The gonihedric paradigm extension of the Ising model

Abstract

We suggest a generalization of the Feynman path integral to an integral over random surfaces. The proposed action is proportional to the linear size of the random surfaces and is called gonihedric. The convergence and the properties of the partition function are analysed. The model can also be formulated as a spin system with identical partition function.The spin system represents a generalisation of the Ising model with ferromagnetic, antiferromagnetic and quartic interactions. Higher symmetry of the model allows to construct dual spin systems in three and four dimensions. In three dimensions the transfer matrix describes the propagation of closed loops and we found its exact spectrum. It is a unique exact solution of the tree-dimensional statistical spin system. In three and four dimensions the system exhibits the second order phase transitions. The gonihedric spin systems have exponentially degenerated vacuum states separated by the potential barriers and can be used as a storage of binary information.

Gravity with a Linear Action and Gravitational Singularities

“…This principle will allow to extend the notion of the Feynman integral over paths to an integral over space-time manifolds so that when a manifold collapses into a single world line the corresponding quantum-mechanical amplitude becomes proportional to the length of the world line. In other words, in this limit the gravitational action should reduce to the relativistic particle action which is equal to the length of the world line and measures it in cm [39].…”

“…The reason is that when the action has a dimension larger than one, that is, the action has dimension cm d , where d > 1, then the geometrical fluctuations of lower dimension will grow uncontrollably on a space-time manifold. This happens because the action is "blind" toward measuring the low dimensional fluctuations [36,37,38,39]. Indeed, let us consider a discretised two-dimensional world sheet surface and a theory in which the action is equal to the area of the surfaces.…”

“…The question is how to measure the perimeter/linear size of the highdimensional manifolds and, in particular, a triangulated two-dimensional world sheet surface in terms of cm, instead of the areas of its triangles. The invariant which characterises the linear size of the discretised two-dimensional surface can be constructed summing the lengths of its edges l ij multiplied by the deficit angle ω ij = |π − α ij | on the corresponding edge < ij > [33,34,35,36,37,38,39] (see Figure 2: The discretised two-dimensional surface describing the propagation of a string from space-time point X to Y . The action (1.3) allows to extend the notion of the Feynman integral over paths to an integral over space-time surfaces, so that when a two-dimensional surface degenerates into a single world line the quantum mechanical amplitude becomes proportional to the length of the world line :…”

Plaquette Ising models, degeneracy and scaling

“…The purely plaquette Hamiltonian of Equation (1) can be thought of as the limiting case, for κ → 0, of a one-parameter family of 3d gonihedric Ising Hamiltonians [23][24][25][26][27][28][29][30][31]. These contain in general nearest-neighbour i, j , next-to-nearest-neighbour i, j and plaquette [i, j, k, l] interactions,…”

Covariant perturbations in the gonihedric string model

“…This field theory, at classical level, predicts the existence of tensionless strings possessing a massless spectrum of higher integer spin gauge fields [4][5][6] whereas, at quantum level, fluctuations generate a nonzero string tension [1][2][3]. Moreover, when the theory is formulated on an Euclidean lattice it has a close relationship with a spin system which generalizes the Ising model with ferromagnetic, antiferromagnetic and quartic interactions [7][8][9].…”