Narrow Results

We give a detailed introduction to the classical Chern–Simons gauge theory, including the mathematical preliminaries. We then explain the perturbative quantization of gauge theories via the Batalin–Vilkovisky (BV) formalism. We then define the perturbative Chern–Simons partition function at any (possibly non-acylic) reference flat connection using the BV formalism, using a Riemannian metric for gauge fixing. We show that it exhibits an anomaly known as the “framing anomaly” when the Riemannian metric is changed, that is, it fails to be gauge invariant. We explain how one can deal with this anomaly to obtain a topological invariant of framed manifolds.

Even though its classical equations of motion are then left invariant, when an action is redefined by an additive total derivative or divergence term (in time, in the case of a mechanical system) such a transformation induces non-trivial consequences for the system’s canonical phase space formulation. This is even more true and then in more subtle ways for the canonically quantized dynamics, with, in particular, an induced transformation in the unitary configuration space representation of the Heisenberg algebra being used for the quantum system. When coupled to a background gauge field, such considerations become crucial for a proper understanding of the consequences for the system’s quantum dynamics of gauge transformations of that classical external background gauge field, while under such transformations, the system’s degrees of freedom, abstract quantum states and quantum dynamics are certainly strictly invariant. After a detailed analysis of these different points in a general context, these are then illustrated specifically in the case of the quantum Landau problem with its classical external background magnetic vector potential for which the most general possible parametrized gauge choice is implemented herein. The latter discussion aims as well to clarify some perplexing statements in the literature regarding the status of gauge choices to be made for the magnetic vector potential for that quantum system. The role of the global spacetime symmetries of the Landau problem and their gauge invariant Noether charges is then also emphasized.

We give a simplified and more complete description of the loop variable approach for writing down gauge-invariant equations of motion for the fields of the open string. A simple proof of gauge invariance to all orders is given. In terms of loop variables, the interacting equations look exactly like the free equations, but with a loop variable depending on an extra parameter, thus making it a band of finite width. The arguments for gauge invariance work exactly as in the free case. We show that these equations are Wilsonian RG equations with a finite worldsheet cutoff and that in the ir limit, equivalence with the Callan–Symanzik β-functions should ensure that they reproduce the on-shell scattering amplitudes in string theory. It is applied to the tachyon–photon system and the general arguments for gauge invariance can be easily checked to the order calculated. One can see that when there is a finite worldsheet cutoff in place, even the U(1) invariance of the equations for the photon, involves massive mode contributions. A field redefinition involving the tachyon is required to get the gauge transformations of the photon into the standard form.

The gauge-invariant loop variable formalism and old covariant formalism for bosonic open string theory are compared in this paper. It is expected that for the free theory, after gauge fixing, the loop variable fields can be mapped to those of the old covariant formalism in bosonic string theory, level by level. This is verified explicitly for the first two massive levels. It is shown that (in the critical dimension) the fields, constraints and gauge transformations can all be mapped from one to the other. Assuming this continues at all levels one can give general arguments that the tree S-matrix (integrated correlation functions for on-shell physical fields) is the same in both formalisms and therefore they describe the same physical theory (at tree level).

The explicit form of linearized gauge invariant interactions of scalar and general higher even spin fields in the AdS_D space is obtained. In the case of general spin-ℓ a generalized "Weyl" transformation is proposed and the corresponding "Weyl" invariant action is constructed. In both cases the invariant actions of the interacting higher even spin gauge field and the scalar field include the whole tower of invariant actions for couplings of the same scalar with all gauge fields of smaller even spin. For the particular value of ℓ = 4, all results are in exact agreement with Ref. 1.

In this work, we study the radiative leptonic decays of B^-, D^- and , including both the short-distance and long-distance contributions. The short-distance contribution is calculated by using the relativistic quark model, where the bound state wave function we used is that obtained in the relativistic potential model. The long-distance contribution is estimated by using vector meson dominance model.

The possible emergence of unparticle has been mooted recently including a mass-like term for gauge field with the Schwinger model at the classical level. A one-loop correction due to bosonization is taken into account and investigation is carried out to study its effect on the unparticle scenario. It is observed that the physical mass, viz., unparticle scale acquires a new definition, i.e. the effect of this correction enters into the unparticle scale in a significant manner. The fermionic propagator is calculated which also agrees with the new scale. It has also been noticed that a novel restoration of the lost gauge invariance reappears when the ambiguity parameter related to the current anomaly acquires a specific expression. We have also observed that a quantum effect can nullify the effect of violation of gauge symmetry caused by some classical terms.

One of the major problems in developing new physics scenarios is that very often the parameters can be adjusted such that in perturbation theory almost all experimental low-energy results can be accommodated. It is therefore desirable to have additional constraints. Field-theoretical considerations can provide such additional constraints on the low-lying spectrum and multiplicities of models. Especially for theories with elementary or composite Higgs particle the Fröhlich–Morchio–Strocchi (FMS) mechanism provides a route to create additional conditions, though showing it to be at work requires genuine non-perturbative calculations. The qualitative features of this procedure are discussed for generic 2-Higgs-doublet models (2HDMs), grand-unified theories (GUTs) and technicolor-type theories.

Gauge-invariant perturbation theory is an extension of ordinary perturbation theory which describes strictly gauge-invariant states in theories with a Brout–Englert–Higgs effect. Such gauge-invariant states are composite operators which have necessarily only global quantum numbers. As a consequence, flavor is exchanged for custodial quantum numbers in the Standard Model, recreating the fermion spectrum in the process. Here, we study the implications of such a description, possibly also for the generation structure of the Standard Model.

In particular, this implies that scattering processes are essentially bound-state–bound-state interactions, and require a suitable description. We analyze the implications for the pair-production process $e^{+}{e}^{-}\to \bar{f}f$ at a linear collider to leading order. We show how ordinary perturbation theory is recovered as the leading contribution. Using a PDF-type language, we also assess the impact of sub-leading contributions. To lowest order, we find that the result is mainly influenced by how large the contribution of the Higgs at large $x$ is. This gives an interesting, possibly experimentally testable, scenario for the formal field theory underlying the electroweak sector of the Standard Model.

Our main interest here is to analyze the gauge invariance issue concerning the noncommutative relativistic particle. Since the analysis of the constraint set from Dirac’s point of view classifies it as a second-class system, it is not a gauge theory. Hence, the objective here is to obtain gauge invariant actions linked to the original one. However, we have two starting points, meaning that firstly we will begin directly from the original action and, using the Noether procedure, we have obtained a specific dual (gauge invariant) action. Following another path, we will act towards the constraints so that we have carried out the conversion of second to first-class constraints through the Batalin–Fradkin–Fradkina–Tyutin formalism, obtaining the second gauge invariant Lagrangian.

Stueckelberg QED with massive photon is known to be renormalizable. But the limit of the mass going to zero is interesting because it brings the resolution to infrared questions through the role of Stueckelberg field at null infinity in addition to providing new asymptotic symmetries. Such symmetries facilitate the soft photon theorems also.

Quantum ChromoDrynamics (QCD) is a quantum field theory, which describes the strong interaction. It explains how quarks and gluons combine to form hadrons. QCD is part of the standard model of particle physics, along with ElectroWeak (EW) theory. The aim of this study is to explore fractional orders in the QCD Lagrangian density using the Atangana–Baleanu fractional derivative. Starting from the fractional Euler–Lagrange equation, we were able to derive the equation of motion of the quark and gluon fields. Then, based on fractional QCD Lagrangian density, the fractional Hamilton equations were obtained. We demonstrate that the principle of local gauge invariance, a fundamental symmetry in QCD, is preserved under a fractional extension of the theory. Our findings indicate that the fractional equations of QCD encompass the classical equations as a specific case, offering a broader perspective on quark-gluon dynamics. The fractional QCD Lagrangian density provides new insights that are not accessible through classical derivatives, potentially enhancing our understanding of quark-gluon plasmas and contributing to advancements in collider phenomenology and precision measurements. This study opens new avenues for exploring the fundamental nature of the strong interaction and its implications for particle physics beyond the Standard Model.

For many systems with second-class constraints, the question posed in the title is answered in the negative. We prove this for a range of systems with two second-class constraints. After looking at two examples, we consider a fairly general proof. It is shown that, to unravel gauge invariances in second-class constrained systems, it is sufficient to work in the original phase space itself. Extension of the phase space by introducing new variables or fields is not required.

Elementary particle scatterings and decays in the presence of a background magnetic field are very common in physics, especially after the observation that the core of the neutron stars can sustain a magnetic field of the order of 10¹³ G. The important point about these calculations is that they are done in a background of a gauge field and as a result the calculations are prone to gauge arbitrariness. In this work we will investigate how this gauge arbitrariness is eradicated in processes where the initial and final particles taking part in the interactions are electrically neutral. Some comments on those processes where the initial or final state consists of electrically charged particles is presented at the end of the paper.

In this work we show that we can obtain dual equivalent actions following the symplectic formalism with the introduction of extra variables which enlarge the phase space. We show that the results are equal as the one obtained with the recently developed gauging iterative Noether dualization method. We believe that, with the arbitrariness property of the zero mode, the symplectic embedding method is more profound since it can reveal a whole family of dual equivalent actions. We illustrate the method demonstrating that the gauge-invariance of the electromagnetic Maxwell Lagrangian broken by the introduction of an explicit mass term and a topological term can be restored to obtain the dual equivalent and gauge-invariant version of the theory.

The issue of space–time gauge invariance for the bosonic string has been earlier addressed using the loop variable formalism. In this paper the question of obtaining a gauge invariant action for the open bosonic string is discussed. The derivative with respect to ln a (where a is a worldsheet cutoff) of the partition function — which is first normalized by dividing by the integral of the two-point function of a marginal operator — is a candidate for the action. Applied to the zero-momentum tachyon it gives a tachyon potential that is similar to those that have been obtained using Witten's background independent formalism. This procedure is easily made gauge invariant in the loop variable formalism by replacing ln a by Σ which is the generalization of the Liouville mode that occurs in this formalism. We also describe a method of resumming the Taylor expansion that is done in the loop variable formalism. This allows one to see the pole structure of string amplitudes that would not be visible in the original loop variable formalism.

We study the connection or equivalence between two well-known extensions of the Standard Model, that is, for the coupling between the familiar massless electromagnetism U(1)_QED and a hidden-sector U(1)_h, and axionic electrodynamics. Our discussion is carried out using the gauge-invariant but path-dependent variables formalism, which is an alternative to the Wilson loop approach. When we compute in this way the static quantum potential for the coupling between the familiar massless electromagnetism U(1)_QED and a hidden-sector U(1)_h, the result of this calculation is a Yukawa correction to the usual static Coulomb potential. Previously,¹⁴, we have shown that axionic electrodynamics has a different structure which is reflected in a confining piece. Therefore, both extensions of the Standard Model are not equivalent. Interestingly, when the above calculation is done inside a superconducting box, the Coulombic piece disappears leading to a screening phase.

Is gauge-invariant complete decomposition of the nucleon spin possible? Although it is a difficult theoretical question which has not reached a complete consensus yet, a general agreement now is that there are at least two physically inequivalent gauge-invariant decompositions (I) and (II) of the nucleon. In these two decompositions, the intrinsic spin parts of quarks and gluons are just common. What discriminate these two decompositions are the orbital angular momentum parts. The orbital angular momenta of quarks and gluons appearing in the decomposition (I) are the so-called "mechanical" orbital angular momenta, while those appearing in the decomposition (II) are the generalized (gauge-invariant) "canonical" ones. By this reason, these decompositions are also called the "mechanical" and "canonical" decompositions of the nucleon spin, respectively. A crucially important question is which decomposition is more favorable from the observational viewpoint. The main objective of this concise review is to try to answer this question with careful consideration of recent intensive researches on this problem.

In an earlier paper, gauge invariant and background covariant equations for closed string modes were obtained from the exact Renormalization Group of the world sheet theory. The background metric (but not the physical metric) had to be flat and hence the method was not manifestly background independent. In this paper, the restrictions on the background metric are relaxed. A simple prescription for the map from loop variables to space–time fields is given whereby for arbitrary backgrounds the equations are generally covariant and gauge invariant. Extra terms involving couplings of the curvature tensor to (derivatives of) the Stueckelberg fields have to be added. The background metric can thus be chosen to be the physical metric without any restrictions. This method thus gives manifestly background independent equations of motion for both open and closed string modes.

In this note we discuss the question of gauge invariance in the presence of a minimal length. This contribution is prepared for the celebration of the 60th anniversary of the Yang–Mills theory.