Universality of the continuum limit of lattice quantum theories

The lattice approximation of the naïve continuum action in quantum mechanics or in field theory is not uniquely determined. We investigate to what extent corrections to the lattice action, which vanish in the naïve continuum limit, affect the continuum limit when taking quantum fluctuations into account. In the quantum mechanical case, modifications of the lattice action may induce non-trivial corrections to the potential of the system and thereby change the structure of the theory completely. We verify this statement analytically as well as numerically by performing a Monte Carlo simulation. In the field theoretical case we argue that the lattice corrections considered do not affect the physics of the continuum limit, at least not for asymptotically free gauge field theories. In four dimensions, one might encounter finite renormalization of CP violating terms.     

Hadron structure on the lattice and in the continuum

Hadron structure physics has in recent years reached a level of precision which allows for a change of perspective. Model-based arguments are often quite unreliable. However, meanwhile they can be more and more replaced by controlled and systematic QCD approaches. The story of the strange electric form factor, which provided much of the motivation for the PAVI Conference series provides a typical example to illustrate this statement. However, high-precision theory is technically very challenging and progress is, therefore, unpleasantly slow. This fact and the present status in general is illustrated by a few typical examples.     

A lattice gas model for thermohydrodynamics

The lattice gas model is extended to include a temperature variable in order to study thermohydrodynamics, the combination of fluid dynamics and heat transfer. The compressible Navier-Stokes equations are derived using a Chapman-Enskog expansion. Heat conduction and convection problems are investigated, including Bénard convection. It is shown that the usual rescaling procedure can be avoided by controlling the temperature.     

A scaling theory of clusters in the lattice gas

It is proposed that the number ofp-particle,k-bond lattice gas cluster configurations is of the form exp{σpf(k/p)} in the limitp→t8. A simple modification permits application to finite clusters, with the consequence that asymptotically the cluster partition function is of the droplet form, i.e., Z _p =exp[kp{itμp^1?1/d }+O(ln p)]. The scaling function for two-dimensional lattices is determined numerically and is found to be qualitatively universal. The scaling theory is used to investigate the size dependence of the surface free energy. The surface tension of small clusters can significantly exceed its limiting value. For intermediate cluster sizes (～100 particles) there is a modest reduction in surface tension, in accord with Tolman's prediction and the results of recent Monte Carlo studies.     

A well-defined transition from continuum to lattice formulation of gauge field theories

In this paper it is shown that the well-known expressions for the action of lattice gauge theories (abelian and nonabelian) can be derived in a well-defined manner from corresponding nonlocal expressions once we take the local limit. First, a nonlocal formulation of the spinorial part of the QED action, accomplished 30 years ago, is extended to the Maxwell field part. Subsequently, the nonlocal action is discretized in the conventional way yielding, in the local limit, the well-known form of the lattice action. The above procedure is successfully repeated for nonabelian gauge fields.     

Hard-square lattice gas

We have studied the hard-square lattice gas, using corner transfer matrices. In particular, we have obtained the first 24 terms of the high-density series for the order parameter ₂– ₁. From these we estimate the critical activity to be 3.7962±0.0001. This is in excellent agreement with the earlier work of Gaunt and Fisher. It conflicts with the value 4.0 given by Müller-Hartmann and Zittartz's formula for the critical point of the antiferromagnetic Ising model in a field, so we conclude that this formula, while a good approximation, is not exact.     

Critical behaviour of the hard-core lattice gas on the honeycomb lattice

The hard triangle lattice-gas model (lattice-gas on the honeycomb lattice with first neighbour exclusion) is studied by the phenomenological renormalization method. The critical activity is found to be z = 7.85 and the critical exponents suggest that this model belongs to the 2-D Ising universality class.     

A lattice gas of prime numbers and the Riemann Hypothesis

In recent years, there has been some interest in applying ideas and methods taken from Physics in order to approach several challenging mathematical problems, particularly the Riemann Hypothesis. Most of these kinds of contributions are suggested by some quantum statistical physics problems or by questions originated in chaos theory. In this article, we show that the real part of the non-trivial zeros of the Riemann zeta function extremizes the grand potential corresponding to a simple model of one-dimensional classical lattice gas, the critical point being located at 1/2 as the Riemann Hypothesis claims.     

Relation between QCD parameters on the lattice and in the continuum

The relation between bare and renormalized coupling constants and quark masses in lattice and continuum QCD is investigated with special attention to numerical values. Within the four flavor staggered fermion formulation it appears that coupling values β = 6/g₀² 6.2 are required for a satisfactory incorporation of realistic quarks masses.     

A continuum model of gas flows with localized density variations

We discuss the kinetic representation of gases and the derivation of macroscopic equations governing the thermomechanical behavior of a dilute gas viewed at the macroscopic level as a continuous medium. We introduce an approach to kinetic theory where spatial distributions of the molecules are incorporated through a mean-free-volume argument. The new kinetic equation derived contains an extra term involving the evolution of this volume, which we attribute to changes in the thermodynamic properties of the medium. Our kinetic equation leads to a macroscopic set of continuum equations in which the gradients of thermodynamic properties, in particular density gradients, impact on diffusive fluxes. New transport terms bearing both convective and diffusive natures arise and are interpreted as purely macroscopic expansion or compression. Our new model is useful for describing gas flows that display non-local-thermodynamic-equilibrium (rarefied gas flows), flows with relatively large variations of macroscopic properties, and/or highly compressible fluid flows.     

A cluster variation method for lattice gas models

Czechoslovak Journal of Physics -     

Towards the continuum coupling in nuclear lattice effective field theory I: A three-particle model

Weakly bound states often occur in nuclear physics. To precisely understand their properties, the coupling to the continuum should be worked out explicitly. As the first step, we use a simple nuclear model in the continuum and on a lattice to investigate the influence of a third particle on a loosely bound state of a particle and a heavy core. Our approach is consistent with the Lüscher formalism.     

A remark on the condensation in the hard-core lattice Bose gas

We point out that Bose-Einstein condensation occurs at sufficiently low temperature in a hard-core ^d-lattice Bose gas for d3 and particle density 1/2, by exploiting its equivalence to a spin-1/2XY model.