Propagation of density perturbations in the deDonder gauge on a gravitational background field II

For the problem of propagation of density waves in a preexisting gravitational field, the advantages of the deDonder gauge over the commonly used synchronous gauge are outlined. In a background matter substratum withp as equation of state there are in the deDonder gauge only decaying modes of the perturbation density contrast with arbitrary large spatial extension, whereas in synchronous gauge there is one growing mode (calculated for vanishing spatial divergency of the perturbation in the 4-velocity, i.e.,usk(1),j/j0). The calculations are extended to the case of finite spatial extensions of the density perturbations. This is done by expanding all perturbations in a power series of the inverse square of the speed of light with the result of getting a recursive set of differential equations in both gauges for the equation of motion of the density perturbations. The lowest orders of this equation are the same in both gauges, but only in the deDonder gauge is the correct Newtonian limit of propagation of density waves in an expanding universe obtained. The correction by the next higher orders in the deDonder gauge are dependent explicitly on the spatial extension of the perturbations; whereas in synchronous gauge this is not the case. For attaining the Newtonian limit this dependence is a necessary condition. At appropriately large spatial extensions, however exact, this dependence in deDonder gauge leads ultimately to a decaying of density contrast modes growing in zeroth order (at least forp=0 andp/3 as equations of state for the background matter substratum). Hence, there are upper boundaries in the spatial extensions of instable growing modes of density contrast.     

Propagation of self-gravitating density waves in the deDonder gauge on a gravitational background field

Arguments are given for using the deDonder instead of the synchronous gauge in describing the propagation of density perturbations in a preexisting gravitational field. Since in the deDonder gauge the corresponding reference frame is fixed on the background, the physical interpretation of results is obvious, while in the synchronous gauge it is at least very difficult to extract the physical consequences from the results. For the propagation of density perturbations, with large spatial extension, a decisive difference is found between the two gauges. While in the synchronous gauge there is a growing mode in the density contrast (at least for adiabatic perturbations on a background matter substratum withp as equation of state), in the deDonder gauge there is not. The calculation in deDonder gauge leads to upper boundaries for the spatial extension of unstable density perturbations, and thus may give a hint for upper boundaries of galaxy masses.     

Stability borders and classification of density perturbation propagations in deDonder-gauge on a cosmological background

We investigate the propagation and the stability borders of density and metric perturbations on a cosmological background in linear perturbation theory in deDonder-gauge. We obtain the algebraic equations for the generally time-dependent stability borders by setting the typical time for perturbation contrasts infinite in the set of differential equations, while all other typical times stay finite. In dD-gauge there are in general three stability borders whereas in synchronous gauge there is only one. In the limiting cases of radiation perturbations and dustlike perturbations we obtain in deDonder-gauge no stability border resp. only one stability border (the ordinary Jeans limit). The first case is in contrast to the synchronous gauge and means that radiation perturbations cannot become unstable. During the recombination there could be three stability borders. We classify the propagation solutions and the systems of differential equations governing them by comparing the characteristic times in the original general system of differential equations, in deDonder-gauge and synchronous gauge. The greatest differences for the propagation of density contrasts arise from the presence of a gravitational wave time scale in deDonder-gauge. This becomes significant if the density perturbations are relativistic with respect to the velocity of sound. Gravitational retardation effects are the origin of the 6-dimensionality of the solution space for density contrasts. This reflects the necessity and physical meaning of gauge solutions.     

Cosmological perturbations: gauge-invariant vs. synchronous gauge analysis

We present a summary of the results, in the large scale, that were derived from the gauge-invariant (GI) cosmological density perturbation analysis. Comparisons are made with the synchronous gauge (SG) results, pointing out past incorrect analyses concerning evolution during the inflationary era and the decaying mode during the matter dominated era. The GI method, using the covariant equations, is recommended for cosmological perturbation analysis, not only because of its gauge-independence, but because of its simplicity and similarity to the Newtonian analysis. Solutions in other specific gauges can simply be recovered from the known results of the GI analysis.     

Infrared saturation and phases of gauge theories with BRST symmetry

We investigate the infrared limit of the quantum equation of motion of the gauge boson propagator in various gauges and models with a BRST symmetry. We find that the saturation of this equation at low momenta distinguishes between the Coulomb, Higgs and confining phase of the gauge theory. The Coulomb phase is characterized by a massless gauge boson. Physical states contribute to the saturation of the transverse equation of motion of the gauge boson at low momenta in the Higgs phase, while the saturation is entirely due to unphysical degrees of freedom in the confining phase. This corollary to the Kugo–Ojima confinement criterion in linear covariant gauges also is sufficient for confinement in general covariant gauges with BRST and anti-BRST symmetry, maximal Abelian gauges with an equivariant BRST symmetry, non-covariant Coulomb gauge and in the Gribov–Zwanziger theory.     

Quantum electrodynamics in the temporal gauge

We canonically quantize electrodynamics in the temporal gauge A₀ = 0. Realizing commutation relations in a Hilbert space containing unphysical longitudinal photons, we pay special attention to the implementation of Gauss's law and the attendant normalization difficulties for physical states. We then formulate the perturbation series and explicitly exhibit equivalence with the standard textbook treatment of the Coulomb gauge.     

A non-linear Rζ gauge condition for the electroweak SU(2) × U(1) model

A non-linear R_ζ gauge condition is presented and explicitly developed in the framework of the SU(2)×U(1) gauge model. We give the corresponding Feynman rules, which are simpler than in R_ζ gauges, because couplings involving unphysical Higgs and gauge bosons disappear or simplify. The Faddeev-Popov sector is more elegant, the ghosts coupling to neutral gauge bosons like in scalar electrodynamics. Finally, as a practical example, the transition Higgs→γγ is considered and compared with the usual calculation in linear gauges.     

Anomalies and gauge symmetry

Anomalies are known to have an intrinsic geometrical meaning. Using a formalism where the gauge condition is never made explicit we reanalyze the gauge theory anomaly problem. By requiring simultaneously the BRS and anti-BRS invariances, we do not need to use in our study the gauge dependent anti-ghost equation of motion. Then all equations definining the anomaly are independent of all parameters specifying the lagrangian. Not only does this stress explicitly the geometrical nature of the anomaly problem, but it allows for a single analysis for all possible BRS and anti-BRS invariant gauges, including those with four-ghost interactions. Our method for solving the anomaly equations is as a new sign of the relevance of the formalism in which the ghost components are unified with those of the classical gauge field, the ghost fields playing the role of a “connection” along unphysical directions. We recover the ABJ anomaly directly from the structure of BRS equations, as a straightforward application of the Chern-Weil theorem in some enlarged space. The method can be formally extended to higher space-time dimensions, and a general formula for “anomalies” in any even dimension is given.     

Quantum gauge equivalence in QED

We discuss gauge transformations in QED coupled to a charged spinor field, and examine whether we can gauge-transform the entire formulation of the theory from one gauge to another, so that not only the gauge and spinor fields, but also the forms of the operator-valued Hamiltonians are transformed. The discussion includes the covariant gauge, in which the gauge condition and Gauss's law are not primary constraints on operator-valued quantities; it also includes the Coulomb gauge, and the spatial axial gauge, in which the constraints are imposed on operator-valued fields by applying the Dirac-Bergmann procedure. We show how to transform the covariant, Coulomb, and spatial axial gauges to what we call “common form,” in which all particle excitation modes have identical properties. We also show that, once that common form has been reached, QED in different gauges has a common time-evolution operator that defines time-translation for states that represent systems of electrons and photons. By combining gauge transformations with changes of representation from standard to common form, the entire apparatus of a gauge theory can be transformed from one gauge to another.     

Fundamental monopoles and multimonopole solutions for arbitrary simple gauge groups

Magnetic monopole solutions for an arbitrary compact simple gauge group are considered in the Prasad-Sommerfield limit. For each group and choice of symmetry breaking there is a set of fundamental monopoles with minimal topological charges and possessing no internal degrees of freedom; the number of these is less than or equal to the rank of the gauge group. It is shown that if the unbroken gauge group is abelian, all solutions with higher topological charges belong to p-parameter families, where p is the number of position and group orientation parameters needed to describe a set of non-interacting fundamental monopoles with the given topological charge. It is argued that these solutions, some examples of which are given, should therefore be interpreted as multimonopole configurations. An extension of these results to the case of a non-albelian unbroken gauge symmetry is conjecture and shown to be valid for a number of examples.     

Spontaneous symmetry breaking of Slavnov symmetry A restriction on the gauge condition

It is shown that the condition of vanishing vacuum expectation value of the gauge operator, using a gauge-fixing term quadratic in this operator, does not necessarily follow from the Slavnov identity, due to a possible spontaneous breakdown of the Slavnov symmetry. For a consistent renormalization such a condition may have to be imposed order by order in perturbation theory, depending on the choice of the gauge. This restriction on non-singular gauges is particularly relevant for the discussion of the spontaneously broken realizations of the gauge symmetry.     

The Kirchhoff gauge

We discuss the Kirchhoff gauge in classical electrodynamics. In this gauge, the scalar potential satisfies an elliptical equation and the vector potential satisfies a wave equation with a nonlocal source. We find the solutions of both equations and show that, despite of the unphysical character of the scalar potential, the electric and magnetic fields obtained from the scalar and vector potentials are given by their well-known retarded expressions. We note that the Kirchhoff gauge pertains to the class of gauges known as the velocity gauge.     

Ghost fields in classical gauge theories

It is shown that ghost fields, characterized as unphysical entities, are a valuable tool in finding numerical solutions of the Euler-Lagrange equations of gauge field theories.     

Perturbation theory for the two-dimensional abelian Higgs modelin the unitary gauge

In the unitary gauge the unphysical degrees of freedom of spontaneously broken gauge theories are eliminated. The Feynman rules are simpler than in other gauges, but it is non-renormalizable by the rules of power counting. On the other hand, it is formally equal to the limit of the renormalizable R -gauge. We consider perturbation theory to one-loop order in the R -gauge and in the unitary gauge for the case of the two-dimensional abelian Higgs model. An apparent conflict between the unitary gauge and the limit of the R -gauge is resolved, and it is demonstrated that results for physical quantities can be obtained in the unitary gauge.Received: 17 July 2003, Revised: 8 August 2003, Published online: 20 November 2003E.E. Scholz: Present address: DESY, Notkestr. 85, 22603 Hamburg, Germany     

各级微扰展开下自发破缺的规范无关性、么正性及可重整性

用微扰论展开明显地讨论了标粒子与规范场矢粒子自发破缺Abel模型,发现在各种可重整规范下同阶各费曼图的内线非物理分量贡献的规范有关部分只与外线的质壳外部分互相依存。在质壳上只剩下物理分量的贡献,亦即转化成了么正规范。这样就明显地验证了么正规范与可重整规范在质壳上的全同,从而说明可重整规范是么正的,以及么正规范下怎样会出现剩余发散,为何剩余发散必然相消,揭穿了么正规范的隐藏可重整性。     

Density perturbations in a cosmological de Donder gauge

Density perturbations are considered during the radiation-dominated and the dust-dominated periods of the expanding universe. The perturbations are taken to have spherical symmetry and the investigation is carried out in the de Donder gauge. In order to guarantee the energy-momentum conservation of the perturbation in the de Donder gauge a compatibility condition is obtained. Equations for the propagation of a spherically symmetric perturbation in linear approximation on a FRW cosmological background are presented. It turns out that the evolutiontendency of the formation is mainly predicted by the state of the cosmic background. A radiation-dominated universe does not stimulate growth processes; the perturbation will be in a frozen state or it will diffuse. It is found that the dust-dominated universe stimulates the perturbation mass to grow. The rate of this cosmic affected growing process is proportional toR ^–1 (R being the scale factor of the universe), so that it seems that almost all galaxies were formed at the beginning of the present dust-dominated era.     

Transition matrices of self-dual solutions to SU(N) gauge theory

We extract self-dual potentials from SU(N) transition matrices of the Atiyah—Ward type. The simplest nontrivial example is studied in detail and we find a topologically nontrivial, regular, self-dual solution. Different gauges can be found for which the corresponding gauge potentials are respectively time-independent or real.     

Quantisation of a gauge field in the temporal-like gauge based on stochastic mechanics

Stochastic mechanics can be applied consistently to quantise gauge field in the temporal-like gauges such as the flow gauges, static gauges and the fully fixed temporal gauges.     

Gauge transformations and quantum mechanics II. Physical interpretation of classical gauge transformations

New gauges are introduced. The potentials, vector and scalar, in these gauges are obtained in closed forms by the Green's function method. These closed form solutions are explicity expressed only in terms of the charge and current densities. The physical interpretation is on how potentials propagate from the charge and current densities. The Coulomb gauge and the Lorentz gauge are special cases of a new gauge defined in this paper. It is called the complete α-Lorentz gauge. The scalar potential propagates at speed αc from the charge density for any positive α. When α is one, the usual solutions for the Lorentz gauge are recovered. When α is not one, our results show that, in order to satisfy the requirement that electromagnetic fields be gauge invariant and in order to conform to Maxwell's interpretation that electromagnetic fields propagate at speed c from the charge and current densities (we only consider the vacuum), the vector potential must contain two mathematically and physically independent gradient components. Furthermore, one such component must propagate at speed αc while the other must at speed c from charge and current densities. Our discussions on the Coulomb gauge are based on the results obtained by letting α go to (positive) infinity. Guided by Maxwell's interpretation, we introduce a new decomposition of the vector potential in the Lorentz gauge into a longitudinal and a transverse component. For an arbitrary charge and current distribution, it is shown that the transverse component will generate all the fields only in the radiation zone. However, for a point charged particle, the transverse component only generates the “free fields”everywhere in the instantaneous rest frame of the charged particle.     

Axial gauge propagators for quarks and gluons on the Polyakov-Wilson lattice

We discuss problems encountered in defining gauge-dependent propagators in a confining theory. For precision we use a finite Polyakov-Wilson lattice to define the Yang-Mills theory and to provide the ultraviolet and infrared regularization. Gauge fixing in a class of superaxial gauges is natural in this framework. A variety of approaches for defining the propagators for quarks and gluons is discussed and the propagators are evaluated explicitly in the strong coupling limit. We speculate upon the infrared behavior of these propagators in the weak coupling limit and upon the utility and validity of the Schwinger-Dyson equations for these propagators. In conclusion we propose that the leading infrared behavior is strongly gauge dependent and governed by the masses of low-lying color singlet states in the hadron spectrum. In the ultraviolet limit, however, with a properly constructed propagator, we find no reason to question the conventional wisdom derived from perturbation theory. Our conclusions should not depend in any fundamental way on the lattice formulation of the gauge theory, except insofar as that formulation serves to give precision to the continuum functional integration.