non-Abelian Gauge Field Theory Research Articles

Local non-abelian gauge symmetry of the Hubbard model

It is shown that the field operators of an electron system on a lattice can be decomposed into direct products of two kinds of operators acting in two separate Hilbert spaces. The Hilbert space of electron states thus becomes a direct product of two Hilbert spaces. By this fact a certain class of electron systems exhibits a formal separation of charge and spin degrees of freedom into two kinds of elementary excitations. A typical example of such a system is given by the Hubbard model. The separation of charge and spin resulting from the new representation of the field operators can be considered as a rigorous realization and generalization of an idea expressed by Anderson concerning the separation of spin and charge degrees of freedom in strongly correlated electron systems. The new representation of electron field operators implies the existence of a localU(2) gauge symmetry in the theory. The theory of superconductivity based on the Hubbard model is then represented by a non-abelian gauge field theory.

Fractal Wilson Loop and confinement in non-compact non-abelian gauge field theory

We discuss properties of the recently proposed fractal Wilson Loop F P as order parameter in non-compact gauge theories. For SU(2), it yields to lowest order of strong coupling expansion an area law, gauge invariance, and has a static quark potential interpretation. For U(1), it yields a perimeter law for all coupling.

On the stability of the Yang-Mills vacuum in the field strength approach

The Yang-Mills sector of non-abelian gauge field theories is reformulated as an effective theory in the field strength. The emerging effective action has non-trivial stationary points describing a homogeneous phase of condensed gauge bosons which corresponds to the non-perturbative vacuum. This vacuum solution is explicitly found in D=3 dimension for the gauge group SU (2). The stability of this Yang-Mills vacuum is investigated. It is found that it is stable at low energies but becomes unstable at sufficient high energies. The critical energy is basically determined by the QCD scale parameter Λ.

Structure of the Gribov horizon for the non-abelian gauge field theory in axial gauges

It is shown that the usual axial gauges, for example A 3=0, are inappropriate for the non-abelian gauge field theory in the periodic boundary condition: This type of gauge-fixing surface does not intercept all gauge orbits. For extended axial gauges, for example ϖA 3 ϖx 3 =0 , the Faddeev-Popov determinant is explicitly calculated and the structure of the Gribov horizon is made clear.

Hamiltonian formulation of QCD in the Schwinger gauge.

The structure of the Hamiltonian related to a regularized non-Abelian gauge field theory is discussed in the light of different choices for gauge-invariant wave functionals (loop space, Coulomb, axial, Schwinger gauge). Arguments are given for the suggestion that the Schwinger gauge offers a specially suited framework for the computation of bound-state (hadron) properties. The most important reasons are the manifest rotation invariance, the lack of a Gribov horizon (giving standard many-body techniques a better chance), and the fact that a regularization analogous to the lattice regularization is easily implementable. Some details of the Schwinger-gauge Hamiltonian theory are discussed.

Non-abelian gauge couplings in thermal gauge field theories

Properties of the effective gauge couplings renormalized at finite temperature and density in thermal non-abelian gauge field theories are studied within one-loop approximations. Strong and severe vertex dependence is shown to come out both in the temperature and chemical potential dependences. Difficulties appearing in the perturbative calculation of physical quantities, indicated by the above disaster, are discussed. Also discussed is what insight might be gained from the present analysis into the “magnetic” screening of effective charge.

Aspects of QED and non-Abelian gauge theories in S1

We study various aspects of Abelian and non-Abelian gauge field theories in flat spacetime with the topology of ${S}^{1}$\ifmmode\times\else\texttimes\fi{}${R}^{n}$ (n=3,4). We first discuss the effective potential for the electromagnetic field in ${S}^{1}$\ifmmode\times\else\texttimes\fi{}${R}^{3}$ spacetime, with the objective of finding the dependence of coupling constants on the size L of the compactified dimension. It is first necessary to determine the stable vacuum. In particular, a charged scalar field satisfies an effective periodic boundary condition, while a charged spinor field satisfies an effective antiperiodic boundary condition at the stable vacuum. Then we study the effective coupling constants which depend on the size L of the compactified dimension at one-loop level for QED in ${S}^{1}$\ifmmode\times\else\texttimes\fi{}${R}^{3}$ spacetime. In the presence of a charged scalar field, the one-loop correction of one of the effective coupling constants for the lowest Fourier component of the electromagnetic field behaves for small L like 1/L, rather than lnL. This is because the lowest Fourier component of the scalar field is constant in the ${S}^{1}$ direction. The effective coupling constants for higher Fourier components have logarithmic L dependence, but due to collinear singularities (i.e., singularities in self-energy occurring in the limit of zero mass when the virtual particles are on shell and have parallel momenta) the coefficient of the lnL term differs from component to component. We also point out some ambiguities in defining the effective coupling constants. Finally we discuss the effective potential of non-Abelian gauge theories with a charged spinor field in ${S}^{1}$\ifmmode\times\else\texttimes\fi{}${R}^{4}$ spacetime. We find that if periodic boundary conditions are imposed on the spinor field in the beginning, the local gauge symmetry will be spontaneously broken in some cases, and we give an explicit model in which this occurs. This spontaneous symmetry breaking depends on the gauge group and its representations, and does not occur for an Abelian gauge field.

The self-dual closed bosonic membranes

Abstract The free string cannot change by itself its own topology. For membranes non-linearities cannot be gauged away. Therefore “free” membranes contain interactions and can change their own topology. This property is analogous to non-abelian gauge field theories and gives the opportunity to introduce the notion of self-duality for the closed bosonic membrane.

Background field method in stochastic quantization

The background field approach to non-abelian gauge field theory in stochastic quantization is presented. We discuss on the basis of the Zwanziger-type stochastic gauge fixing, in which we need not introduce the Faddeev-Popov ghost field. We can choose the Zwanziger function, and thus the Langevin equation, to be manifestly covariant under the background gauge transformation. Due to the background gauge covariance, we can easily find a simple Ward identity by which the β-function calculation in stochastic quantization can be greatly simplified. We give the one-loop result of the β-function which is equivalent to the normal one. We also give another β-function, β γ, which will give the relation between the energy scale and the fictitious time scale.

Renormalization of Path Ordered Phase Factors and Related Hadron Operators in Gauge Field Theories

AbstractWe give a review of the renormalization and short distance properties of path ordered phase factors in nonabelian gauge field theories. It includes nonlocal gauge invariant meson, baryon and gluonium operators constructed with the help of such phase factors. Furthermore, the renormalization properties of functional derivatives of phase factors as they are needed in dynamical equations are considered. The discussion is based on an one dimensional auxiliary field formalism which enables the application of the usual language of local Green's functions.

Операторные калибровочные преобразования в нерелятивистской квантовой электродинамике Применения к мультипольному Гамильтониану

Operator gauge transformations on the electromagnetic potentials in quantum electrodynamics are obtained by requiring form invariance of the Schrodinger equation of a single particle under arbitrary space- and time-dependent unitary transformations. These operator gauge transformations are found to have the same form as in non-Abelian gauge field theories. By considering the Lorentz force on the particle, we show that the definition of the electromagnetic-field strength operator in terms of the potentials must also have the same form as in non-Abelian gauge field theory. Since the Hamiltonian is gauge dependent, a distinction is made between the Hamiltonian and the gauge-invariant energy operator of the particle. Local conservation of energy for the particle is derived and it is shown that the particle gains (loses) energy at the same rate as expected classically. An operator gauge transformation is made to obtain potentials in the multipolar gauge and the multipolar Hamiltonian is shown to be the minimally coupled Hamiltonian in the multipolar gauge.

Monopole mass in supersymmetric gauge theories

The one-loop quantum corrections to the mass of a monopole in a non-abelian supersymmetric gauge field theory are calculated. The corrections due to Bose and Fermi fluctuations around the monopole are shown to the governed essentially by the eigenvalues of two related second order differential operators, with the same nonzero mode spectrum but different densities of the continuum eigenmodes. However, the one-loop monopole mass, when expressed in terms of one-loop renormalized parameters, does not obtain any corrections.

Propagators and renormalization transformations for lattice gauge theories. I

Lattice gauge theories may be looked at as perturbations of the theory of a vector field with a Gaussian action. We study this theory here and in following papers obtaining crucial results for understanding the renormalization group method in more complicated non-Abelian gauge field theories.

Theory and renormalization of the gauge-invariant effective action

The different methods for constructing a gauge-invariant effective action (GIEA) for quantum non-Abelian gauge field theories proposed by 't Hooft, DeWitt, Boulware, and Abbott are all shown to be equivalent. In the course of proving this equivalence we show how to extend the usual background-field method so as to construct what may be considered the prototypical GIEA and discuss in some detail the invariance and gauge transformation properties of both the usual theory and the new theory using the GIEA. All solutions to the GIEA field equations are shown to be physical---being solutions to the usual field equations with an arbitrary gauge condition. The renormalization program based upon the GIEA is shown to differ from the standard theory and we outline the modifications which are needed in the present proof of renormalizability. In particular we prove that the physical renormalization is independent of any gauge-fixing choice. Finally, we prove that the $S$-matrix elements derived from the GIEA for an arbitrary background-field solution to the field equations are the same as those derived using the usual effective action.

Renormalization and scaling behavior of non-Abelian gauge fields in curved spacetime

In this article we discuss the one loop renormalization and scaling behavior of non-Abelian gauge field theories in a general curved spacetime. A generating functional is constructed which forms the basis for both the perturbation expansion and the Ward identities. Local momentum space representations for the vector and ghost particles are developed and used to extract the divergent parts of Feynman integrals. The one loop diagram for the ghost propagator and the vector-ghost vertex are shown to have no divergences not present in Minkowski space. The Ward identities insure that this is true for the vector propagator as well. It is shown that the above renormalizations render the three- and four-vector vertices finite. Finally, a renormalization group equation valid in curved spacetimes is derived. Its solution is given and the theory is shown to be asymptotically free as in Minkowski space.

Planar rotor: Theθ-vacuum structure, and some approximate methods in quantum mechanics

The quantum planar rotor is chosen as a model to test different approximate techniques, emphasizing the analogies between this simple system and the non-Abelian quantum gauge field theories (instantons, $\ensuremath{\theta}$ vacuum, etc.). In particular, variational and perturbative methods, path-integral techniques, and Monte Carlo simulations are applied. It is pointed out that the maximal destructive interference between instantons takes place in the $\ensuremath{\theta}$-vacuum realization of the interacting rotor system with $\ensuremath{\theta}=\frac{1}{2}$, where the tunneling effects are shown to be severely diminished.

Approaches to a non-abelian antisymmetric tensor gauge field theory

Two distinct attempts at constructing a theory of non-abelian antisymmetric tensor gauge fields (ATGF's) are considered. First, a recently proposed geometry of abelian ATGF's is reviewed and then generalized to the non-abelian case. The resulting geometric action is non-local and is invariant under non-local gauge transformations; in the local limit the action describes free fields. Lattice actions for both the abelian and non-abelian ATGF theories are also presented. In the second approach, a lattice action for non-abelian ATGF's is constructed using a plaquette variables that carry four internal indices. The continuum limit is also a non-interacting theory.

Canonical operator theory of non-Abelian gauge fields

Non-Abelian gauge theory in a manifestly covariant gauge is formulated as a theory of canonical field operators and embedded in an indefinite metric space. A gauge-fixing field is included and every field component has a nonvanishing adjoint momentum with which it has canonical commutation (or anticommutation) relations. Faddeev-Popov fields are represented as scalar fermion fields with ghost particle excitations. Feynman rules are derived from the canonical formulation. A discussion is given of the relation between the existence of a subsidiary condition and the existence of « pure gauge » states that dynamically detach from observable states.

Quantization of fields on principal bundles

A constructive method of quantization of free gauge fields is presented. In particular, by using a geometric approach (classical gauge fields as connections on principal fibre bundles) and the Borchers-Uhlmann method, an algebraic quantum field theory of gauge fields is constructed. Thus, a linearized version of a quantum field theory of non-Abelian gauge fields is obtained generalizing Bongaarts' results for the Abelian U(1)-theory.

Quarkonium decays: Testing the 3-gluon vertex

We study the 3-jet decays of S- and P-wave quarkonia with C=+. If observed, some of these will offer a way of seeing the 3G vertex of QCD via 1S 0, 3P 0, 3P 2(Q Q ) → GGG + Gq q → 3 jets. (As is well-known, cancellations reduce 3 P 1( Q Q )→ GGG.) We elaborate in detail the S-wave decay, as it is expected to show all the characteristic features of orthoquarkonium decays into 4 jets, 3S 1(Q Q ) → GGGG + GGq q → 4 jets, which we will comment upon. These quarkonium decays offer a very clear signal for QCD as a non-abelian local gauge field theory with color-charged gluons.