriate vacuum structure of SUF(3) gauge symmetry breaking. By choosing the SUF(3) gauge fixing condition to possess a residual Z2 symmetry and requiring the vacuum structure of spontaneous symmetry breaking to have approximate global U(1) family symmetries, we obtain naturally the tri-bimaximal mixing matrix and largely degenerate neutrino masses in the neutrino sector and the small mixing matrix in the charged-lepton sector. With a simple ansatz that all the smallness due to the approximate global U(1) family symmetries is characterized by a single Wolfenstein parameter λ≃0.22, and the charged-lepton mixing matrix has a similar hierarchy structure as the CKM quark mixing matrix, we arrive at a consistent prediction for the MNSP lepton-flavor mixing with a maximal spontaneous CP violation: δ=π/2, sin2θ13≃12λ2≃0.024 (sin22θ13≃0.094), sin2θ12≃13(1−2λ3)≃0.326 and sin2θ23≃12(1−λ2)≃0.48, which agree well with the current experimental data. The CP-violating Jarlskog-invariant is obtained to be JCP≃16λ(1−λ2/2−λ3)sinδ≃0.035, which is detectable in next generation neutrino experiment. The small masses of the neutrinos and charged leptons are simply attributed to the standard seesaw mechanism. The largely degenerate neutrino masses with the normal hierarchy and inverse hierarchy are discussed and found to be at the order mνi≃O(λ2)≃0.04–0.06 eV with a total mass ∑mν∼0.15 eV, which is testable in future precision astrophysics and cosmology.

A construction of Spin(7)-instantons

Joyce constructed examples of compact eight-manifolds with holonomy Spin(7), starting with a Calabi-Yau four-orbifold with isolated singular points of a special kind. That construction can be seen as the gluing of ALE Spin(7)-manifolds to each singular point of the Calabi-Yau four-orbifold divided by an anti-holomorphic involution fixing only the singular points. On the other hand, there are higher-dimensional analogues of anti-self-dual instantons in four dimensions on Spin(7)-manifolds, which are called Spin(7)-instantons. They are minimizers of the Yang-Mills action, and the Spin(7)-instanton equation together with a gauge fixing condition forms an elliptic system. In this article, we construct Spin(7)-instantons on the examples of compact Spin(7)-manifolds above, starting with Hermitian-Einstein connections on the Calabi-Yau four-orbifolds and ALE spaces. Under some assumptions on the Hermitian-Einstein connections, we glue them together to obtain Spin(7)-instantons on the compact Spin(7)-manifolds. We also give a simple example of our construction.

Validity of gauge-fixing conditions and the structure of propagators in open superstring field theory

We make a detailed analysis on validity of gauge-fixing conditions and the structure of propagators in the Wess-Zumino-Witten-type open superstring field theory. First, we generalize the gauge-fixing conditions considered in JHEP 03 (2012) 030 [arXiv:1201.1761] by the present author et al., and propose a large class of conditions characterized by zero modes of world-sheet oscillators. Then we demonstrate its validity: we prove that gauge degrees of freedom allow us to impose the conditions, and that the conditions fix the gauges completely. Moreover, we elucidate how the information about the gauge choices is reflected in the structure of propagators. The results can be readily extended to the case in which gauge-fixing conditions involve linear combinations of the world-sheet oscillators, including nonzero modes. We investigate also such extended gauges, which are the counterpart of linear $b$-gauges in bosonic string field theory, and obtain the corresponding propagators.

Gauge Fixing of Modified Cubic Open Superstring Field Theory

The gauge-fixing problem in the modified cubic open superstring field theory is discussed in detail for both the Ramond and Neveu-Schwarz (NS) sectors in the Batalin-Vilkovisky (BV) framework. We prove for the first time that the same form of action as the classical gauge-invariant one with the ghost-number constraint on the string field relaxed, gives the master action satisfying the BV master equation. This is achieved by identifying independent component fields by analyzing the kernel structure of the inverse picture-changing operator. The explicit gauge-fixing conditions for the component fields are discussed. In a kind of b0 = 0 gauge, we explicitly obtain an NS propagator that has poles at zeros of the Virasoro operator L0.

Large adiabatic scalar perturbations in a regular bouncing universe

It has been shown that a contracting universe with a dust-like ($w \approx 0$) fluid may provide an almost scale invariant spectrum for the gravitational scalar perturbations. As the universe contracts, the amplitude of such perturbations are amplified. The gauge invariant variable $\Phi$ develops a growing mode which becomes much larger than the constant one around the bounce phase. The constant mode has its amplitude fixed by Cosmic Background Explorer (COBE) normalization, thus the amplitude of the growing mode can become much larger than 1. In this paper, we first show that this is a general feature of bouncing models, since we expect that general relativity should be valid in all scales away from the bounce. However, in the Newtonian gauge, the variable $\Phi$ gives the value of the metric perturbation $\phi$, raising doubts on the validity of the linear perturbative regime at the bounce. In order to address this issue, we obtain a set of necessary conditions for the perturbative series to be valid along the whole history of the model, and we show that there is a gauge in which all these conditions are satisfied, for a set of models, if the constant mode is fixed by COBE normalization. As a by-product of this analysis, we point out that there are sets of solutions for the perturbation variables where some gauge-fixing conditions are not well defined, turning these gauges prohibited for those solutions.

Quantum treatment of the Weyl vector meson

The Weyl meson arises in theories with local scale invariance. It acts as a candidate for dark matter in a generalized Standard Model with local scale invariance. The Higgs particle is absent from the physical spectrum in this theory. We consider the quantization of this theory in detail, imposing suitable gauge fixing conditions. We also consider a further generalization of this model which includes an additional real scalar field. In this theory a Higgs like particle remains in the particle spectrum. Since this particle couples to the Weyl meson, it can lead to interesting phenomenology involving this vector field in particle colliders.

Nonquadratic gauge fixing and ghosts for gauge theories on the hypersphere

It has been suggested that using a gauge fixing Lagrangian that is not quadratic in a gauge fixing condition is most appropriate for gauge theories formulated on a hypersphere. We reexamine the appropriate ghost action that is to be associated with gauge fixing, applying a technique that has been used for ensuring that the propagator for a massless spin-two field is transverse and traceless. It is shown that this nonquadratic gauge fixing Lagrangian leads to two pair of complex Fermionic ghosts and two Bosonic real ghosts.

Gauge fixing in (2+1)-gravity: Dirac bracket and spacetime geometry

We consider (2+1)-gravity with a vanishing cosmological constant as a constrained dynamical system. By applying Dirac's gauge fixing procedure, we implement the constraints and determine the Dirac bracket on the gauge-invariant phase space. The chosen gauge fixing conditions have a natural physical interpretation and specify an observer in the spacetime. We derive explicit expressions for the resulting Dirac brackets and discuss their geometrical interpretation. In particular, we show that specifying an observer with respect to two point particles gives rise to conical spacetimes, whose deficit angle and time shift are determined, respectively, by the relative velocity and minimal distance of the two particles.

Canonical path-integral measures for Holst and Plebanski gravity: II. Gauge invariance and physical inner product

This paper serves as a continuation for the discussion in Engle et al (2010, Class. Quantum Grav. 27 245014). We analyze the invariance properties of the gravity path-integral measure derived from canonical framework and discuss which path-integral formula may be employed in the concrete computation e.g. constructing a spin-foam model, so that the final model can be interpreted as a physical inner product in the canonical theory. This paper is divided into two parts, the first part is concerning the gauge invariance of the canonical path-integral measure for gravity from the reduced phase space quantization. We show that the path-integral measure is invariant under all the gauge transformations generated by all the constraints. These gauge transformations are the local symmetries of the gravity action, which is implemented without anomaly at the quantum level by the invariant path-integral measure. However, these gauge transformations coincide with the spacetime diffeomorphisms only when the equations of motion are imposed. But the path-integral measure is not invariant under spacetime diffeomorphisms, i.e. the local symmetry of spacetime diffeomorphisms become anomalous in the reduced phase space path-integral quantization. In the second part, we present a path-integral formula, which formally solves all the quantum constraint equations of gravity, and further results in a rigging map in the sense of refined algebraic quantization (RAQ). Then we give a formal path-integral expression of the physical inner product in loop quantum gravity (LQG). This path-integral expression is simpler than the one from reduced phase space quantization, since all the gauge-fixing conditions are removed except the time gauge. The resulting path-integral measure is different from the product Lebesgue measure up to a local measure factor containing both the spacetime volume element and the spatial volume element. This formal path-integral expression of the physical inner product can be a starting point for constructing a spin-foam model.

Global symmetries and renormalizability of Lee-Wick theories

In this paper we discuss the global symmetries and the renormalizability of Lee-Wick (LW) scalar QED. In particular, in the ``auxiliary-field'' formalism we identify softly broken $SO(1,1)$ global symmetries of the theory. We introduce $SO(1,1)$ invariant gauge-fixing conditions that allow us to show in the auxiliary-field formalism directly that the number of superficially divergent amplitudes in a LW Abelian gauge theory is finite. To illustrate the renormalizability of the theory, we explicitly carry out the one-loop renormalization program in LW scalar QED and demonstrate how the counterterms required are constrained by the joint conditions of gauge and $SO(1,1)$ invariance. We also compute the one-loop beta functions in LW scalar QED and contrast them with those of ordinary scalar QED.

Light-Front Hamiltonian, Path Integral and BRST Formulations of the Chern-Simons Theory under Appropriate Gauge-Fixing

The Chern-Simons theory in two-space one-time dimensions is quantized on the light-front under appropriate gauge-fixing conditions using the Hamiltonian, path integral and BRST formulations.

Decoupling of unphysical states in the minimal pure spinor formalism I

This is the first of a series of two papers where decoupling of unphysical states in the minimal pure spinor formalism is investigated. The multi-loop amplitude prescription for the minimal pure spinor superstring formulated in hep-th/0406055 involves the insertion of picture changing operators in the path integral. These operators are BRST closed in a distributional sense and depend on a number of constant tensors. One can trace the origin of these insertions to gauge fixing, so the amplitudes are formally independent of the constant tensors. We show however by explicit tree-level and one-loop computations that the picture changing operators are not BRST closed inside correlators and the amplitudes do depend on these constant tensors. This is due to the fact that the gauge fixing condition implicit in the existing minimal amplitude prescription is singular and this can lead to Lorentz violation and non-decoupling of BRST exact states. As discussed in hep-th/0406055, a manifestly Lorentz invariant prescription can be obtained by integrating over the constant tensors and in the sequel to this paper, it is shown that when one includes these integrations unphysical states do decouple to all orders despite the fact that the PCO's are not BRST closed inside correlators.

Decoupling of unphysical states in the minimal pure spinor formalism II

This is the first of a series of two papers where decoupling of unphysical states in the minimal pure spinor formalism is investigated. The multi-loop amplitude prescription for the minimal pure spinor superstring formulated in hep-th/0406055 involves the insertion of picture changing operators in the path integral. These operators are BRST closed in a distributional sense and depend on a number of constant tensors. One can trace the origin of these insertions to gauge fixing, so the amplitudes are formally independent of the constant tensors. We show however by explicit tree-level and one-loop computations that the picture changing operators are not BRST closed inside correlators and the amplitudes do depend on these constant tensors. This is due to the fact that the gauge fixing condition implicit in the existing minimal amplitude prescription is singular and this can lead to Lorentz violation and non-decoupling of BRST exact states. As discussed in hep-th/0406055, a manifestly Lorentz invariant prescription can be obtained by integrating over the constant tensors and in the sequel to this paper, it is shown that when one includes these integrations unphysical states do decouple to all orders despite the fact that the PCO's are not BRST closed inside correlators.

CONFORMAL FIELD THEORY ON R × S3 FROM QUANTIZED GRAVITY

Conformal algebra on R × S3 derived from quantized gravitational fields is examined. The model we study is a renormalizable quantum theory of gravity in four dimensions described by a combined system of the Weyl action for the traceless tensor mode and the induced Wess–Zumino action managing nonperturbative dynamics of the conformal factor in the metric field. It is shown that the residual diffeomorphism invariance in the radiation+ gauge is equal to the conformal symmetry, and the conformal transformation preserving the gauge-fixing condition that forms a closed algebra quantum mechanically is given by a combination of naive conformal transformation and a certain field-dependent gauge transformation. The unitarity issue of gravity is discussed in the context of conformal field theory. We construct physical states by solving the conformal invariance condition and calculate their scaling dimensions. It is shown that the conformal symmetry mixes the positive-metric and the negative-metric modes and thus the negative-metric mode does not appear independently as a gauge invariant state at all.

Functional integration and gauge ambiguities in generalized abelian gauge theories

We consider the covariant quantization of generalized abelian gauge theories on a closed and compact n -dimensional manifold whose space of gauge invariant fields is the abelian group of Cheeger–Simons differential characters. The space of gauge fields is shown to be a non-trivial bundle over the orbits of the subgroup of smooth Cheeger–Simons differential characters. Furthermore each orbit itself has the structure of a bundle over a multi-dimensional torus. As a consequence there is a topological obstruction to the existence of a global gauge fixing condition. A functional integral measure is proposed on the space of gauge fields which takes this problem into account and provides a regularization of the gauge degrees of freedom. For the generalized p -form Maxwell theory closed expressions for all physical observables are obtained. The Green’s functions are shown to be affected by the non-trivial bundle structure. Finally the vacuum expectation values of circle-valued homomorphisms, including the Wilson operator for singular p -cycles of the manifold, are computed and selection rules are derived.

ON THE 1-LOOP CALCULATIONS OF SOFTLY BROKEN FERMION-TORSION THEORY IN CURVED SPACE USING THE STÜCKELBERG PROCEDURE

The soft breaking of gauge or other symmetries is the typical Quantum Field Theory phenomenon. In many cases one can apply the Stückelberg procedure, which means introducing some additional field (or fields) and restore the gauge symmetry. The original softly broken theory corresponds to a particular choice of the gauge fixing condition. In this paper we use this scheme for performing quantum calculations for fermion-torsion theory, softly broken by the torsion mass in arbitrary curved spacetime.

Numerical Evaluation of Gauge Invariants for a-Gauge Solutions in Open String Field Theory

We evaluate gauge invariants (vacuum energy and gauge invariant overlap) for numerical classical solutions in cubic open string field theory under Asano-Kato's $a$-gauge fixing condition. We propose an efficient iterative procedure for solving the equations of motion so that the BRST invariance of the solutions is numerically ensured. The resulting gauge invariants are numerically stable and almost equal to those of Schnabl's tachyon vacuum solution in the well-defined region of a gauge parameter. These results provide further evidence that the numerical and analytical solutions are gauge equivalent.

Hamiltonian, path integral and BRST formulations of the Chern–Simons–Higgs theory under appropriate gauge fixing

The Hamiltonian, path integral and Becchi–Rouet–Stora and Tyutin (BRST) formulations of the Chern–Simons–Higgs theory in two-space one-time dimensions are investigated under appropriate gauge-fixing conditions.

Desingularization of G2 manifolds with isolated conical singularities

We present a method to desingularize a compact G_2 manifold with isolated conical singularities by cutting out a neighbourhood of each singular point and glueing in an asymptotically conical G_2 manifold. Controlling the error on the overlap glueing region enables us to use a result of Joyce to conclude that the resulting compact smooth 7-manifold admits a torsion-free G_2 structure, with full G_2 holonomy. There are topological obstructions for this procedure to work, which arise from the degree 3 and degree 4 cohomology of the asymptotically conical G_2 manifolds which are glued in at each conical singularity. When a certain necessary topological condition on the manifold with isolated conical singularities is satisfied, we can introduce correction terms to the glueing procedure to ensure that it still works. In the case of degree 4 obstructions, these correction terms are trivial to construct, but in the case of degree 3 obstructions we need to solve an elliptic equation on a non-compact manifold. For this we use the Lockhart-McOwen theory of weighted Sobolev spaces on manifolds with ends. This theory is also used to obtain a good asymptotic expansion of the G_2 structure on an asymptotically conical G_2 manifold under an appropriate gauge-fixing condition, which is required to make the glueing procedure work.