Gauge Condition Research Articles

A consistent theory of D → 4 Einstein-Gauss-Bonnet gravity

We investigate the D→4 limit of the D-dimensional Einstein-Gauss-Bonnet gravity, where the limit is taken with α˜=(D−4)α kept fixed and α is the original Gauss-Bonnet coupling. Using the ADM decomposition in D dimensions, we clarify that the limit is rather subtle and ambiguous (if not ill-defined) and depends on the way how to regularize the Hamiltonian or/and the equations of motion. To find a consistent theory in 4 dimensions that is different from general relativity, the regularization needs to either break (a part of) the diffeomorphism invariance or lead to an extra degree of freedom, in agreement with the Lovelock theorem. We then propose a consistent theory of D→4 Einstein-Gauss-Bonnet gravity with two dynamical degrees of freedom by breaking the temporal diffeomorphism invariance and argue that, under a number of reasonable assumptions, the theory is unique up to a choice of a constraint that stems from a temporal gauge condition.

Second-order effective energy-momentum tensor of gravitational scalar perturbations with perfect fluid

We investigate the second-order gravitational scalar perturbations for a barotropic fluid. We derive the effective energy-momentum tensor described by the quadratic terms of the gravitational and the matter perturbations. We show that the second-order effective energy-momentum tensor is gauge dependent. We impose three gauge conditions (longitudinal, spatially-flat, and comoving gauges) for dust and radiation. The resulting energy-momentum tensor is described only by a gauge invariant variable, but the functional form depends on the gauge choice. In the matter-dominated epoch with dust-like fluid background, the second-order effective energy density and pressure of the perturbations evolve as 1/a^2 in all three gauge choices, like the curvature density of the Universe, but they do not provide the correct equation of state. The value of this parameter depends also on the gauge choice. In the radiation-dominated epoch, the perturbations in the short-wave limit behave in the same way as the radiation-like fluid in the longitudinal and the spatially-flat gauges. However, they behave in a different way in the comoving gauge. As a whole, we conclude that the second-order effective energy-momentum tensor of the scalar perturbation is strictly gauge dependent.

Restricted Weyl symmetry

We discuss the physics of a restricted Weyl symmetry in a curved space-time where a gauge parameter $\Omega(x)$ of Weyl transformation satisfies a constraint $\Box \Omega = 0$. First, we present a model of QED where we have a restricted gauge symmetry in the sense that a $U(1)$ gauge parameter $\theta(x)$ obeys a similar constraint $\Box \theta = 0$ in a flat Minkowski space-time. Next, it is precisely shown that a global scale symmetry must be spontaneously broken at the quantum level. Finally, we discuss the origin of the restricted Weyl symmetry and show that its symmetry can be derived from a full Weyl symmetry by taking a gauge condition $R = 0$ in the BRST formalism.

Magnetohydrodynamics with Post-Newtonian Corrections

Abstract Using the fully nonlinear and exact perturbation formulation with magnetohydrodynamics (MHD) in the Minkowski background, we derive first-order, post-Newtonian (1PN) equations without imposing the slicing (temporal gauge) condition. The 1PN-MHD formulation is complementary to our recently presented fully relativistic MHD combined with 0PN gravity available only in the maximal slicing. We present the 1PN-MHD equations in two gauge conditions previously used in the literature and provide gauge transformation relations between different gauges. We derive the PN effects on MHD waves in a static homogeneous medium.

Formal Global AKSZ Gauge Observables and Generalized Wilson Surfaces

We consider a construction of observables by using methods of supersymmetric field theories. In particular, we give an extension of AKSZ-type observables constructed in Mnev (Lett Math Phys 105:1735–1783, 2015) using the Batalin–Vilkovisky structure of AKSZ theories to a formal global version with methods of formal geometry. We will consider the case where the AKSZ theory is “split” which will give an explicit construction for formal vector fields on base and fiber within the formal global action. Moreover, we consider the example of formal global generalized Wilson surface observables whose expectation values are invariants of higher-dimensional knots by using BF field theory. These constructions give rise to interesting global gauge conditions such as the differential quantum master equation and further extensions.

Inside the final black hole: puncture and trapped surface dynamics

A popular approach in numerical simulations of black hole binaries is to model black holes as punctures in the fabric of spacetime. The location and the properties of the black hole punctures are tracked with apparent horizons, namely outermost marginally outer trapped surfaces (MOTSs). As the holes approach each other, a common apparent horizon suddenly appears, engulfing the two black holes and signaling the merger. The evolution of common apparent horizons and their connection with gravitational wave emission have been studied in detail with the framework of dynamical horizons. We present a study of the dynamics of the MOTSs and their punctures in the interior of the final black hole. The study focuses on head-on mergers for various initial separations and mass ratios. We find that MOTSs intersect for most of the parameter space. We show that for those situations in which they do not, it is because of the singularity avoidance property of the moving puncture gauge condition used in the study. Although we are unable to carry out evolutions that last long enough to show the ultimate fate of the punctures, our results suggest that MOTSs always intersect and that at late times their overlap is only partial. As a consequence, the punctures inside the MOTSs, although close enough to each other to act effectively as a single puncture, do not merge.

Unimodular gauge in perturbative gravity and supergravity

This paper explains the Unimodular gauge fixing of gravity and supergravity in the framework of a perturbative BRST construction. The unphysical sector contains additional BRST-exact quartets to suppress possible ambiguities and impose both the Unimodular gauge fixing condition on the metric and a gauge condition for the reparametrization symmetry of the unimodular part of the metric. The Unimodular gauge choice of the metric must be completed by a γ-Traceless gauge condition for the Rarita–Schwinger field in the case of supergravity. This gives an interesting new class of gauges for gravity and supergravity.

Perturbations and linearization stability of closed Friedmann universes

We consider perturbations of closed Friedmann universes. Perturbation modes of two lowest wavenumbers ($L=0$ and $1$) are generally known to be fictitious, but here we show that both are physical. The issue is more subtle in Einstein static universes where closed background space has a time-like Killing vector with the consequent occurrence of linearization instability. Proper solutions of the linearized equation need to satisfy the Taub constraint on a quadratic combination of first-order variables. We evaluate the Taub constraint in the two available fundamental gauge conditions, and show that in both gauges the $L\geq 1$ modes should accompany the $L=0$ (homogeneous) mode for vanishing sound speed, $c_{s}$. For $c_{s}^{2}>1/5$ (a scalar field supported Einstein static model belongs to this case with $c_s^2 = 1$), the $L\geq 2$ modes are known to be stable. In order to have a stable Einstein static evolutionary stage in the early universe, before inflation and without singularity, although the Taub constraint does not forbid it, we need to find a mechanism to suppress the unstable $L=0$ and $L=1$ modes.

Hypersymmetry field rotation angle

The paper determines a limit energy under which hypersymmetry (HySy) is broken. According to gauge theories, interaction mediating spin-0 bosons must be massless. The theory of HySy predicted massive intermediate bosons. Hypersymmetry field rotation, described in this paper, justifies the mass of the HySy mediating boson. The mass of intermediate bosons must arise from dynamical spontaneous breaking of the group of HySy. HySy rotation is performed in the velocity-dependent D field. The derived rotation of the field is defined by the spontaneous symmetry breaking and precession of the velocity v around its third projection in the D field (that produced the mass of the field's bosons). The latter represents the real- and effective velocities of a boson-emitting particle in the direction towards a target particle. The mass of the discussed (fictitious) Goldstone bosons can be removed by the unitarity gauge condition through Higgs (BEH) mechanism. According to the simultaneous presence of a Standard Model (SM) interaction's symmetry group and the (beyond SM) HySy group, their bosons should be transformed together. Spontaneous breakdown of HySy may allow performing a transformation that does not influence the SM physical state of the investigated system. The paper describes a field transformation that eliminates the mass of the intermediate bosons, rotates the SM- and HySy bosons’ masses together while leaving the SM bosons intact. The result is an angle that characterises the HySy by a precession mechanism of the velocity that generates the field. In contrast to the known SM intermediate bosons, the HySy intermediate bosons have no fixed mass. The mass of the HySy intermediate bosons (that appear as quanta of a velocity-dependent gauge field D) depends on the relative velocity of the particles whose interaction they mediate. So, the derived precession angle is a function of that velocity.

Infinitesimal Gribov copies in gauge-fixed topological Yang-Mills theories

We study the Gribov problem in four-dimensional topological Yang-Mills theories following the Baulieu-Singer approach in the (anti-)self-dual Landau gauges. This is a gauge-fixed approach that allows to recover the topological spectrum, as first constructed by Witten, by means of an equivariant (or constrained) BRST cohomology. As standard gauge-fixed Yang-Mills theories suffer from the gauge copy (Gribov) ambiguity, one might wonder if and how this has repercussions for this analysis. The resolution of the small (infinitesimal) gauge copies, in general, affects the dynamics of the underlying theory. In particular, treating the Gribov problem for the standard Landau gauge condition in non-topological Yang-Mills theories strongly affects the dynamics of the theory in the infrared. In the current paper, although the theory is investigated with the same gauge condition, the effects of the copies turn out to be completely different. In other words: in both cases, the copies are there, but the effects are very different. As suggested by the tree-level exactness of the topological model in this gauge choice, the Gribov copies are shown to be inoffensive at the quantum level. To be more precise, following Gribov, we discuss the path integral restriction to the Gribov horizon. The associated gap equation, which fixes the so-called Gribov parameter, is however shown to only possess a trivial solution, making the restriction obsolete. We relate this to the absence of radiative corrections in both gauge and ghost sectors. We give further evidence by employing the renormalization group which shows that, for this kind of topological model, the gap equation indeed forbids the introduction of a massive Gribov parameter.

A note on harmonic gauge(s) in massive gravity

We consider the harmonic gauge condition in linearized gravity, seen as a gauge theory for a symmetric tensor field. Once the harmonic gauge condition is implemented, as customary, according to the Faddeev-Popov procedure, the gauge fixed action still depends on one gauge parameter. Consequently, the harmonic gauge appears to be a class of conditions, rather than a particular one. This allows to give a physical motivation for the covariant harmonic gauge(s), which emerges when the gravitational perturbation is given a mass term. In fact, for a particular choice of harmonic gauge, we find a theory of linearized massive gravity displaying five degrees of freedom, as it should, and which is not affected by the vDVZ discontinuity, differently from what happens in the standard Fierz-Pauli theory.

Gauge-invariant spherical linear perturbations of wormholes in Einstein gravity minimally coupled to a self-interacting phantom scalar field

Recently, there has been quite a lot of interest in static, spherical wormhole spacetimes and the question of their stability with respect to time-dependent perturbations. The consideration of linearized perturbations usually leads to a master wave equation with effective potential which can then be analyzed using standard tools from quantum mechanics. However, in the wormhole case, particular care must be taken with the gauge conditions when formulating the master equation. A poor coordinate choice, based for example on fixing the areal radial coordinate, may lead to singularities at the throat which complicate the stability analysis or might even lead to erroneous conclusions regarding the stability of the underlying wormhole configuration. In this work, we present a general method for deriving a gauge-invariant wave system of linearized perturbation equations in the spherically symmetric case, assuming that the matter supporting the wormhole is a phantom scalar field, that is, a self-interacting scalar field whose kinetic energy has the reversed sign. We show how this system can be decoupled and reduced to a single master wave equation with a regular potential, with no intermediate steps involving singularities at the throat. Two applications of our formalism are given. First, we rederive the master equation for the linearly perturbed Ellis-Bronnikov wormhole using our new, singularity-free method. Second, we derive the master equation describing the linear perturbations of a certain Anti de Sitter wormhole, provide a detailed analysis of the spectral properties of the underlying operator and prove that this wormhole is linearly unstable. In the final part of the paper, we consider a wormhole with de Sitter-type ends, whose spacetime presents horizons and admits a nonstatic extension beyond them; for this system we derive partial results of linear instability.

Chern–Simons–Schrödinger theory on a one-dimensional lattice

We propose a gauge-invariant system of the Chern-Simons-Schrodinger type on a one-dimensional lattice. By using the spatial gauge condition, we prove local and global well-posedness of the initial-value problem in the space of square summable sequences for the scalar field. We also study the existence region of the stationary bound states, which depends on the lattice spacing and the nonlinearity power. A major difficulty in the existence problem is related to the lack of variational formulation of the stationary equations. Our approach is based on the implicit function theorem in the anti-continuum limit and the solvability constraint in the continuum limit.

From complex holomorphic systems to real systems

Symmetries in modern physics are a fundamental subject of high relevance that allows appreciating more deeply the physical structure of a theory. In this paper, we analyze a gauge symmetry that appears in complex holomorphic systems. We show that a complex system can be reduced to different real systems, using different gauge conditions, and the gauge transformations connect several real systems in the complex space. We prove that the space of solutions of one system is related using a gauge transformation to another one. Gauge transformations are, in some cases, canonical transformations. However, in other cases, these are more general transformations that change the symplectic structure, but there is still a map between systems. We establish a construction to extend the analysis to the quantum case using path integrals through the Batalin–Fradkin–Vilkovisky theorem and within the canonical formalism, where we show explicitly that solutions of the Schrödinger equation are gauge-related.

Giving mass to the mediating boson of HyperSymmetry by a field transformation applying Higgs mechanism beyond the Standard Model

According to gauge theories, interaction mediating spin-0 bosons must be massless. Theory of hypersymmetry (HySy) predicted massive intermediate bosons. Hypersymmetry field rotation, to be described in this paper, gives mass to the HySy mediating boson. Hysy rotation is performed in the velocity dependent D field – a gauge field defined beyond the Standard Model (SM). Its angle is something similar to the weak mixing Weinberg angle that explains the surplus mass to the neutral weak vector boson; as well as it is similar to the fermion flavour mixing Cabibbo-Kobayashi-Maskawa angles that justify the mass change under weakly interacting quarks’ mixing, respectively. Mass of intermediate bosons must arise from dynamical spontaneous breaking of the group of HySy. The mass of the discussed (fictitious) Goldstone bosons can be removed by the unitarity gauge condition through the Higgs (BEH) mechanism. According to the simultaneous presence of a SM interaction’s symmetry group and the HySy group, their bosons should be transformed together. Spontaneous breakdown of HySy may allow to perform a transformation that does not influence the SM physical state of the investigated system. The paper describes a field transformation that eliminates the mass of the intermediate bosons by the application of the BEH mechanism, rotates the SM and HySy bosons’ masses together, while leaves the SM bosons intact. The result is an angle of precession inclination that characterises the HySy mechanism. In contrast to the known SM intermediate bosons, the HySy intermediate bosons have no fix mass. The mass of the HySy intermediate bosons (that appear as quanta of a velocity dependent gauge field D) depends on the relative velocity of the particles whose interaction they mediate, therefore the derived transformation angle is a function of that velocity.

Canonical quantization of massive symmetric rank-two tensor in string theory

The canonical quantization of a massive symmetric rank-two tensor in string theory, which contains two Stueckelberg fields, was studied. As a preliminary study, we performed a canonical quantization of the Proca model to describe a massive vector particle that shares common properties with the massive symmetric rank-two tensor model. By performing a canonical analysis of the Lagrangian, which describes the symmetric rank-two tensor, obtained by Siegel and Zwiebach (SZ) from string field theory, we deduced that the Lagrangian possesses only first class constraints that generate local gauge transformation. By explicit calculations, we show that the massive symmetric rank-two tensor theory is gauge invariant only in the critical dimension of open bosonic string theory, i.e., d=26. This emphasizes that the origin of local symmetry is the nilpotency of the Becchi-Rouet-Stora-Tyutin (BRST) operator, which is valid only in the critical dimension. For a particular gauge imposed on the Stueckelberg fields, the gauge-invariant Lagrangian of the SZ model reduces to the Fierz-Pauli Lagrangian of a massive spin-two particle. Thus, the Fierz-Pauli Lagrangian is a gauge-fixed version of the gauge-invariant Lagrangian for a massive symmetric rank-two tensor. By noting that the Fierz-Pauli Lagrangian is not suitable for studying massive spin-two particles with small masses, we propose the transverse-traceless (TT) gauge to quantize the SZ model as an alternative gauge condition. In the TT gauge, the two Stueckelberg fields can be decoupled from the symmetric rank-two tensor and integrated trivially. The massive spin-two particle can be described by the SZ model in the TT gauge, where the propagator of the massive spin-two particle has a well-defined massless limit.

Electromagnetic Wave Equations of Dyon in Arbitrary Media

Dyon is a hypothetical particle in high energy physics that carries simultaneously both electric and magnetic charge. A dyon with zero electric charge is referred to a magnetic monopole. The paper, reports a simple reformulation of Maxwell equations for dyon in arbitrary media. The Lorentz, Coulomb gauge conditions and the wave equations of dyon in arbitrary media are derived in a simple and compact manner.

Faddeev–Jackiw quantization of Christ–Lee model

We analyze the constraints of Christ–Lee model by means of modified Faddeev–Jackiw formalism in Cartesian as well as polar coordinates. Further, we accomplish quantization à la Faddeev–Jackiw by choosing appropriate gauge conditions in both the coordinate systems. Finally, we establish gauge symmetries of Christ–Lee model with the help of zero-modes of the symplectic matrix.

Modeling and stabilization of current-controlled piezo-electric beams with dynamic electromagnetic field

Piezoelectric materials can be controlled with current (or charge) as the electrical input, instead of voltage. The main purpose of this paper is to derive the governing equations for a current-controlled piezo-electric beam and to investigate stabilizability. The magnetic permeability in piezo-electric materials is generally neglected in models. However, it has a significant qualitative effect on properties of the control system such as stabilizability. Besides the consideration of current control, there are several new aspects to the model. Most importantly, a fully dynamic magnetic model is included. Also, electrical potential and magnetic vector potential are chosen to be quadratic-through thickness to include the induced effects of the electromagnetic field. Hamilton’s principle is used to derive a boundary value problem that models a single piezo-electric beam actuated by a current (or charge) source at the electrodes. Two sets of decoupled system of partial differential equations are obtained; one for stretching of the beam and another one for bending motion. Since current (or charge) controller only affects the stretching motion, attention is focused on control of the stretching equations in this paper. It is shown that the Lagrangian of the beam is invariant under certain transformations. A Coulomb type gauge condition is used. This gauge condition decouples the electrical potential equation from the equations of the magnetic potential. A semigroup approach is used to prove that the Cauchy problem is well-posed. Unlike voltage actuation, a bounded control operator in the natural energy space is obtained. The paper concludes with analysis of stabilizability and comparison with other actuation approaches and models.

Spontaneous symmetry breakdown of scale symmetries

We discuss spontaneous symmetry breakdown (SSB) of both global and local scale symmetries in scalar-tensor gravity with two scalar fields, one of which couples nonminimally to scalar curvature while the other is a normal scalar field. In case of a global scale symmetry, by moving from the Jordan frame to the Einstein frame, a normal scalar field becomes massive while the dilaton remains massless after the SSB. In case of a local scale symmetry, we take a gauge fixing condition for the local scale invariance, $a R + b \phi^2 = k$, which was found in our previous study of a Weyl's quadratic gravity. Together with locally scale invariant potential terms in a classical action, this gauge condition generates a Higgs potential whose vacuum expectation value (VEV) produces the Einstein-Hilbert action with an $R^2$ term in the lowest level of approximation. One interesting aspect in this model is that a massless Nambu-Goldstone boson associated with a scale invariance is absorbed into the metric tensor field and consequently an $R^2$ term is induced in the action.