Residual gauge invariance of Hamiltonian lattice gauge theories

The time-independent residual gauge invariance of Hamiltonian lattice gauge theories is considered. Eigenvalues and eigenfunctions of the unperturbed Hamiltonian are found in terms of Gegenbauer's polynomials. Physical states which satisfy the subsidiary condition corresponding to Gauss' law are constructed systematically.     

Anomalous commutators in two-dimensional chiral gauge theories

It is shown that in two-dimensional models of chiral fermions coupled with background gauge fields, anomalous commutators of currents as well as of generators of time-independent gauge transformations are completely determined by the anomaly if the gauge group is semisimple. The four-dimensional case is commented on.     

Anomalous commutators in vector and chiral gauge theories

A fixed-time point-splitting regularisation employing suitable phase factors is used to derive various anomalous commutators - both in usual vector gauge theories and non-abelian chiral gauge theories in four dimensions. In the latter case, new structures for the commutators involving currents have been found. Consistency of these expressions is established by reproducing some familiar results.     

Symmetries in constrained Hamiltonian systems

We derive an algorithm for the construction of all the gauge generators of a constrained hamiltonian theory. Dirac's conjecture that all secondary first-class constraints generate symmetries is revisited and replaced by a theorem. The algorithm is applied to Yang-Mills theories and metric gravity, and we find generators which operate on the complete set of canonical variables, thus producing the correct transformation laws also for the unphysical coordinates. Finally we discuss the general structure of the Hamiltonian for constrained theories. We show how in most cases one can read off the first-class constraints directly from the Hamiltonian.     

Guided random walks for solving Hamiltonian lattice gauge theories

Motivated by developments for many-particle quantum systems, a Monte Carlo method for solving Hamiltonian lattice gauge theories without fermions is presented in which a stochastic random walk is guided by a trial wave function. To the extent that a substantial portion of the local structure of the theory can be incorporated in the trial function, the method offers significant advantages relative to existing techniques. The method is applicable to the study of SU(N) lattice gauge theories, and its utility is demonstrated by solving the compact U(1) gauge theory in three spatial dimensions.     

Anomalous cross section for photon-photon scattering in gauge theories

We consider the deep inelastic structure functions of the photon in an asymptotically free gauge theory. In contrast to the case of a hadronic target, we find that the shortdistance analysis determines the shape and magnitude and not merely the Q² dependence of the structure functions. The structure functions of the free quark theory are renormalized by finite, calculable factors. For example, at x = 0.1, we find that F₂ will, at large Q², exceed the free quark result by a factor 1.751, while for x = 0.5, F₂ is suppressed asymptotically, relative to the free quark theory, by a factor 0.964, and at x = 0.8, by a factor 0.611.     

Classical gauge theories as dynamical systems—Regularity and chaos

In this review we present the salient features of dynamical chaos in classical gauge theories with spatially homogeneous fields. The chaotic behaviour displayed by both abelian and non-abelian gauge theories and the effect of the Higgs term in both cases are discussed. The role of the Chern-Simons term in these theories is examined in detail. Whereas, in the abelian case, the pure Chern-Simons-Higgs system is integrable, addition of the Maxwell term renders the system chaotic. In contrast, the non-abelian Chern-Simons-Higgs system is chaotic both in the presence and the absence of the Yang-Mills term. We support our conclusions with numerical studies on plots of phase trajectories and Lyapunov exponents. Analytical tests of integrability such as the Painlevé criterion are carried out for these theories. The role of the various terms in the Hamiltonians for the abelian Higgs, Yang-Mills-Higgs and Yang-Mills-Chern-Simons-Higgs systems with spatially homogeneous fields, in determining the nature of order-disorder transitions is highlighted, and the effects are shown to be counter-intuitive in the last-named system.     

Incomplete Dirac reduction of constrained Hamiltonian systems

First-class constraints constitute a potential obstacle to the computation of a Poisson bracket in Dirac’s theory of constrained Hamiltonian systems. Using the pseudoinverse instead of the inverse of the matrix defined by the Poisson brackets between the constraints, we show that a Dirac–Poisson bracket can be constructed, even if it corresponds to an incomplete reduction of the original Hamiltonian system. The uniqueness of Dirac brackets is discussed. The relevance of this procedure for infinite dimensional Hamiltonian systems is exemplified.     

Classical integrable systems and gauge field theories

In this review we consider the Hitchin integrable systems and their relations with the self-duality equations and twisted super-symmetric Yang-Mills theory in four dimensions. We define the Symplectic Hecke correspondence between different integrable systems. As an example we consider the Elliptic Calogero-Moser system and integrable Euler-Arnold top on coadjoint orbits of the group GL (N, C) and explain the Symplectic Hecke correspondence for these systems. The text was submitted by the author in English.     

Partial and complete observables for Hamiltonian constrained systems

We will pick up the concepts of partial and complete observables introduced by Rovelli in Conceptional Problems in Quantum Gravity, Birkhäuser, Boston (1991); Class Quant Grav, 8:1895 (1991); Phys Rev, D65:124013 (2002); Quantum Gravity, Cambridge University Press, Cambridge (2007) in order to construct Dirac observables in gauge systems. We will generalize these ideas to an arbitrary number of gauge degrees of freedom. Different methods to calculate such Dirac observables are developed. For background independent field theories we will show that partial and complete observables can be related to Kucha?’s Bubble-Time Formalism (J Math Phys, 13:768, 1972). Moreover one can define a non-trivial gauge action on the space of complete observables and also state the Poisson brackets of these functions. Additionally we will investigate, whether it is possible to calculate Dirac observables starting with partially invariant partial observables, for instance functions, which are invariant under the spatial diffeomorphism group.     

Lie symmetries and invariants of constrained Hamiltonian systems

According to the theory of the invariance of ordinary differential equations under the infinitesimal transformations of group, the relations between Lie symmetries and invariants of the mechanical system with a singular Lagrangian are investigated in phase space. New dynamical equations of the system are given in canonical form and the determining equations of Lie symmetry transformations are derived. The proposition about the Lie symmetries and invariants are presented. An example is given to illustrate the application of the result in this paper.     

Hamiltonian reduction and unfolding of dynamical systems with gauge symmetries

We investigate the reduction and unfolding of dynamical systems with gauge symmetries. An application is provided by a non relativistic point charge in the field of a Dirac monopole. The corresponding dynamical system possessing a Kepler type symmetry is associated with the Taub-NUT metric using a reduction procedure of symplectic manifolds with symmetries. The reverse of the reduction procedure is done by stages performing the unfolding of the gauge transformation followed by the Eisenhart lift in connection with scalar potentials.     

Branes as solutions of gauge theories in gravitational field

In this paper we study the behavior of the jet quenching parameter in a background metric with hyperscaling violation at finite temperature. The background metric is covariant under a generalized Lifshitz scaling symmetry with the dynamical exponent \(z\) and hyperscaling exponent \(\theta \) . We have evaluated the jet quenching parameter for a certain range of these parameters which are consistent with the Gubser bound conditions in terms of \(T\) , \(z\) , and \(\theta \) . The results are compared with those of experimental data as well as conformal and the non-conformal cases. Finally, we add a constant electric field to the background and find its effect on the jet quenching parameter.     

Magnetically extended gauge theories

The gauge fields of two distinct O(3) gauge theories with spontaneous symmetry breaking are constrained to share an invariantly defined abelian-like “photon” sector. This magnetically extended lagrangian theory has O(3) × O(3) gauge invariance but only five gauge bosons: the “photon” and four further bosons that provide the electric and magnetic currents to which the “photon” couples. Generalizations to higher groups and comparison with a recent proposal of Montonen and Olive are made.     

Noncommutative two-dimensional gauge theories

We elaborate on the dynamics of noncommutative two-dimensional gauge field theories. We consider U(N) gauge theories with fermions in either the fundamental or the adjoint representation. Noncommutativity leads to a rather non-trivial dependence on theta (the noncommutativity parameter) and to a rich dynamics. In particular the mass spectrum of the noncommutative U(1) theory with adjoint matter is similar to that of ordinary (commutative) two-dimensional large-NSU(N) gauge theory with adjoint matter. The noncommutative version of the ?t Hooft model receives a non-trivial contribution to the vacuum polarization starting from three-loops order. As a result the mass spectrum of the noncommutative theory is expected to be different from that of the commutative theory.