Gauge invariance and renormalization

The renormalization of Abelian and non-Abelian local gauge theories is discussed. It is recalled that whereas Abelian gauge theories are invariant to local c-number gauge transformations δA_μ(x) = ?_μ,…, with □Λ = 0, and to the operator gauge transformation δA_μ(x) = ?_μφ(x), …, δφ(x) = α^?1?·A(x), with □φ = 0, non-Abelian gauge theories are invariant only to the operator gauge transformations $\text{δA}{}_{\mathrm{\mu }}\text{(x) ～}\textcolor{red}{}{}_{\mathrm{\mu }}\text{C(x)}$, …, introduced by Becchi, Rouet and Stora, where

_μ is the covariant derivative matrix and C is the vector of ghost fields. The renormalization of these gauge transformation is discussed in a formal way, assuming that a gauge-invariant regularization is present. The naive renormalized local non-Abelian c-number gauge transformation $\text{δA}{}_{\mathrm{\mu }}\text{(x) = (Z}{}_1\text{/Z}{}_3\text{)gA}{}_{\mathrm{\mu }}\text{(x) ×}\text{Λ}\text{(x)+?}{}_{\mathrm{\mu }}\text{Λ}\text{(x)}$, …, is never a symmetry transformation and is never finite in perturbation theory. Only for $\text{Λ}\text{(x) = (Z}{}_3\text{/Z}{}_1\text{)L}$ with L finite constants or for $\text{Λ}\text{(x) = Ω}\text{z}\text{?}{}_3\text{C(x)}$ with Ω a finite constant does it become a finite symmetry transformation, where $\text{z}\text{?}{}_3$ is the ghost field renormalization constant. The renormalized non-Abelian Ward-Takahashi (Slavnov-Taylor) identities are consequences of the invariance of the renormalized gauge theory to this formation. It is also shown how the symmetry generators are renormalized, how photons appear as Goldstone bosons, how the (non-multiplicatively renormalizable) composite operator A_μ × C is renormalized, and how an Abelian c-number gauge symmetry may be reinstated in the exact solution of many asymptotically fr ee non-Abelian gauge theories.     

Gauge invariance and daisies

Perturbative calculation of effective potentials based on fine-tuning of coupling constants must be carefully done in order to preserve its gauge invariant contents.     

Gauge invariance and fock states

The Fock-space formulation is extended to nonabelian gauge theories. Using a kinematical potential instead of the Yang-Mills field, we construct invariant creation operators. Physical states are selected by the requirement that they remain invariant under a new gauge-invariant global transformation.     

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 and resonant transitions

It is shown that resonant transitions can be used to demonstrate experimentally the effects of the requirement of gauge invariance in quantum mechanics.     

Two-center problem for the dirac equation

The ground-state wave function and the energy term of a relativistic electron moving in the field of two fixed Coulomb centers are calculated analytically by the LCAO method. The resulting analytic formula is used to calculate the critical internuclear distance at which the energy term crosses the boundary of the lower continuum.     

Gauge invariance and the generalized Bondi-Metzner algebra

We examine the algebraic meaning of the Electromagnetic gauge invariance and show that it leads to the new concepts of gauged operators, gauged representations and hence to infinite dimensional extensions of Lie algebras. In particular we prove that the generalized Bondi-Metzner algebra can be interpreted as a gauged Lorentz algebra related to the Electromagnetic gauge.     

Gauge invariance of the Euler-Lagrange expressions

We prove, for a Lagrangian density L(g_ij;A _i ⁱ ;A _i ^j ), that the gauge invariance of the Euler-Lagrange expressionsE ⁱ (L) implies the existence of a gauge-invariant scalar densityL ₁, such thatE ⁱ (L) =E ⁱ (L₁). We then prove the uniqueness of the Yang-Mills field equations.     

Gauge invariance and a nonlocal gauge

It is shown that the longitudinal part of the gluon propagator D_μν^L in path-dependent gauges is crucially dependent on the order of taking the limit X → ± ∞. X is the starting point of the smooth path γ(x, X). The limit should be taken only after cancelling noninvariant terms. In our treatment the propagator D_μν^L runs to zero when |x₀-x′|→ ∞. The asymptotic growth of the propagator D_μν^L (shown by Slavnov and Frolov) is the price for the trasition from field configurations with undetermined gauge at one point to configurations with unfixed gauge on a three-dimensional surface at infinity.     

Supercriticality and transmission resonances in the dirac equation

It is shown that a Dirac particle of mass m and arbitrarily small momentum will tunnel without reflection through a potential barrier V = U(c)(x) of finite range provided that the potential well V = -U(c)(x) supports a bound state of energy E = -m. This is called a supercritical potential well.     

Minimization methods for the one-particle dirac equation

Taking into account relativistic effects in quantum chemistry is crucial for accurate computations involving heavy atoms. Standard numerical methods can deal with the problem of variational collapse and the appearance of spurious roots only in special cases. The goal of this Letter is to provide a general and robust method to compute particle bound states of the Dirac equation.     

Gauge invariance for extended objects

Just as the vector potential (one-form) couples to charged point-particles, antisymmetric tensor fields of higher rank (p-forms) couple to elementary objects of higher dimensionality (strings, membranes, …). It is shown that the only possible gauge invariant interaction of such an extended object with a gauge field in spacetime is based on the abelian group U(1). This is unlike the situation for particles where Yang-Mills actions based on any gauge group may be written down. The properties of the abelian theory are explored. It is pointed out that a compact object is analogous to a particle-antiparticle pair and its quantum rate of production in a constant external field is calculated semiclassically. The analysis is performed keeping generic both the dimension of the object and that of spacetime.     

Gauge invariance in braneworld scenarios

Rubakov and Shaposhnikov (RSH), in a seminal paper, discussed the possibility that particles are confined in a potential well. This is considered as the first mention to the today?s idea that we live in a brane, i.e., the braneworld concept. In this work we show precisely that the proposed RSH model has a gauge invariant equivalent action and we discuss it in the light of braneworld structure. We analyzed the intrinsic features of both models trying to disclose new properties within RSH braneworld theory.     

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 invariance and fermion doubling on the lattice

(1) We consider a possible chiral invariant solution of the lattice fermion doubling problem. This makes the unwanted states decouple in the continuum limit, at least in the non-interacting theory. The introduction of gauge interactions restores doubling. We examine how local gauge invariance makes all the species in a doubled spectrum act alike. (2) We generalise earlier results to show how gauge invariance forces a doubled spectrum on us even when other considerations do not.     

Gauge functions and Galilean invariance of Lagrangians

A novel method to make Lagrangians Galilean invariant is developed. The method, based on null Lagrangians and their gauge functions, is used to demonstrate the Galilean invariance of the Lagrangian for Newton's law of inertia. It is suggested that this new solution of an old physics problem may have implications and potential applications to all gauge-based theories of physics.     

Gauge invariance structure of quantum chromodynamics

Perturbative QCD may be subdivided into separately gauge-invariant sectors according to the projection of non-abelian color weights onto linearly independent basis elements. We exploit the general Lie group structure of the theory to give an algorithm for finding these gauge-invariant sets and present several examples of its use. The planar sector and the systematics of the non-planar corrections are defined for any gauge theory. Our gauge set classification has implications for QCD bound states, finite order perturbative QCD calculations, the study of QCD infrared singularities and for the question of convergence of the perturbation series.