Systematic study of the completeness of two-dimensional classical ϕ4 theory

The completeness of some classical statistical mechanical (SM) models is a recent result that has been developed by quantum formalism for the partition functions. In this paper, we consider a 2D classical [Formula: see text] filed theory whose completeness has been proved in [V. Karimipour and M. H. Zarei, Phys. Rev. A 85 (2012) 032316]. We give a new and general systematic proof for the completeness of such a model where, by a few simple steps, we show how the partition function of an arbitrary classical field theory can be derived from a 2D classical [Formula: see text] model. To this end, we start from various classical field theories containing models on arbitrary lattices and also [Formula: see text] lattice gauge theories. Then we convert them to a new classical field model on a nonplanar bipartite graph with imaginary kinetic terms. After that, we show that any polynomial function of the field in the corresponding Hamiltonian can approximately be converted to a [Formula: see text] term by adding enough numbers of vertices to the bipartite graph. In the next step, we give a few graphical transformations to convert the final nonplanar graph to a 2D rectangular lattice. We also show that the number of vertices which should be added grows polynomially with the number of vertices in the original model.

Supergeometry and Quantum Field Theory, or: What is a Classical Configuration?

We discuss the conceptual difficulties connected with the anticommutativity of classical fermion fields, and we argue that the "space" of all classical configurations of a model with such fields should be described as an infinite-dimensional supermanifold M. We discuss the two main approaches to supermanifolds, and we examine the reasons why many physicists tend to prefer the Rogers approach although the Berezin–Kostant–Leites approach is the more fundamental one. We develop the infinite-dimensional variant of the latter, and we show that the superfunctionals considered in [44] are nothing but superfunctions on M. We propose a programme for future mathematical work, which applies to any classical field model with fermion fields. A part of this programme will be implemented in the successor paper [45].

A classical field model for charged particles

A classical field model is proposed, involving a scalar field interacting with the electromagnetic field, that has discrete particle-like solutions corresponding to any desired mass spectrum, all such solutions having exactly the same electric charge.