Analytic approximations to Hamiltonian lattice field theories (II). The weak-coupling behavior of U(1) theories

It is shown that at weak coupling physical quantities in hamiltonian U(1) lattive gauge (or global symmetric) theories of arbitrary dimension are provided as expectation values in a d ? 1 dimensional lagrangian Z(2) gauge (or spin) theory with calculable long-range interactions.Confinement and the existence of a magnetic mass gap are equivalent to the existence of infinite-range plaquette-plaquette (or link-link) correlations in the spin field. The existence of infinite range correlations is simply related to the dimension of the lattice and the transformation property of the order parameter. As expected, only the d = 2+1 U(1) gauge theory confines electric charges at all non-vanishing coupling.     

A hamiltonian structure from gauge transformations of the Zakharov-Shabat system

We study a hamiltonian structure for nonlinear evolution equations arising from a class of gauge transformations of the Zakharov-Shabat system.     

Variational method in hamiltonian lattice gauge theories

A general expression for the expectation value of the hamiltonian of a d + 1 dimensional lattice gauge theory as a function of the norm of the variational state (that itself has the form of a partition function of a d-dimensional lattice gauge theory) is given. Applications include U(1), SU(2), U(2) and U(N) gauge theories for large N in d = 2 + 1 dimensions. It is also demonstrated that the deconfining phase transition is of first order in every dimension above the critical one, provided it is of first or second order at the critical dimension.     

Geometrical string and dual spin systems

We are able to perform the duality transformation of the spin system which was found before as a lattice realization of the string with linear action. In four and higher dimensions this spin system can be described in terms of a two-plaquette gauge hamiltonian. The duality transformation is constructed in geometrical and algebraic language. The dual hamiltonian represents a new type of spin system with local gauge invariance. At each vertex ξ there are d (d − 1) /2 Ising spins ∧_{μ, η} = ∧_{η,μ N}. ≠ P = 1, … , d and one Ising spin Γ on every linkξ ξ + e,). For the frozen spin Γ  1 the dual hamiltonian factorizes into d (d − 1) /2 two-dimensional Ising ferromagnets and into antiferromagnets in the case Γ  −1. For fluctuating F it is a sort of spin-glass system with local gauge invariance. The generalization to p-membranes is given.     

Low-energy Compton scattering by a proton: Comparison of effective hamiltonians with relativistic corrections

We discuss low-energy Compton scattering by a proton using two different methods of obtaining “effective 2 × 2 hamiltonians” of the electromagnetic interaction which include relativistic corrections. One is the standard Foldy-Wouthuysen transformation which we compare with the hamiltonian obtained from a direct reduction of the matrix element of the interaction hamiltonian between positive-energy solutions of the free Dirac equation. It is found that the Foldy-Wouthuysen hamiltonian yields the same result as a covariant calculation. However, an identification of the Z-diagrams of the usual Dirac representation with the contact graphs of the Foldy-Wouthuysen representation is incorrect beyond the order of the low-energy theorem. The direct-reduction method will, in general, lead to incorrect results. Although part of the problem may be cured by restoring the gauge invariance property of the hamiltonian, gauge invariance alone is not sufficient to reproduce the correct result.     

A string-like equation for a U(N) gauge theory

A change of variables is made in the hamiltonian of a U(N) gauge theory so that the independent variables are the path dependent phase factors. The resulting hamiltonian is similar in form to that of the Nambu-Gato string     

Hamiltonian BRST quantization of the interacting antisymmetric tensor field

We study the hamiltonian BRST quantization of the non-abelian antisymmetric tensor field. We find the constrained system which arises from the standard action by Dirac's procedure, and eliminate the second-class constraints by introducing Dirac brackets. Having isolated the underlying first-class constrained system, we quantize it using the hamiltonian BRST techniques of Batalin and Fradkin. We study the Lorentz covariant gauge fixing of this system, and discuss the relationship between our results and other recent studies of the interacting antisymmetric tensor field.     

Origins of global anomalies in quantum mechanics

Simple and tractable examples of abelian and non-abelian gauge systems with global anomalies are presented in quantum mechanics. Explicit calculations are done both in the path-integral and hamiltonian formalism. Algebraic criteria are given for the existence of global gauge anomalies. These criteria are applied for every gauge group and many representations. The inconsistency of theories with a global gauge anomaly is discussed.     

Density dependence of 129Xe N.M.R. chemical shifts in O2 and NO

For neutral species it is possible to choose a gauge in which the molecular hamiltonian is translationally invariant. Here it is shown that for charged species this is not entirely possible, but that, if molecular coordinates are employed, the lack of invariance may be confined to the translational part of the hamiltonian.     

Quantum string representation for Yang-Mills fields

A new way to describe the dynamics of Yang-Mills fields is proposed. The quantum strings with internal degrees of freedom are the principle objects here. A hamiltonian of the system has been obtained in a single string sector. The consideration has been carried out for the SU(2) gauge group.     

Ward identities in four-dimensional string theory

A four-dimensional heterotic string theory is envisaged in the background of its massless gauge Ward identities are derived. The spontaneous generating functional is constructed in the hamiltonian phase space and gauge Ward identities are derived. The spontaneous breaking of gauge symmetry is discussed in this approach.     

Simulation of two dimensional systems with two dimensional electrostatics

Some of the problems associated with Monte Carlo and molecular dynamics simulations of three dimensional ionic and dipolar systems are discussed, with emphasis on the use of periodic boundary conditions (PBC). It is shown that analogous problems may arise in two dimensional systems provided that the interactions are two dimensional electrostatic interactions, that is, interactions derived from the two dimensional Laplace equation. The PBC hamiltonian is evaluated by considering the appropriate two dimensional lattice sums, and a computable form for the effective pair interactions in PBC developed. The idea of an external dielectric constant is introduced and its effects included in the PBC hamiltonian. Formulae for evaluating the dielectric constant from a simulation with any external dielectric constant are given. Perturbation formulae showing the effects on the structure and mean square dipole moment of a dipolar system which are caused by a change in external dielectric constant are derived. A formula for the shift in mean square dipole moment of an ionic system is also developed. The problem of interpreting results from such simulations is discussed briefly.     

Hot gluon propagator

In order to match two complementary approaches to quark gluon plasma, namely the classical hamiltonian lattice gauge field simulation which uses the temporal axial gauge, and hot perturbative QCD which rather uses the Coulomb or the covariant gauge, we obtain a general expression for the hard thermal loop gluon propagator for a variety of non-background gauge fixing conditions. The Coulomb energy is shown to be independent of the gauge fixing condition.     

Soliton equations and coherent states

The τ-functions, which represent the totality of solutions for hierarchies of equations in soliton theory, are identified with the coherent states of the infinite dimensional Lie algebra gl(∞). The associated quantum system can be realized by an infinite set of harmonically interacting fermionic modes. The soliton dynamical evolution is thus mapped into a quantum hamiltonian evolution, and the latter back into a classical hamiltonian flow corresponding to a succession of infinitesimal contact Bäcklund transformations.     

The coherent state variational algorithm: A numerical method for solving large-N gauge theories

This paper presents a new approach for studying large-N gauge theories which directly exploits the classical nature of the N → ∞ limit. This method supplies a practical algorithm for computing and minimizing the classical hamiltonian (or effective action) which governs N = ∞ dynamics, and allows one to calculate physical quantities such as the mass spectrum or scattering amplitudes of glueballs or mesons. Two different implementations of the basic ideas are discussed; one variant provides an algorithm for constructing N = ∞ master field matrices, while the other works directly with a list of expectation values of physical operators. Algorithms are developed for both the hamiltonian and euclidean formulations of lattice gauge theories. The inclusion of fermions in the hamiltonian version is also described. Detailed tests of the method in the context of the exactly solvable one-plaquette model are presented.     

Hamiltonian formulation of QED in the superaxial gauge

We present a hamiltonian formulation of QED in a fully fixed axial gauge. The equal-time commutators for all field variables are computed and are shown to lead to the correct equations of motion. The constraints and gauge conditions hold as strong operator relations.     

Finite matrix models with continuous local gauge invariance

We construct a hamiltonian lattice gauge theory which possesses local SU (2) gauge invariance and yet is defined on a Hilbert space of 5-dimensional real vectors for every link. This construction does not allow for generalization to arbitrary SU(N), but a small variation of it can be generalized to an SU(N) × U(1) local gauge invariant model. The latter is solvable in simple gauge sectors leading to trivial spectra. We display these by studying a U(1) local gauge invariant model with similar characteristics.     

On the Gribov ambiguity in the Faddeev-Popov procedure

The standard Faddeev-Popov procedure for quantization of a gauge theory is modified so as to be valid even when the Gribov problem is present. The hamiltonian is employed and a definite expression for the path integral is obtained for a wide class of gauges.     

Becchi-Rouet-Stora-Tyutin quantization and Hamiltonian formalism

An introductory review of BRST hamiltonian formalism is presented. The method of quantization of gauge and string theories is discussed. A few simple examples are presented to illustrate the BRST techniques.