Usually the discrete geometry is considered as a geometry on a lattice. In other words, discreteness appears due to properties of the pointset, where the geometry is given. In this case, the geometry cannot be uniform and isotropic. However, one can try to obtain discreteness due to properties of metric. Is it possible, that in this case the geometry will be uniform and isotropic?
I has a colleague recently that looked a little bit into this and it didn't seem so straightforward.. the discrete nature of quantum space wasn't as clear cut regular it seemed.. here follows a link. I have to ask him again about the details of his search.
http://en.wikipedia.org/wiki/Quantum_gravity
Dear Eric, I do not think, that it is correct to explain such simple property as discreteness via sych complicated things as quantum fields and quantum properties. Besides, the quantum gravitation is a wrong direction of the physics development. The geometric fields do not needs quantization. Besides the dynamic equations for the gravitational field do not contain quantum constant.
Dear Robert, the tolerance spaces are very mistical spaces. It is not clear, what is it in reality. what is a connection between the discreteness and tolerance.
Dear Robert, discrete geometry is such a geometry, where there are no close points. On one hand, any generalized geometry (and a discrete geometry) is a result of a modification of the proper Euclidean geometry. On the other hand, a discrete geometry and the Euclidean geometry have only one common quantity: distance (metric). Let us describe the Euclidean geometry in terms of distance and only in terms of distance. The discrete geometry is also can be described in terms of only distance. If the distance dd of the discrete geometryis a function of the Euclidean distance dE,then the discrete geometry will be uniform and isotropic as well as the proper Euclidean geometry, because a function of invariant with respect to coordinat transformations will be invariant with respect the same transformations. You do not need tolerant spaces, topology and other such things. You must only describe a geometry in terms of only distance. It is possible for the proper Euclidean geometry. It is possible also for other generalized geometries, if they are defined properly.
Choosing points randomly, with some probability distribution, it is, of course, possible to obtain samples, where the points are distributed isotropically (by having a distribution be invariant under rotations) and uniformly. There will be finite size effects, due to the finite number of points, of course. However, there will, also, be a dependence of any observables on the probability distribution itself, more precisely, on the class to which it belongs. Random lattices have been studied in many contexts and ways, e.g. using matrix models-cf. http://arxiv.org/abs/hep-th/9306153, for example. Physical quantities shouldn't depend on such data and this statement can be given a precise meaning through the renormalization group.
Dear Stam, you begin to consider a discrete geometry from the consideration of a lattice. You believe that a discrete geometry is a geometry on a lattice (maybe on a stochastic lattice). It is a wrong approach to discrete geometry. Definition of a discrete geometry looks at follows |d(P,Q)| does not belong to interval (0,\lambda) for any points P,Q. Here d is a distance between points P,Q, and \lambda is a minimal distance. This definition contains only such quantity as the distance function d (metric). This condition is considered usually as a restriction on the pointset \Omega, where the geometry is given. Then one obtains a lattice for \Omega.
There is another approach, when this condition is considered as a restriction on the distance function d, whereas \Omega is taken as a manifold in the geometry of Minkowski.
The proper Euclidean geometry (including a dimension) can be presented in terms of Euclidean metric dE. The Euclidean geometry is a monistic conception, which is described completely in terms of one quantity: Euclidean distance dE. It means that all geometric relations of the Euclidean geometry can be expressed in terms of metric dE. For instance, the straight line segment T[PQ] between the points P,Q can be presented in the form
T[PQ] = {R|dE(P,R) + dE(R,Q) = dE(P,Q)}
It is a coordinateless expression. It has the same form in any other generalized (discrete) geometry G, which is obtained from the Euclidean geometry by a replacement of dE by the distance function d of the geometry G in all expressions of GE. It is of no importance that the straight line segment is a surface, but not one-dimensionsl line. It is the strong coordinateless definition of geometrical object in the discrete geometry.
The proper Euclidean geometry is the first geometry, which has been constructed. It is a logical construction, because the Euclidean geometry GE has unlimited divisibility, and any geometrical object of GE can be constructed of blocks. A discrete geometry has a limited divisibility. It can not be constructed of blocks. If one tries to construct the discrete geometry, deducing it from axioms, it is a waste of time. The space-time geometry cannot be a logical construction, because it is multivariant. Construction of a generalized geometry by means of a deformation of GE ( i.e. by a replacement dE by d in all geometrical expressions) is the most simple construction method, which is used the monistic presentation of GE in terms of dE.
Such a construction of the discrete space-time geometry is useful in the relation, that it admits one to explain quantum phenomena via discreteness of the space-time geometry, and \lambda appears to be connected with the quantum constant \hbar.
Tendency of mathematicians to preserve a generalized geometry as a logical construction is preconception. If one cannot present a geometry in the coordinateless form, then one does not understand the geometry.
No, you don't need a lattice to consider a discrete geometry. The lattice imposes certain discrete symmetries, that's all. But my point is that by choosing a countable set of points from a continuous probability distribution, it is possible to deduce isotropy and uniformity. You do introduce a non-trivial dependence on the distribution, however. But it isn't necessary to embark in philosophical speculation-it is possible to compute the consequences. And the metric need not be Euclidian-that's an assumption, not a requirement.
Dear Stam, maybe, I have written about some redundant things. I wanted only to say that there are two different methods of the space-time geometry construction.
The first method: Euclidean one. The geometrical statements are deduced from a system of axioms. This method was used for a construction of the proper Euclidean geometry. Mathematician believe that this method can be used for construction of any generalized geometry. One needs to invent a system of new axioms, to test their compatibility and, proving necessary theorems, to derive geometrical properties of the generalized geometry. The obtained generalized geometry appears to be single-variant. It means, that the equivalence relation is transitive, and the obtained generalized geometry is a logical construction. It is very complicated method. Test of the axiom compatibility is especially complicated.
Another method of the generalized geometry construction: deformation method is much simpler. It use the Euclidean geometry itself (not the method of its construction). It uses two facts (1) Euclidean geometry is described completely by the Euclidean metric dE and (2) the Euclidean geometry is a monistic conception, where: all geometrical relation are expressed via metric dE. The deformation method supposes, that the two properties are true for any generalized geometry. One replaces the Euclidean metric dE by the metric d of the generalized geometry in all relations of the Euclidean geometry. One obtains all relations of the generalized geometry. This method is very simple. One does not need to prove theorems and to test compatibility of axioms, because there is only one essential quantity: metric. Such quantities as coordinate system, topology, dimension are not introduced. They appear as derivative concepts, if one needs to use them. The obtained generalized geometry is multivariant, generally speaking, because any logic was not used at the geometry construction (deformation violates logical connection between statements of the Euclidean geometry). Discrete geometry is a partial case of the generalized geometry, constructed by the deformation method, and the discrete geometry is uniform and isotropic, if d is a function of only dE. As to probability distribution, lattices, and other metageometrical concepts, they has no relation the discrete geometry.
Dear Amirt, I think that it is useless to explain simple things (like a space-time geomery) via complicated things (like a quantum vacuum). In general, the term vacuum means simply an empty space. The relativistic quantum theory fills the vacuum with virtual particles, because at the second quantization the vacuum appears to be nonstationary. But this vacuum nonstationarity is simply a mistake appeared in the conventional procedure of the second quantization. See, for instance, "On quantization of non-linear relativistic field without recourse to perturbation theory". Int. J. Theor. Phys. 6, 181-204.
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