Discrete Field Theory Research Articles

THE FERMION DOUBLING PROBLEM AND NONCOMMUTATIVE GEOMETRY

We propose a resolution for the fermion doubling problem in discrete field theories based on the fuzzy sphere and its Cartesian products. Its relation to the Ginsparg–Wilson approach is also clarified.

Pulses in the zero-spacing limit of the GOY model

We study the propagation of localised disturbances in a turbulent, but momentarily quiescent and unforced shell model (an approximation of the Navier–Stokes equations on a set of exponentially spaced momentum shells). These disturbances represent bursts of turbulence travelling down the inertial range, which is thought to be responsible for the intermittency observed in turbulence. Starting from the GOY shell model, we go to the limit where the distance between succeeding shells approaches zero (“the zero spacing limit” ) and helicity conservation is retained. We obtain a discrete field theory which is numerically shown to have pulse solutions travelling with constant speed and with unchanged form. We give numerical evidence that the model might even be exactly integrable, although the continuum limit seems to be singular and the pulses show an unusual super exponential decay to zero as exp(− constant σ n) when n→∞, where σ is the golden mean. For finite momentum shell spacing, we argue that the pulses should accelerate, moving to infinity in a finite time. Finally, we show that the maximal Lyapunov exponent of the GOY model approaches zero in this limit.

Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs

This paper presents a geometric-variational approach to continuous and discrete mechanics and field theories. Using multisymplectic geometry, we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle. In particular, we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function, and use this structure to prove covariant generalizations of conservation of symplecticity and Noether's theorem. Natural discretization schemes for PDEs, which have these important preservation properties, then follow by choosing a discrete action functional. In the case of mechanics, we recover the variational symplectic integrators of Veselov type, while for PDEs we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries. We show that the usual notion of symplecticity along an infinite-dimensional space of fields can be naturally obtained by making a spacetime split. All of the aspects of our method are demonstrated with a nonlinear sine-Gordon equation, including computational results and a comparison with other discretization schemes.

Principles of discrete time mechanics: III. Quantum field theory

We apply the principles discussed in earlier papers to the construction of discrete time quantum field theories. We use the Schwinger action principle to find the discrete time free field commutators for scalar fields, which allows us to set up the reduction formalism for discrete time scattering processes. Then we derive the discrete time analogue of the Feynman rules for a scalar field with a cubic self interaction and give examples of discrete time scattering amplitude calculations. We find overall conservation of total linear momentum and overall conservation of total theta parameters, which is the discrete time analogue of energy conservation and corresponds to the existence of a Logan invariant for the system. We find that temporal discretisation leads to softened vertex factors, modifies propagators and gives a natural cutoff for physical particle momenta.

Conformal invariance in two-dimensional discrete field theory

A discretized massless wave equation in two dimensions, on an appropriately chosen square lattice, exactly reproduces the solutions of the corresponding continuous equations. We show that the reason for this exact solution property is the discrete analog of conformal invariance present in the model, and find more general field theories on a two-dimensional lattice that exactly solve their continuous limit equations. These theories describe in general non-linearly coupled bosonic and fermionic fields and are similar to the Wess-Zumino-Witten model.

Principles of discrete time mechanics: II. Classical field theory

We apply the principles discussed in an earlier paper to the construction of discrete time field theories. We derive the discrete time field equations of motion and Noether's theorem and apply them to the Schrödinger equation to illustrate the methodology. Stationary solutions to the discrete time Schrödinger wave equation are found to be identical to standard energy eigenvalue solutions except for a fundamental limit on the energy. Then we apply the formalism to the free neutral Klein - Gordon system, deriving the equations of motion and conserved quantities such as the linear momentum and angular momentum. We show that there is an upper bound on the magnitude of linear momentum for physical particle-like solutions. We extend the formalism to the charged scalar field coupled to Maxwell's electrodynamics in a gauge invariant way. We apply the formalism to include the Maxwell and Dirac fields, setting the scene for second quantization of discrete time mechanics and discrete time quantum electrodynamics.

The 1/N-expansion as a perturbation about the mean field theory: A one-dimensional fermi model

We examine a family of probability measures onR L with real parameter ζ>0 and integer parametersN,L>0. Each such measure is equivalent to the lattice version of a one-dimensional discrete chiral-invariant fermionic quantum field theory with quartic interaction, withN the number of flavours. After applying the Matthews-Salam formula, the model becomes a statistical mechanical model of a chain of continuous Gaussian spins, coupled with a certain non-standard weight function. Finally, the model can also be considered as a probability measure on the set of tridiagonal matrices with fixed off-diagonal and random diagonal entries. Our analysis shows how to develop an asymptotic expansion in 1/N, valid for allL and ζ, for the fundamental expectation values. From this it follows that the two point fermion correlation function decays with a mass which agrees to the leading order in 1/N with the mean field value calculated by the argument of Gross-Neveu. The analytical technique we develop in essence combines a transfer matrix method with the Laplace method (steepest descent) for asymptotics of integrals.

Flip-moves and graded associative algebras

The relation between discrete topological field theories on triangulations of two-dimensional manifolds and associative algebras was worked out recently. The starting point for this development was the graphical interpretation of the associativity as flip of triangles. We show that there is a more general relation between flip-moves with two $n$-gons and $Z_{n-2}$-graded associative algebras. A detailed examination shows that flip-invariant models on a lattice of $n$-gons can be constructed {}from $Z_{2}$- or $Z_{1}$-graded algebras, reducing in the second case to triangulations of the two-dimensional manifolds. Related problems occure naturally in three-dimensional topological lattice theories.

Anyons in discrete gauge theories with Chern-Simons terms

A gauge theory with a discrete group H in (2 + 1)-dimensional space-time is known to describe (non-abelian) anyons. We study the effect of adding a Chern-Simons term to such a theory. As in a previous paper, we emphasize the algebraic structure underlying a discrete H gauge theory, namely the Hopf algebra D(H). For H ⋍ Z N , we argue on physical grounds that a Chern-Simons term in the action leads to a non-trivial 3-cocycle on D(H). Accordingly, the physically inequivalent models are labeled by the elements of the cohomology group H 3(H, U(1)). It depends periodically on the coefficient of the Chern-Simons term which model is realized. This establishes a relation with the discrete topological field theories of Dijkgraaf and Witten. We extrapolate these results to non-abelian H, and work out the representative example H ⋍ D 2 .

Direct Discretization of Planar Div-Curl Problems

A control volume method is proposed for planar div-curl systems. The method is independent of potential and least squares formulations, and works directly with the div-curl system. The novelty of the technique lies in its use of a single local vector field component and two control volumes rather than the other way around. A discrete vector field theory comes quite naturally from this idea and is developed. Error estimates are proved for the method, and other ramifications investigated.

Continuum strings from discrete field theories

We demonstrate that one of the phases of a light-cone lattice gauge theory with an infinite number of colors exactly describes free fundamental strings. The lattice spacing does not have to be taken to zero. Thus, exact rotation and translation invariance can coexist with discrete space.

TEM studies of line defects in interfaces

Line defects are ubiquitious features in interfaces, and have important structural and mechanistic role. Recently, a crystallographic theory of such defects has been presented which appears to offer a comprehensive framework for their classification. The object of the present paper is firstly to outline the characterisation and classification of defects according to this treatment. Secondly, we illustrate examples of defects in the distinctive classes observed using tern, and discuss the various imaging techniques which have been employed.In the absence of a rigorous treatment of line defects in single crystals and interfaces, which would require the development of a discrete field theory, approximate methods of defect characterisation are used. The most popular method involves mapping a contour, initially constructed around a defect of interest, into a reference space. For defeats in single crystals this Burgers circuit method, introduced by Frank, is very helpful, but suffers from certain procedural inconveniences in the case of interfacial defects.

Quantum field theory and algorithmic complexity

Almost any discrete field theory can be regarded as a Turing machine operating on programs that are weighted accoriding to a universal form of distribution. Certain generic field configurations are rare regardless of what the lagrangian is.