Gauge invariance and Nielsen identities

The one-loop contribution to the effective potential and mass are computed within the context of scalar electrodynamics for the class of generalR gauges in the \(\overline {MS} \) scheme. These calculations are performed in order to construct a non-trivial verification of the corresponding Nielsen identities within the context of the Higgs model. Some brief comments on the Coleman-Weinberg model are also included.     

Gauge invariance,lorentz covariance and the observer

We discuss a number of questions related to the role of the observer in classical and quantum theories of fields, in particular electrodynamics. We find the gauge-independent parts of the electromagnetic potential, which are classical observables, both in a non-covariant manner and in a Lorentz covariant, observer-dependent way. We present an analysis of the probabilistic interpretation of relativistic quantum mechanics, similar to that of the nonrelativistic theory, and discuss the gauge invariance of the corresponding probability amplitudes.     

Gauge fixing,BRS invariance and Ward identities for randomly stirred flows

The Galilean invariance of the Navier–Stokes equation is shown to be akin to a global gauge symmetry familiar from quantum field theory. This symmetry leads to a multiple counting of infinitely many inertial reference frames in the path integral approach to randomly stirred fluids. This problem is solved by fixing the gauge, i.e., singling out one reference frame. The gauge fixed theory has an underlying Becchi–Rouet–Stora (BRS) symmetry which leads to the Ward identity relating the exact inverse response and vertex functions. This identification of Galilean invariance as a gauge symmetry is explored in detail, for different gauge choices and by performing a rigorous examination of a discretized version of the theory. The Navier–Stokes equation is also invariant under arbitrary rectilinear frame accelerations, known as extended Galilean invariance (EGI). We gauge fix this extended symmetry and derive the generalized Ward identity that follows from the BRS invariance of the gauge-fixed theory. This new Ward identity reduces to the standard one in the limit of zero acceleration. This gauge-fixing approach unambiguously shows that Galilean invariance and EGI constrain only the zero mode of the vertex but none of the higher wavenumber modes.     

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.     

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

It is shown that various Green function identities in quantized gauge theories may be viewed as arising from the local conformal groupSO(N+1, 1) of motions of theN-dimensional gauge fixing surface. Translations and rotations correspond respectively to the usual Slavnov-Taylorinvariance and anew ‘dual’ analogue. Dilations give rise to thecovariance identities, and in axial type gauges we obtain a closed form for the covariance relationships. It is shown that the generalized Slavnov-Taylor identities, and the BRST identities, are equivalent, as are their duals.     

Formal structures,the concepts of covariance,invariance, equivalent reference frames,and the principle Relativity

In this paper a given spacetime theoryT is characterized as the theory of a certainspecies of structure in the sense of Bourbaki [1]. It is then possible to clarify in a rigorous way the concepts ofpassive andactive covariance ofT under the action of the manifold mapping groupG _M . For eachT, we define also aninvariance groupG _I T and, in general,G _I T ≠G _M . This group is defined once we realize that, for eachτ ∈ModT, each explicit geometrical object defining the structure can be classified as absolute or dynamical [2]. All spacetime theories possess alsoimplicit geometrical objects that do not appear explicitly in the structure. These implicit objects are not absolute nor dynamical. Among them there are thereference frame fields, i.e., “timelike” vector fieldsX ∈TU, \(U \subseteq M\) M, whereM is a manifold which is part ofST, a substructure for eachτ ∈ModT, called spacetime. We give a physically motivated definition of equivalent reference frames and introduce the concept of theequivalence group of a class of reference frames of kind X according to T, G _X T. We define thatT admits aweak principle of relativity (WPR) only ifG _X T ≠ identity for someX. IfG _X T =G _I T for someX, we say thatT admits a strong principle of relativity (PR). The results of this paper generalize and clarify several results obtained by Anderson [2], Scheibe [3], Hiskes [4], Recami and Rodrigues [5], Friedman [6], Fock [7], and Scanavini [8]. Among the novelties here, there is the realization that the definitions ofG _I T andG _X T can be given only when certain boundary conditions for the equations of motion ofT can be physically realizable in the domainU \(U \subseteq M\) M, where a given reference frame is defined. The existence ofphysically realizable boundary conditions for eachτ ∈ModT (in ?U), in contrast with the mathematically possible boundary condition, is then seen to be essential for the validity of a principle of relativity forT. The methodology of the present paper has been applied to several topics of spacetime physics with very interesting results. Here we mention:

1. The Newtonian concepts of absolute space and absolute time can be presented in a very elegant way as “species of structure”. One of the surprising results is that we succeeded in finding a Lorentzian structure [9] in Newtonian spacetime without introducing any new explict geometrical object in the original structure. The Newtonian spacetime structure and its relation to the relativistic spacetime structure and to the structure of the spacetime of the so-called Lorentz aether theories [11,12] is fully discussed in [13].
2. It is possible to present in a novel and unified way the question concerning experiments designed to detect a possible breakdown of Lorentz invariance, a subject we already dedicated attention to in Rodrigues and Tiomno [11,12] and Rodrigues [14,15]. A full account of this subject will be published elsewhere.
3. In Rodrigues and Scanavini [16], we proved that there are models of General Relativity that contain a canonical privileged locally inertial reference frame that can be physically distinguished from any other frame by experiments doneinside the frame.

Although the formalism of this paper may at first sight look very abstract, actually it is easy to aplly it to specific theories. We present an example at the end of the paper which is sufficiently general to show “in action” almost all concepts introduced in this paper.     

Conformal invariance,current algebra and modular invariance

A large class of conformally invariant models in two dimensions is realised by constraining free fermion theories. The Fock spaces of the constrained theories are described, using the representation theory of affine Kac-Moody algebras. The results are extended to superconformally invariant theories. Projections of the models, producing consistent two-dimensional field theories, are discussed.     

Synchrotron radiation under conditions of violated lorentz invariance

We have obtained an exact solution to the Dirac equation for an electron in a constant uniform magnetic field by taking into account the anomalous magnetic moment and Lorentz invariance violation in minimal CPT-odd form. Based on the solution found, we have calculated synchrotron radiation characteristics and predicted possible observable effects attributable to the Lorentz invariance violation.     

Tractors,mass, and Weyl invariance

Deser and Nepomechie established a relationship between masslessness and rigid conformal invariance by coupling to a background metric and demanding local Weyl invariance, a method which applies neither to massive theories nor theories which rely upon gauge invariances for masslessness. We extend this method to describe massive and gauge invariant theories using Weyl invariance. The key idea is to introduce a new scalar field which is constant when evaluated at the scale corresponding to the metric of physical interest. This technique relies on being able to efficiently construct Weyl invariant theories. This is achieved using tractor calculus—a mathematical machinery designed for the study of conformal geometry. From a physics standpoint, this amounts to arranging fields in multiplets with respect to the conformal group but with novel Weyl transformation laws. Our approach gives a mechanism for generating masses from Weyl weights. Breitenlohner–Freedman stability bounds for Anti-de Sitter theories arise naturally as do direct derivations of the novel Weyl invariant theories given by Deser and Nepomechie. In constant curvature spaces, partially massless theories—which rely on the interplay between mass and gauge invariance—are also generated by our method. Another simple consequence is conformal invariance of the maximal depth partially massless theories. Detailed examples for spins s?2$s?2$ are given including tractor and component actions, on-shell and off-shell approaches and gauge invariances. For all spins s?2$s?2$ we give tractor equations of motion unifying massive, massless, and partially massless theories.     

The Gross-Neveu Model under the conditions of violated Lorentz invariance

The aim of this work is to calculate the effective potential in the two-dimensional Gross-Neveu model with violated Lorentz invariance, to study the properties of the obtained potential, and to find extreme points by solving the gap equation and to compare of the results with the theory in the case of nonviolated Lorentz invariance. It is shown that, in the model under study, the effects that make it crucially different from the well-studied model without violation are observed.     

Foams in contact with solid boundaries: Equilibrium conditions and conformal invariance

A liquid foam in contact with a solid surface forms a two-dimensional foam on the surface. We derive the equilibrium equations for this 2D foam when the solid surface is curved and smooth, generalising the standard case of flat Hele-Shaw cells. The equilibrium conditions at the vertices in 2D, at the edges in 3D, are invariant by conformal transformations. Regarding the films, conformal invariance only holds with restrictions, which we explicit for 3D and flat 2D foams. Considering foams confined in thin interstices between two non-parallel plates, normal incidence and Laplace’s law lead to an approximate equation relating the plate profile to the conformal map. Solutions are given for the logarithm and power laws in the case of constant pressure. The paper concludes on a comparison with available experimental data.     

An application of functional equations to the analysis of the invariance identities of classical gauge field theory

The equations of motion for a particle in a classical gauge field are derived from the invariance identities ² and basic assumptions about the Lagrangian. They are found to be consistent with the equations of some other approaches to classical gauge-field theory, and are expressed in terms of a set of undetermined functions E. The functions E are found to satisfy a system of differential equations which has the same formal structure as a system of equations from Yang-Mills theory. ³ These results are obtained by a new method which applies techniques from the theory of functional equations to deduce the way in which the arguments of the Lagrangian must combine. The method constitutes an aid for obtaining the equations of motion when a non-gauge-invariant Lagrangian is chosen, and it is assumed that the equations of motion can be written in a gauge-invariant manner.     

Computation of discrete transparent boundary conditions for the 2D Helmholtz equation

We present a family of non-local transparent boundary conditions for the 2D Helmholtz equation. The whole domain, on which the Helmholtz equation is defined, is decomposed into an interior and an exterior domain. The corresponding interior Helmholtz problem is formulated as a variational problem in a standard manner, representing a boundary value problem, whereas the exterior problem is posed as an initial value problem in the radial variable. This problem is then solved approximately by means of the Laplace transformation. The derived boundary conditions are asymptotically correct, model inhomogeneous exterior domains and are simple to implement.     

Theta functions,modular invariance,and strings

We use Quillen's theorem and algebraic geometry to investigate the modular transformation properties of some quantities of interest in string theory. In particular, we show that the spin structure dependence of the chiral Dirac determinant on a Riemann surface is given by Riemann's theta function. We use this result to investigate the modular invariance of multiloop heterotic string amplitudes.     

On Weyl invariance conditions in string theory coupled with two-dimensional gravity

The unified theory of string and two-dimensional quantum gravity is considered. We introduce nontrivial dynamics for the two-dimensional metric g_μν from the very beginning and calculate the path integral over the string coordinates and g_μν without taking into account the order of integrations. Throughout the paper we use two different kinds of gauges - the covariant one of the harmonic type and also the conformal gauge, where the original (D + 1)-dimensional sigma model with quantum gravity becomes the (D+2)-dimensional sigma model on the classical background of g_μν The general symmetries of the theory consist in the reparametrizations of the target space coordinates, in the conformal transformations of the metric and in the usual 2d diffeomorphisms. These symmetries do not disturb the structure of the background fields in the (D+2) -dimensional formulation. On the other hand the related arbitrariness of the renormalization does not affect the qualitative structure of the loop contributions to the Weyl anomaly. In the theory with quantum gravity the parameter a′ does not play as the parameter of the loop expansion. That is why the one-loop conditions of the Weyl invariance differs from the well known effective equations which arise in the standard approach when g_μν is not quantized simultaneously with the string coordinates. Therefore, despite the new conditions of the Weyl invariance for the background fields are different from the standard effective equations, our result does not contradict to the standard approach. The new one-loop conditions of the Weyl invariance are much more complicated and contain the higher derivatives in the dilaton sector.