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 al… Show more
“…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].…”
mentioning
confidence: 99%
“…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.…”
mentioning
confidence: 99%
“…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 :…”
Motivated by quantum-mechanical considerations we earlier suggested an alternative action for discretised quantum gravity which measures the perimeter of the space-time and has a dimension of length. It is the so called perimeter action, since it is a "square root" of the area action in gravity and has a new constant of dimension one in front. The physical reason to introduce the perimeter/linear action was to suppress singular configurations "spikes" in the quantum-mechanical integral over geometries. Here we shall consider the continuous limit of the discretised perimeter/linear action.We shall demonstrate that in the modified theory during the time evolution of a large massive star, when a star undergoes a collapse and develops an event horizon which confines the light, a smaller space-time region will be created behind the event horizon which is unreachable by test particles.These regions are located in the places where a standard theory of gravity has singularities. We are confronted here with a drastically new concept that during the time evolution of a massive star a space-time region is created which is excluded from the physical scene, being physically unreachable by test particles or observables. If this concept is accepted, then it seems plausible that the gravitational singularities are excluded from the modified theory.
“…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].…”
mentioning
confidence: 99%
“…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.…”
mentioning
confidence: 99%
“…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 :…”
Motivated by quantum-mechanical considerations we earlier suggested an alternative action for discretised quantum gravity which measures the perimeter of the space-time and has a dimension of length. It is the so called perimeter action, since it is a "square root" of the area action in gravity and has a new constant of dimension one in front. The physical reason to introduce the perimeter/linear action was to suppress singular configurations "spikes" in the quantum-mechanical integral over geometries. Here we shall consider the continuous limit of the discretised perimeter/linear action.We shall demonstrate that in the modified theory during the time evolution of a large massive star, when a star undergoes a collapse and develops an event horizon which confines the light, a smaller space-time region will be created behind the event horizon which is unreachable by test particles.These regions are located in the places where a standard theory of gravity has singularities. We are confronted here with a drastically new concept that during the time evolution of a massive star a space-time region is created which is excluded from the physical scene, being physically unreachable by test particles or observables. If this concept is accepted, then it seems plausible that the gravitational singularities are excluded from the modified theory.
“…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,…”
Abstract. We review some recent investigations of the 3d plaquette Ising model. This displays a strong first-order phase transition with unusual scaling properties due to the size-dependent degeneracy of the low-temperature phase. In particular, the leading scaling correction is modified from the usual inverse volume behaviour ∝ 1/L 3 to 1/L 2 . The degeneracy also has implications for the magnetic order in the model which has an intermediate nature between local and global order and gives rise to novel fracton topological defects in a related quantum Hamiltonian.
“…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].…”
We provide a covariant framework to study classically the stability of small perturbations on the so-called gonihedric string model by making precise use of variational techniques. The local action depends of the square root of the quadratic mean extrinsic curvature of the worldsheet swept out by the string, and is reparametrization invariant. A general expression for the worldsheet perturbations, guided by Jacobi equations without any early gauge fixing, is obtained. This is manifested through a set of highly coupled nonlinear differential partial equations where the perturbations are described by scalar fields, Φ i , living in the worldsheet. This model contains, as a special limit, to the linear model in the mean extrinsic curvature. In such a case the Jacobi equations specialize to a single wave-like equation for Φ.
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