Gauge invariant geometric variables for Yang-Mills theory

In a previous publication, local gauge invariant geometric variables were introduced to describe the physical Hilbert space of Yang-Mills theory. In these variables, the electric energy involves the inverse of an operator which can generically have zero-modes, and thus its calculation is subtle. In the present work, we resolve these subtleties by considering a small deformation in the definition of these variables, which in the end is removed. The case of spherical configurations of the gauge invariant variables is treated in detail, as well as the inclusion of infinitely heavy point color sources, and the expression for the associated electric field is found explicitly. These spherical geometries are seen to correspond to the spatial components of instanton configurations. The related geometries corresponding to Wu-Yang monopoles and merons are also identified.     

Geometric regularization of Yang-Mills theory

A continuum regularization of Yang-Mills theory with a possible non-perturbative interpretation is introduced in the euclidean formalism. The regularization preserves Lorentz covariance and gauge invariance and involves not only a regularization of the action but also a regularization of the functional measure in terms of a nuclear-riemannian structure of the space of gauge orbits.     

Exact solutions of supergauge invariant Yang-Mills equations

We point our a new class of solutions of the supersymmetric Yang-Mills equations. This class provides solutions which cannot be generated from the solutions of the ordinary Yang-Mills equations by finite supersymmetry transformations and contains the supersymmetric generalization of the non-abelian plane waves.     

规范,相位因子和杨-米尔斯场--规范场理论在中国续记

文章是在杨-米尔斯场50周年学术报告会上的报告的精简部分.它和作者发表的一组评述互相补充,目的是引介非亚贝尔规范场和量子不可积相位因子,解说作为20世纪物理学主旋律之一的相位因子的物理意义.     

Analogue of Black Strings in Yang-Mills Gauge Theory

The classical Yang–Mills equations areanalyzed within the geometrical framework of aneffective gravity theory. Exact analytical solutions arederived for the cylindrically symmetric configurationsof the coupled gauge and isoscalar fields. Itturns out that there is an infinite family of solutionsparametrized by two real parameters, one of whichdetermines the asymptotic behavior of fields near the symmetry axis and in infinity, while the secondlocates the singularity. These configurations have asimple pole at a finite value of the radial coordinate,and physically they represent thickstring-like objects which possess the confinementproperties. It is demonstrated that the particles withgauge charge cannot move classically and quantummechanically out of the interior region. Such objectsare thus direct analogues of the blackstring gravitational configurations reportedrecently in the literature.     

Gauge invariant fractional electromagnetic fields

Fractional derivatives and integrations of non-integers orders was introduced more than three centuries ago but only recently gained more attention due to its application on nonlocal phenomenas. In this context, several formulations of fractional electromagnetic fields was proposed, but all these theories suffer from the absence of an effective fractional vector calculus, and in general are non-causal or spatially asymmetric. In order to deal with these difficulties, we propose a spatially symmetric and causal gauge invariant fractional electromagnetic field from a Lagrangian formulation. From our fractional Maxwell?s fields arose a definition for the fractional gradient, divergent and curl operators.     

Conformally invariant Yang-Mills theory realizing the Virasoro-Kac-Moody algebra

We will discuss a conformally invariant Yang-Mills theory in two dimensions, which realizes the Virasoro-Kac-Moody algebra as the BRST symmetry. The theory will have the Virasoro anomaly with the central charge −26, but no anomaly for the Kac-Moody algebra.     

Symmetries of SU(2) invariant Yang-Mills theories

It is shown that both the field equations and the self-duality equations of SU(2) invariant Yang-Mills theories do not allow any other Lie symmetries than those they have by construction, i.e., the 15 generators of the conformal group and the 3 generators of the gauge group. This is true in Euclidean as well in Minkowskian space-time.     

Gauge invariant field theory of free strings

We present the gauge invariant and covariant equations of motion for free strings and superstrings, using a simple method of successive gauge transformations. Gauge invariance is enforced by introducing both subsidiary and Stueckelberg fields. We use methods of cohomology associated with the various Virasoro algebras, and derive the cohomology algebras associated with the boson and fermion sector of the open and closed superstrings.     

Massive Yang-Mills fields: Gauge invariance and one-loop counterterm

One-loop counterterms for the massive Yang-Mills lagrangian are computed, using the background field method. Gauge invariance off-mass-shell and an invariant loop expansion scheme are discussed in detail.     

A Generalized Yang-Mills Model and Dynamical Breaking of Gauge Symmetry

A generalized Yang-Mills model, which contains, besides the vector part V_μ, also a scalar part S, is constructed and the dynamical breaking of gauge symmetry in the model is also discussed. It is shown, in terms of Nambu-Jona-Lasinio (NJL) mechanism, that the gauge symmetry breaking can be realized dynamically in the generalized Yang-Mills model. The combination of the generalized Yang-Mills model and the NJL mechanism provides a way to overcome the difficulties related to the Higgs field and the Higgs mechanism in the usual spontaneous symmetry breaking theory.     

Maximally Generalized Yang-Mills Model and Dynamical Breaking of Gauge Symmetry

A maximally generalized Yang-Mills model, which contains, besides the vector part V_μ, also an axial-vector part A_μ, a scalar part S, a pseudoscalar part P, and a tensor part T_μν, is constructed and the dynamical breaking of gauge symmetry in the model is also discussed. It is shown, in terms of the Nambu-Jona-Lasinio mechanism, that the gauge symmetry breaking can be realized dynamically in the maximally generalized Yang-Mills model. The combination of the maximally generalized Yang-Mills model and the NJL mechanism provides a way to overcome the difficulties related to the Higgs field and the Higgs mechanism in the usual spontaneous symmetry breaking theory.     

On the invariant regularization of the standard model

We show that the gauge invariant regularization of the Standard Model proposed by Frolov and Slavnov describes a nonlocal theory with quite simple Lagrangian.     

Gauge invariant electromagnetic coupling with torsion potential

Electromagnetism is coupled to torsion in a gauge invariant manner by relaxing minimal coupling and introducing into the Lagrangian a term bilinear in the electromagnetic field tensor and the torsion potential. The resulting coupling between electromagnetism and torsion is examined and a solution corresponding to traveling coupled waves is given.     

ζ-function regularization of the quantum fluctuations around the Yang-Mills pseudoparticle

We use the hypersphere stereographic projection and the ζ-function regularization procedure to compute the one loop correction around the Yang-Mills pseudoparticle with scalars and fermions in an arbitrary representation of the SU(2) gauge group.     

Gauge transformations and quantum mechanics I. Gauge invariant interpretation of quantum mechanics

Quantum mechanical operators are interpreted according to their equations of motion. Operators representing physical quantities which have classical analog are constructed by requiring that the quantum and the classical (i.e., Newtonian) equations of motion have a term by term correspondence. Of special importance to the interpretation of quantum mechanics is the particle's energy operator. In the presence of a time-varying electric fieldE, the particle's energy operator is constructed so that its time derivative is the power operatorJ · E (J being the current operator). This interpretation of operators, such as the particle's energy operator, is gauge invariant despite the possible explicit dependence on electromagnetic potentials of the operators concerned. A gauge invariant interpretation of quantum mechanics is obtained by expanding the wave-function (in an arbitrary gauge) in the orthonormal set of eigenfunctions of the particle's energy operator (in the same gauge) and by interpreting the resulting expansion coefficients as probability amplitudes. This formulation possesses all the traditional gauge freedom and contains no gauge ambiguity. (Here, by gauge invariance we also mean that the dependence on paths in the DeWitt-Mandelstam formalism and on the procedures for path averaging in the Belinfante-Rohrlich-Strocchi formalism does not occur.) In particular, probability amplitudes and transition matrix elements are gauge invariant, and the transition matrix elements between states of different energies are proportional to the corresponding matrix elements of J · E, rather than J_μA_μ. Lamb found experimental evidence that led to the conclusion that “the usual interpretation of probability amplitudes” was gauge dependent and was correct only in the gauge in which the interaction Hamiltonian was of the form of the electric dipole interaction −er · E(0, t), instead of the usual −eA(0, t) ·p/mc. It is shown here that the gauge invariant formulation for bound systems derives the electric dipole interaction in any arbitrary gauge as the result of the long wavelength and lowest order approximation of fields. For a quantum system interacting with a precessing magnetic field, the Güttinger-Schwinger procedure of quantizing the system along the instantaneous magnetic field has been known to yield the correct transition probabilities during the interaction. This quantization procedure follows directly from the gauge invariant formulation. The electric and the magnetic multipole interactions appearing in the gauge invariant formulation directly correspond to terms in the classical Poynting theorem. The gauge invariant magnetic multipole interactions differ from their counterparts in the conventional formalism. For example, the gauge invariant magnetic dipole interaction involves the time derivative of the magnetic field. This result is shown to be consistent with the Poynting theorem. Although the gauge invariant interpretive scheme proposed here is formulated for a nonrelativistic, spinless charged particle, the extension to the Dirac equation is straightforward.     

Elliptic-type solutions to the scale invariant Yang-Mills and sigma-model hierarchies

We outline the construction of non-self-dual elliptic solutions by relating the spherically symmetric subsystem of the (scale invariant) Yang-Mills and sigmamodel hierarchies to the hierarchies of ⁴ and Sine-Gordon models in one dimension respectively. The construction is carried out explicitly for the usual Yang-Mills model on ₄, and the first two sigma-models on ₂ and ₄. The solution to the first member of the Yang-Mills hierarchy is related to elliptic solutions found previously.     

Gauge invariant Green's function in the bloch-nordsieck model

The gauge invariant two-point Fermion Green's function is calculated within the framework of the Bloch-Nordsieck approximation. Its dependence on the contour is investigated. It is shown that in the case of a straight contour the gaugeinvariant fermion propagator is equal to the free one.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 65–70, January, 1985.In conclusion the author thanks V. I. Savrin and N. B. Skachkov for interest in this work and for valuable remarks, and also A. A. Afonin, V. N. Kapshai, and O. P. Solovtsova for useful discussions of the results.     

Gauge invariant green function in the Schwinger model

In two-dimensional quantum electrodynamics with massless fermions (Schwinger model), the gauge invariant one-particle fermion Green function is computed. It is shown that this Green function is independent of the choice of integration contour in the gauge exponential, and coincides with the free fermionic propagator.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 49–52, December, 1984.The authors are deeply grateful to V. B. Belyaev and N. B. Skachkov for their stimulating interest in this work and useful comments.