Local Gauge Transformations Research Articles

Gauge Orbit Types for Theories with Gauge Group O(n), SO(n) or Sp(n)

We determine the orbit types of the action of the group of local gauge transformations on the space of connections in a principal bundle with structure group O(n), SO(n) or $Sp(n)$ over a closed, simply connected manifold of dimension 4. Complemented with earlier results on U(n) and SU(n) this completes the classification of the orbit types for all classical compact gauge groups over such space-time manifolds. On the way we derive the classification of principal bundles with structure group SO(n) over these manifolds and the Howe subgroups of SO(n).

REFLEXIVE MODULES OF RANK ONE OVER WEYL ALGEBRAS OF NON-ZERO CHARACTERISTICS

Many of the properties of a Weyl algebra An over a base field of non-zero characteristic are explained in terms of connections and curvatures on a vector bundle on an affine space X = A2n. In particular, it is known that an algebra endomorphism φ of An gives rise to a symplectic endomorphism f of X with a gauge transformation g. In this paper we study the converse problem of finding φ from an arbitrary symplectic endomorphism f of X = A2n. It is shown that, given such f , we may construct a projective left An-module (which corresponds to ‘the sheaf of local gauge transformations' ) such that its triviality is equivalent to the existence of the ‘lift' φ. Some properties of such a module will be discussed using the theory of reflexive sheaves.

On the gauge orbit types for theories with classical compact gauge group

We classify principal O( n )-bundles over 4-dimensional closed manifolds for n = 3 and n ≥ 5 and SO(2 n )-bundles over 2 n -dimensional closed manifolds. This is part of a program to determine the orbit types of the action of the group of local gauge transformations on gauge potentials for an arbitrary compact classical gauge group and an arbitrary 4-dimensional space-time.

Gauge theories under incorporation of a generalized uncertainty principle

There is considered an extension of gauge theories according to the assumption of a generalized uncertainty principle which implies a minimal length scale. A modification of the usual uncertainty principle implies an extended shape of matter field equations like the Dirac equation. If there is postulated invariance of such a generalized field equation under local gauge transformations, the usual covariant derivative containing the gauge potential has to be replaced by a generalized covariant derivative. This leads to a generalized interaction between the matter field and the gauge field as well as to an additional self interaction of the gauge field. Since the existence of a minimal length scale seems to be a necessary assumption of any consistent quantum theory of gravity, the gauge principle is a constitutive ingredient of the standard model and even gravity can be described as gauge theory of local translations or Lorentz transformations, the presented extension of gauge theories appears as a very important consideration.

Gauge invariance and vacuum energies of non-Abelian string-configurations

We improve on a method to compute the fermion contribution to the vacuum polarization energy of string-like configurations in a non-Abelian gauge theory. We establish the new method by numerically verifying the invariance under (a subset of) local gauge transformations. This also provides further support for the use of spectral methods to compute vacuum polarization energies in general. We confirm that the vacuum energy in the MS¯ renormalization scheme is tiny as compared to the mass of the fluctuating fermion field. Numerical results for the physical on-shell scheme are also presented.

Operators in ultraviolet completions for collective symmetry breaking

We investigate the Lagrangian terms of scalar-gauge interactions in a weakly-coupled gauge theory beyond the ultraviolet momentum cutoff of models where the collective symmetry breaking mechanism protects the Higgs mass squared against having quadratic divergence. We also propose a five-dimensional model in which the gauge symmetry breaking is achieved by boundary conditions consistent with local gauge transformations. It is shown that the scalar-gauge operators in the two models differ in the absence of quadratic divergence.

Chiral currents and static properties of nucleons in holographic QCD

We analyze static properties of nucleons in the two flavor holographic QCD model of Sakai and Sugimoto described effectively by a five-dimensional $U(2)$ Yang-Mills theory with the Chern-Simons term on a curved background. The baryons are represented in this model as a soliton, which at a time slice is approximately the Belavin-Polyakov-Shvarts-Tyupkin (BPST) instanton with a fixed size. First, we construct a chiral current in four dimensions from the Noether current of local gauge transformations which are nonvanishing on the boundaries of the extra dimension. We examine this chiral current for nucleons with quantized collective coordinates to compute their charge distribution, charge radii, magnetic moments, and axial-vector coupling. Most of the results are better close to the experimental values than in the Skyrme model. We discuss the problems of our chiral current, nonuniqueness of the local gauge transformation for defining the current, and its gauge-noninvariance.

Towards a quantization of gauge fields on de Sitter group by functional integral method

A formulation of the de Sitter symmetry as a purely inner symmetry defined on a fixed Minkowski space-time is presented. We define the generators of the de Sitter group and write the structure equations using a constant deformation parameter λ. The conserved gauge currents are calculated, and their physical meaning is given. Local gauge transformations and the corresponding covariant derivative depending on the gauge fields are also obtained. We study the behavior of gauge fields, the torsion and curvature tensors and give a regularization technique in terms of the ζ function.

Geometric foundations of classical Yang–Mills theory

We analyze the geometric foundations of classical Yang–Mills theory by studying the relationships between internal relativity, locality, global/local invariance, and background independence. We argue that internal relativity and background independence are the two independent defining principles of Yang–Mills theory. We show that local gauge invariance—heuristically implemented by means of the gauge argument—is a direct consequence of internal relativity. Finally, we analyze the conceptual meaning of BRST symmetry in terms of the invariance of the gauge fixed theory under general local gauge transformations.

A projector formulation of the Galilean covariant Duffin–Kemmer–Petiau field

We construct in this paper a projector formulation of the Galilean covariant Duffin–Kemmer–Petiau field in five dimensions. Such an approach allows us to select the scalar and vector sectors of the theory through the use of the appropriate operators. As an application, we study a non-minimal coupling associated to the DKP harmonic oscillator, naturally in the non-relativistic regime and with the selection of the spin-0 sector in a general representation. We also discuss the local gauge invariance and the anomalous term which appear in the wave equations due to the minimal coupling. In both questions, our results were carried out as in the relativistic case, i.e. the consistent local gauge transformations can be obtained through the choice of the right form of the non-relativistic DKP field.

Spin‐charge rotating local reference frames: a unified U(2) = U(1) ⊗ SU(2) approach to the interacting electrons

AbstractA spin‐charge unifying description for the Hubbard model based on the time dependent local gauge transformations is developed. The collective variables for charge and spin are isolated in the form of the space‐time fluctuating U(1) phase field and rotating spin quantization axis governed by the SU(2) symmetry, respectively. As a result interacting electrons appear as a composite objects consisting of bare fermions with attached U(1) and SU(2) gauge fields. We elaborate on the microscopic origins of the effective action with the Coulomb interaction that contain topological theta terms. Furthermore, we unravel the link between non‐trivial multiply‐connected topological structure of the U(2) = U(1) ⊗ SU(2) configurational space for gauge fields and the instanton contribution to the statistical sum. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Various Full Green Functions in QED

We are interested in deriving various full Green functions through general Ward–Takahashi identities (WTIs) for quantized field theories. With the help of a postulate of gauge group parameter, the general local gauge transformation laws preserving the gauge-invariance of the generating functional itself of QED model have been established successfully. By using path-integral technique, the various WTIs with resulting anomaly terms are derived under the gauge transformations. The arising of Jacobian factor from the integration measure gives a viable possibility to express full Green function. As a consequence, the complete expressions of the full vector, the full axial-vector, the full tensor vertex functions and so on are presented respectively by solving the complete set of the WTIs in the momentum space without considering the constraint imposing any Ansatz. In addition, anomaly function also provides an effective means to judge the divergence of variant coupling currents on fields.

Gauge principle and variational formulation for ideal fluids with reference to translation symmetry

Following the gauge principle in the field theory of physics, a new variational formulation is presented for flows of an ideal fluid. In the present gauge-theoretical analysis, it is assumed that the field of fluid flow is characterized by a translation symmetry (group) and in addition that the fluid itself is a material in motion characterized thermodynamically by mass density and entropy (per unit mass). Local gauge transformation in the present case is local Galilean transformation (without rotation) which is a subgroup of a generalized local Galilean transformation group between non-inertial frames. In complying with the requirement of local gauge invariance of Lagrangians, a gauge-covariant derivative with respect to time is defined by introducing a gauge term. Galilean invariance requires that the covariant derivative should be the convective derivative, i.e. the so-called Lagrange derivative. Using this gauge-covariant operator, a free-field Lagrangian and Lagrangians associated with gauge fields are defined under the gauge symmetry. Euler's equation of motion is derived from the action principle. Simutaneously, the equation of continuity and equation of entropy conservation are derived from the variational principle. It is found that general solution thus obtained is equivalent to the classical Clebsch solution. If entropy of the fluid is non-uniform, the flow will be rotational. However, if the entropy is uniform throughout the space (i.e. homentropic), then the flow field reduces to that of a potential flow. Discussions are given on the issue. From the gauge invariance with respect to translational transformations, a differential conservation law of momentum is deduced as Noether's theorem.

Electromagnetic and gravitational interactions of the spinning particle

We consider the invariance of the spinning free particle Lagrangian under the global coordinate transformations for the classical model of the electron with internal degrees of freedom and obtain the conservation of the energy-momentum, total angular momentum, and electric charge. The local gauge transformations give the electromagnetic and gravitational interactions of the spinning particle in the Riemann-Cartan space from the generalized spin connections. We show that the covariant constancy of the Dirac matrices gives; (i) the form invariance of the classical equations of motion, except the gravitational force terms in nongeodesic equation, (ii) the conservation of the electromagnetic current, (iii) the quantum Hamiltonian and equations of motion from the classical ones without the quantum ordering corrections, and (iv) the minimal coupling of the gravitation with the spinning particle in the Hamiltonian and in wave equations in the Riemann-Cartan space–time.

The Metric of Quantum States Revisited.

A generalised definition of the metric of quantum states is proposed by using the techniques of differential geometry. The metric of quantum state space derived earlier by Anandan, is reproduced and verified here by this generalised definition. The metric of quantum states in the configuration space and its possible geometrical framework is explored. Also, invariance of the metric of quantum states under local gauge transformations, coordinate transformations, and the relativistic transformations is discussed.

The Quantal Symmetries in the Composite Boson's System

The composite Boson's system is quantizated in Faddeev-Senjanovic (FS) path integral quantizated formalism. The canonical Ward identities for proper vertices under local gauge transformation are derived. The fractional spins and fractional statistics are obtained by using the quantum Noether theorem.

QUANTUM FIELD THEORY TOOLS: A MECHANISM OF MASS GENERATION OF GAUGE FIELDS

We present a simple mechanism for mass generation of gauge fields for the Yang–Mills theory, where two gauge SU (N)-connections are introduced to incorporate the mass term. Variations of these two sets of gauge fields compensate each other under local gauge transformations with the local gauge transformations of the matter fields, preserving gauge invariance. In this way the mass term of gauge fields is introduced without violating the local gauge symmetry of the Lagrangian. Because the Lagrangian has strict local gauge symmetry, the model is a renormalizable quantum model. This model, in the appropriate limit, comes from a class of universal Lagrangians which define a new massive Yang–Mills theories without Higgs bosons.

FIELD ALGEBRA, TOPOLOGICAL GAUGE CLASSES AND VACUUM STRUCTURE IN TWO-DIMENSIONAL GAUGE THEORIES WITH FERMIONIC QUARTIC-SELF INTERACTION

We analyze the field algebra, the unitary operators implementing local gauge transformations, the existence of inequivalent gauge classes and the vacuum structure of QED2 generalized to include Thirring-type interactions.

Quantal Symmetries in the Non-Linear Sigma Model with Maxwell–Chern–Simons Term

The quantal symmetry property in the CP1 nonlinear sigma model with Abelian–Maxwell–Chern–Simons (AMCS) term in 2 + 1 dimensions is studied. In the Coulomb gauge, the system is quantized in the Faddeev–Senjanovic (FS) path-integral formalism. The canonical Ward identities for proper vertices under local gauge transformation are derived. Based on the quantal symmetries of a constrained Hamiltonian system, the fractional spin at the quantum level of this system is also presented as those of the system without Maxwell term.

The symmetries in the Maxwell–Chern–Simons theory coupled to matter fields

The Maxwell–Chern–Simons gauge theory coupled to a complex scalar field is quantized in the Becchi–Rouet–Stora–Tyutin path integral formalism. Based on the symmetries of a constrained canonical (Hamiltonian) system, we obtain the quantal conserved angular momentum of the system under the global symmetry transformation. It is shown that fractional spin also appears at the quantum level. The canonical Ward identities for this system are derived under local gauge transformation.