Dependence of physical interpretations on the choice of electromagnetic gauge

The principle of gauge invariance requires that the values of physically measurable quantities will be preserved upon changing the gauge in which electromagnetic quantities are expressed. It is emphasized here that physical interpretations do not depend only on laboratory measurables, but also upon other quantities that are altered by a gauge transformation. It is shown by a variety of simple examples that different gauges can lead to major changes in physical interpretations even though the electromagnetic fields are unaltered. The usual hypothesis that the radiation gauge (also known as the Coulomb gauge) is the “physical” gauge, in the sense that it meets the expectations of a laboratory interpretation, is supported by the various cases considered.     

Spaces of regular gauge field configurations on a lattice and gauge fixing conditions

We consider spaces of lattice gauge field configurations satisfying gauge invariant regularity conditions, and intersections of these spaces with a surface given by gauge fixing conditions. We prove that if these conditions are chosen properly then configurations belonging to the intersection are small and regular.Research supported in part by the National Science Foundation under Grant PHY-82-03669     

Gauge fields and interactions in matrix string theory

It is shown that all possible N sheeted coverings of the cylinder are contained in type IIA matrix string theory as non-trivial gauge field configurations. Using these gauge field configurations as backgrounds the large N limit is shown to lead to the type IIA conformal field theory defined on the corresponding Riemann surfaces. The sum over string diagrams is identified as the sum over non-trivial gauge backgrounds of the SYM theory.     

Statistical mechanics and stability of random lattice field theory

The averaging procedure in the random lattice field theory is studied by viewing it as a statistical mechanics of a system of classical particles. The corresponding thermodynamic phase is shown to determine the random lattice configuration which contributes dominantly to the generating function. The non-abelian gauge theory in four (space plus time) dimensions in the annealed and quenched averaging versions is shown to exist as an ideal classical gas, implying that macroscopically homogeneous configurations dominate the configurational averaging. For the free massless scalar field theory with O(n) global symmetry, in the annealed average, the pressure becomes negative for dimensions greater than two when n exceeds a critical number. This implies that macroscopically inhomogeneous collapsed configurations contribute dominantly. In the quenched averaging, the collapse of the massless scalar field theory is prevented and the system becomes an ideal gas which is at infinite temperature. Our results are obtained using exact scaling analysis. We also show approximately that SU(N) gauge theory collapses for dimensions greater than four in the annealed average. Within the same approximation, the collapse is prevented in the quenched average. We also obtain exact scaling differential equations satisfied by the generating function and physical quantities.     

The interaction of classical Yang-Mills charges

Beginning with Khriplovich's formulation of the two point charge Yang-Mills problem, the gauge freedom is used to reduce by one the number of quantities entering the field equations. While a Coulomb solution is shown to exist for any orientation of the color charges, any solution which has a magnetic field present will have lower energy. When the field equations are solved for the simple charge configurations of parallel plates and cylindrical capacitors, one finds screening due to the vector potential. This screening is shown to be complete for any isolated charged plate or line charge. This result may be an indication that isolated charge cannot occur physically for the classical Yang-Mills theory.     

Topological field patterns in the Yang-Mills theory

We present a formalism where the topological configurations of pure Yang-Mills theory are characterised using gauge fields alone. Here, we obtain an expression for the charges of these topologicalSO(3) gauge field configurations in terms of the Abelian vector potentials. In this formalism we analyse the ’t Hooft-Polyakov monopole solution.     

Cooling of SU(3) lattice gauge field configurations and the η' mass

A method is proposed to calculate the topological susceptibility in SU(3) lattice gauge theory by measuring the asymptotic action of cooled configurations for an ensemble of configurations. The results of a pilot study are presenred.     

Coulomb gauge vacua of SU(3) gauge theory

SU(3) gauge field theory is studied in the Coulomb gauge, and the topologically distinct, but gauge equivalent, vacuum configurations are analysed. Considering the gauge transformations of the form U ε U(2) ?SU(3)/U(2), we have obtained a new class of vacuum fields characterized by the topological quantum number η = ±1.     

High energy behavior of the electromagnetic singlet current in the Glashow-Weinberg-Salam model

The electromagnetic singlet lepton form factors are shown to exhibit “anomalous” short distance behavior which spoils perturbative renormalizability of the physical electromagnetic current. Implications for formulations of gauge theories in terms of gauge invariant quantities are pointed out.     

Topological Charges in Gauge Theories

Topological and geometric aspects of gauge theories are examined. The geometry of the fiber-bundle formulation of gauge theories is discussed and compared with the formalism of general relativity. The basic role played by the parallel displacement operator of this geometry is examined. With this operator a gauge independent characterization of various topological singularities and non-singular soliton configurations is carried out.     

Averaging operations for lattice gauge theories

Usually renormalization group transformations are defined by some averaging operations. In this paper we study such operations for lattice gauge fields and for gauge transformations. We are interested especially in characterizing some classes of field configurations on which the averaging operations are regular (e.g., analytic). These results will be used in subsequent papers on the renormalization group method in lattice gauge theories.Research supported in part by the National Science Foundation under Grant PHY-82-03669     

Fermions in random gauge fields and chiral-symmetry breaking

Random superpositions of gauge fields such that a fermion can propagate in them along the same one-dimensional trajectory in four-dimensional space over arbitrary distances without reduction of the amplitude are considered. Conditions are found under which such structures possess a finite density of fermion zero modes. The possibility of chiral-symmetry breaking in these configurations of gauge fields is explored.     

Instantons in non-Cartesian coordinates

The multi-instanton solutions of ’t Hooft and of Jackiw, Nohl, and Rebbi are generalized to the case of curvilinear coordinates. The resulting formulas are considerably simplified if the transformation of coordinates is supplemented with a gauge transformation. As a result, the gauge field develops a term that has the same form as in Cartesian coordinates and which describes pseudoparticles and a compensating addition of a geometric origin (it is determined by the coordinate frame used). The singularities of the compensating field are irrelevant to physical quantities, but they can affect gauge-dependent quantities.     

Configurations of SO(3) monopoles with multiple magnetic charge

In this paper we consider configurations of monopoles with multiple magnetic charge in nonabelian gauge theory. Explicit gauge transformations are constructed which eliminate the string singularities for single monopoles of multiple magnetic charge, and for an arbitrary number of monopoles lying along a line. The question of finding nonsingular solutions is discussed.     

VII. Difficulties in fixing the gauge in non-Abelian gauge theories

Some problems arising from the use of the Coulomb gauge in SU(2) Yang-Mills theory are discussed. It is shown that: i) the transversality condition does not fix the gauge uniquely (Gribov ambiguity); ii) there exist physical configurations that cannot be described by a continuous A_μ in the Coulomb gauge.     

Global aspects of gauged Wess-Zumino-Witten models

A study of the gauged Wess-Zumino-Witten models is given focusing on the effect of topologically non-trivial configurations of gauge fields. A correlation function is expressed as an integral over a moduli space of holomorphic bundles with quasi-parabolic structure. Two actions of the fundamental group of the gauge group is defined: One on the space of gauge invariant local fields and the other on the moduli spaces. Applying these in the integral expression, we obtain a certain identity which relates correlation functions for configurations of different topologies. It gives an important information on the topological sum for the partition and correlation functions.     

Massless monopoles and multi-pronged strings

We investigate the role of massless magnetic monopoles in the N=4 supersymmetric Yang-Mills Higgs theories. They can appear naturally in the 1/4-BPS dyonic configurations associated with multi-pronged string configurations. Massless magnetic monopoles can carry nonabelian electric charge when their associated gauge symmetry is unbroken. Surprisingly, massless monopoles can also appear even when the gauge symmetry is broken to abelian subgroups.     

Why Gauge?

The world appears to be well described by gauge theories; why? I suggest that gauge is more than mathematical redundancy. Gauge-dependent quantities can not be predicted, but there is a sense in which they can be measured. They describe “handles” though which systems couple: they represent real relational structures to which the experimentalist has access in measurement by supplying one of the relata in the measurement procedure itself. This observation leads to a physical interpretation for the ubiquity of gauge: it is a consequence of a relational structure of physical quantities.     

Quantization of gauge theories

A path-integral procedure for quantizing gauge theories is proposed (on a heuristic level). The Hilbert space of physical states is constructed. Each physical state is represented by an infinite set of gauge equivalent configurations. All physical transition amplitudes are defined. In this approach, the “natural” value of parameter θ is zero.