Continuum Gauge Theory Research Articles

On Multidisciplinary Potential Applications of Gauge Theories

We review how gauge theories, initial introduced for classical mechanics and quantum led theory, seem to apply to many fields of research such as information theory, computer science, economy, biology. This non-exhaustive list raises natural question on future developments of this theory. Short Communication How gauge theories appear in information theory and economics Basically, the mathematical structure of groups is among the best adapted for describing transformations and moves. A path on a group can encode the evolution of a dynamical system, or the moves of an exterior observer with respect to a given system. In the theory of quantum fields, more general objects called principal bundles, which consist in a total space P with fibers isomorphic to a (Lie) group G over a base (simplicial complex or manifold). M For one of the most simple settings, one can see when the base is a manifold (we call them continuum gauge theories) [1-5]. When the base is a simplicial complex (or a lattice), the models are called discrete gauge theories. They stand formally as an integrated version of the previous ones which (more or less formally again). The physical study consists in solving equations on P or after quantization into fields, in minimizing so-called action functionals. These action functionals are in fact acting on connections, which are infinitesimal expressions of local slices. They can be understood also as differential operators. These delicate settings coming from physics have, in the last twenty years, found two applications to our knowledge: In so-called geometric arbitrage theory» for managing portfolios in financial markets [2,6-10], where fiber bundles or principal bundles describe, along their fibers and on the base, the external and internal parameters for the prediction of the evolution of the portfolio. Fiber bundles over a manifold described in [9] are related to principal bundles in a way described in [10]. In so-called approximate reasoning in pairwise comparisons where complete studies can be performed when * G + =  and with normalized objects, with very recent axiomatization [10-19]. However, the need of so-called non-numerical ranking appeals the use of non-abelian groups G and of mathematical tools coming from discrete gauge theories [12,18]. Gauge wraith in biology and neurology The aim of this short article, giving the opinion of the sole author, is to mention that there are some hidden aspects of neurology and biology which reveal, in some kind of matching mirror process, the presence of gauge effects. This aspect is not actually truly confirmed, it is more-or-less at a step of conceptualization, and that is the reason why we talk of gauge wraith. Other authors have already mentioned the similarities of models coming from biology and neuronal science [11,15,16], each time with very mild interpretations of classical facts. From a heuristic viewpoint, this is not so surprising since «physics of life» deal with entities interacting with each other, exactly like elementary particles of nuclear physics interact. At the microscopic scale, a simplicial structure arise naturally by linking an entity (neuron, molecule, cell) with each of its neighbours making discrete gauge theories appropriate, and at macroscopic level, a first approximation of phenomena can rise form continuum gauge theories. However, as a last remark, the mathematical theory of discretization of gauge theories has to be questioned very seriously, because the classical scheme based on leads to mathematical problems while other still non studied ways to discretize a continuum model can be proposed [17].

Remarks on a New Possible Discretization Scheme for Gauge Theories

We propose here a new discretization method for a class continuum gauge theories which action functionnals are polynomials of the curvature. Based on the notion of holonomy, this discretization procedure appears gauge-invariant for discretized analogs of Yang-Mills theories, and hence gauge-fixing is fully rigorous for these discretized action functionnals. Heuristic parts are forwarded to the quantization procedure via Feynman integrals and the meaning of the heuristic infinite dimensional Lebesgue integral is questionned.

Lattice gauge fixing as quenching and the violation of spectral positivity

Lattice Landau gauge and other related lattice gauge fixing schemes are known to violate spectral positivity. The most direct sign of the violation is the rise of the effective mass as a function of distance. The origin of this phenomenon lies in the quenched character of the auxiliary field $g$ used to implement lattice gauge fixing, and is similar to quenched QCD in this respect. This is best studied using the PJLZ formalism, leading to a class of covariant gauges similar to the one-parameter class of covariant gauges commonly used in continuum gauge theories. Soluble models are used to illustrate the origin of the violation of spectral positivity. The phase diagram of the lattice theory, as a function of the gauge coupling $\beta$ and the gauge-fixing parameter $\alpha$, is similar to that of the unquenched theory, a Higgs model of a type first studied by Fradkin and Shenker. The gluon propagator is interpreted as yielding bound states in the confined phase, and a mixture of fundamental particles in the Higgs phase, but lattice simulation shows the two phases are connected. Gauge field propagators from the simulation of an SU(2) lattice gauge theory on a $20^4$ lattice are well described by a quenched mass-mixing model. The mass of the lightest state, which we interpret as the gluon mass, appears to be independent of $\alpha$ for sufficiently large $\alpha$.

Test of the Kugo-Ojima confinement criterion in the lattice landau gauge

We present the first results of numerical test of the Kugo-Ojima confinement criterion in the lattice Landau gauge. The Kugo-Ojima criterion of colour confinement in the BRS formulation of the continuum gauge theory is given by u b a (0) = − δ b a , where ∫dXe ip(x−)〈O|TD μc a(X)g(A v× c) b(Y)|O〉=(g μv P μP v p 2 )U b a(p 2). (∗) We measured the lattice version of u b a (0) in use of 1/(− ∂D ( A)) where D μ ( A) is a lattice covariant derivative in the new definition of the gauge field as U = e A . We obtained that u b a (0) is consistent with − cδ b a , c = 0.7 in SU(3) quenched simulation data of β = 5.5, on 8 4 and 12 4. We report the β dependence and finite-size effect of c.

Existence of a confinement phase in quantum electrodynamics

We show that the four-dimensional U(1) gauge theory in the continuum formulation has a confining phase (exhibiting area law of the Wilson loop) in the strong coupling region above a critical coupling $g_c$. This result is obtained by taking into account topological non-trivial sectors in U(1) gauge theory. The derivation is based on the reformulation of gauge theory as a deformation of a topological quantum field theory and subsequent dimensional reduction of the D-dimensional topological quantum field theory to the (D-2)-dimensional nonlinear sigma model. The topological quantum field theory part of the four-dimensional U(1) gauge theory is exactly equivalent to the two-dimensional O(2) nonlinear sigma model. The confining (resp. Coulomb) phase of U(1) gauge theory corresponds to the high (resp. low) temperature phase of O(2) nonlinear sigma model and the critical point $g_c$ is determined by the Berezinskii-Kosterlitz-Thouless phase transition temperature. The quark (charge) confinement in the strong coupling phase is caused by the vortex condensation. Thus the continuum gauge theory has the direct correspondence to the compact formulation of lattice gauge theory.

Instantons in the Maximally Abelian gauge

We investigate the Maximally Abelian (MA) Projection for a single SU(2) instanton in continuum gauge theory. We find that there is a class of solutions to the differential MA gauge condition with circular monopole loops of radius R centered on the instanton of width ϱ. However, the MA gauge fixing functional G decreases monotonically as R/ ϱ → 0. Its global minimum is the instanton in the singular gauge. We point out that interactions with nearby anti-instantons are likely to excite these monopole loops.

Vacuum polarization and chiral lattice fermions

The vacuum polarization due to chiral fermions on a 4-dimensional Euclidean lattice is calculated according to the overlap prescription. The fermions are coupled to weak and slowly varying background gauge and Higgs fields, and the polarization tensor is given by second order perturbation theory. In this order the overlap constitutes a gauge-invariant regularization of the fermion vacuum amplitude. Its low-energy-long-wavelength behaviour can be computed explicitly and we verify that it coincides with the Feynman graph result obtainable, for example, by dimensional regularization of continuum gauge theory. In particular, the Standard Model Callan-Symanzik RG functions are recovered. Moreover, there are no residual lattice artefacts such as a dependence on Wilson-type mass parameters.

Fermion induced SU( N) Yang-Mills theory

We investigate the gauge interaction induced by heavy fermions using both dimensional and lattice regularization. We study the condition under which heavy fermions induce a continuum gauge theory.

Son of gauge fixing on the lattice

The gauge fixing (Gribov) ambiguity on the lattice is investigated in simple models. The global understanding of this phenomenon comes from singer's analysis of the bundle of connections. This bundle in continuum gauge theory is infinite dimensional and of non-trivial topology, however on the lattice it is finite dimensional and tractable. Theoretical expectations from Morse theory are supported by numerical evidence, and the structure of lattice gauge orbit space is displayed. Finally, shortcomings of standard gauge fixing prescriptions are addressed.

Phase-transition behavior of the continuum SU(2) gauge theory at finite temperature.

The effective systems corresponding to the infrared sector of the original continuum gauge theory are studied. The emphasis is on the phase-transition behavior. The effective potential is shown to display the features known in the Ginzburg-Landau theory. The phase transition depends on the value of the (renormalized) coupling constant. The numerical value of the critical coupling or the critical temperature is in agreement with lattice calculations.

Plaquette percolation for lattice gauge theory

A new Monte Carlo dynamics is proposed for Z 2 lattice gauge based on the percolation of spanning surfaces, analogous to the percolation of clusters used in the Swendsen-Wang algorithm for the Potts model. In three dimensions, the resulting dynamical exponent for critical slowing down is reduce to z = 0.61 ± 0.05 relative to the heat bath exponent of z = 2.1 ± 0.2. Extensions to continuum gauge theories in higher dimensions are discussed.

Gauge theory and general relativity

The purpose of this paper is to prove that most of the differential geometry of mathematical physics today is based on a confusion between two differential sequences, namely, the Janet sequence (1920) and the Spencer sequence (1970) that can be found in the formal theory of systems of partial differential equations and Lie pseudogroups. Correcting this confusion by means of new homological methods (diagram chasing) brings at once the need for visiting again the foundations of well established theories, including continuum mechanics, gauge theory and general relativity. In particular, we give for the first time a modern exposition of E. and F. Cosserat elasticity theory along the work of D. C. Spencer, while extending it to electromagnetism along the idea of H. Weyl for using the conformal group of space-time. Finally, exactly as Newton theory of gravitation is not coherent with pseudogroup theory. More precisely, Yang-Mills equations and Einstein equations are not coherent with Spencer sequence!

Gauge theory of defects in the elastic continuum

A gauge theory of defects in an elastic continuum is developed after providing the necessary background in continuum elasticity and gauge theories. The gauge group is the three-dimensional (3D) Euclidean group [semi-direct product of the translation group T (3) with the rotation group SO (3)]. We obtainboth dislocations and disclinations by breaking of the translational invariance. Breaking of the rotational invariance is shownnot to lead to any interesting effects in a linear analysis. These results are shown to be consistent with the topological analysis which is briefly discussed at the end of the paper. Any defect given by the present theory acquires acore which removes the singularity of the stress field at the origin. The stress field agrees with the continuum result asymptotically, as is expected. Geometrical aspects of the deformed state of condensed matter are also briefly touched upon.

Non-Grassmann formulation of regularized gauge theory with fermions

Regularized continuum gauge theory coupled to quadratic matter simplifies significantly on integration of the matter fields. As an illustration, we discuss in some detail the resulting non-Grassmann formulation of regularized gauge theory with Dirac fermions.

Evidence against asymptotic freedom from SU(2) lattice gauge theory.

A recent result in continuum Yang-Mills theories that exposes the ''essential instability'' of the perturbative ground state is shown to be reproduced by a Monte Carlo SU(2) lattice calculation. A discussion is given of the adverse consequences of our findings upon the validity of asymptotic freedom and the reliability of the lattice strategy for computing the physics of continuum gauge theories.

Continuum regularization of QCD

We introduce a new stochastic regularization for continuum gauge theory. The scheme is a covariant derivative regularization of the Parisi-Wu Langevin equation, or equivalently, the Schwinger-Dyson equations. The regularized formulation is manifestly Lorentz invariant, gauge-invariant, ghost-free and finite to all orders. We verify a vanishing gluon mass at one loop.

Surface description of gauge theories

The partition function of non-abelian gauge theory is expressed, in the continuum limit, as a sum over surfaces which are swept out by the propagation of electric flux rings. Each flux surface is described by a two-dimensional continuum gauge theory, confined to that particular surface. The gauge field can then be integrated out; however, for closed and intersecting surfaces interesting complications arise, which reveal an algebraic structure typical of strong-coupling lattice gauge theory.

Plaquette formulation and the Bianchi identity for lattice gauge theories

We extend Halpern's field-strength formulation and dual potentials (for continuum gauge theories) to abelian and non-abelian lattice gauge theories. New results include: (i) plaquette formulation of all lattice gauge theories, (ii) the strong coupling expansion is seen as (a) a perturbation in dual links or (b) a gradual restoration of the lattice Bianchi identity. To leading order in the strong coupling expansion the lattice Bianchi identity is completely ignored. Geometrical interpretation of the lattice Bianchi identity is presented along with a discussion of the “abelianization” of the non-abelian identity and its connection with gauge-invariant variables. For abelian theories we also show that the dual potential is Fourier conjugate to the Bianchi identity and that the Coulomb gas representation of these theories is easily obtained in this formulation.

A quenched momentum prescription for large- N theories

A general quenched momentum prescription is presented for treating theories with a global, or local gauge, U( N) symmetry, both on the lattice and in the continuum, in the limit of infinite N. It is shown that the quenched theory produces, to all orders in perturbation theory, the integrands of the standard Feynman diagrams for invariant Green functions. In the case of gauge theories the quenching of the momentum must be accompanied by a constraint on the eigenvalues of the covariant derivative. In ultraviolet finite theories integration over the values of the quenched momenta reproduces ordinary perturbation theory, and it is suggested that for theories with logarithmic ultraviolet divergences a quenched momentum cutoff can be used a gauge-invariant regularization. The quenched lattice gauge theory is shown to be equivalent to the quenched Eguchi-Kawai model, and is proven to be equivalent, to all orders in perturbation theory, to the standard model. The loop equations of this model are derived, shown to be true equations of the standard model, and are argued to have a smooth continuum limit. The coupling constant renormalization is calculated in the quenched continuum gauge theory to one-loop order, and the formal loop equations are derived. Application of the method to numerical calculations of continuum QCD are discussed, as well as the possibility of achieving a gauge-invariant continuum regularization of chiral gauge and supersymmetric theories.

Order parameters in a modified lattice gauge theory

Mack and Petkova, and Yaffe have recently considered a modified lattice gauge theory, in which the 't Hooft operator has an unexpected area law and is not complementary to the Wilson loop operator. We find that this occurs because the modified theory has a conserved flux not present in the usual lattice theory; as a result, it has two pairs of dual order parameters rather than one. The 't Hooft operator, which probes the physics of interest in the weak-coupling, continuum limit, and which is complementary to the Wilson operator, is not the one considered by the earlier authors. We discuss the order parameters at weak and strong coupling, the corresponding topological fluxes, and the implications for other (lattice or continuum) gauge theories.