The God has created a man in order that he creates that the God fails to do

Thursday, 9 August 2012

Discrete space-time (from my Scientific Biography)

In 1930. D.Ivanenko and V.Ambarzumian put forward the idea of discreteness of space-time inside an atomic nuclear that, by means of introducing the fundamental length, to solve some of the problems encountered at that time in nuclear physics.

The idea of discreteness then was not constructively developed, but many scientists from time to time turned to this concept from general considerations, but restricted themselves, as a rule, to examination of lattices. A kind of its embodiment is theory of gauge fields on a lattice, which enabled one to do some incentive calculations. In 1965, the historically-review book "Discrete space-time" (M., Science) by A.Vialtcev came out. D.Ivanenko himself repeatedly returned to this idea. In the 1970s, his Ph.D. student G.Gorelik was engaged in this subject, but nothing new has happened. Of course, all of this was discussed at the seminars of Ivanenko, and these discussions have led me to think about the problem.

From the beginning, I refused the discrete space-time as a discrete topological space (which lattices belong to). Then what? I suggested totally disconnected topological spaces that occur in a number of theoretical models (such is, for example, the set of rational numbers). D.Ivanenko again enjoyed it, and we published a few works. Moreover, it was so well-defined mathematical model that my article "Discrete space-time" was taken in very responsible "Mathematical encyclopedia":

“One conceivable hypothesis on the structure of space in the microcosmos, conceived as a collection of disconnected elements in space (points) which cannot be distinguished by observations. An acceptable formalization of discrete space-time can be given in terms of topological spaces Y in which the connected component of its point y is its closure, and, in a Hausdorff space Y, is this point itself (totally disconnected spaces). Examples of Y include a discrete topological space, a rational straight line, analytic manifolds, and Lie groups over fields with ultra-metric absolute values.

The discrete space-time hypothesis was originally developed as a variation of a finite totally-disconnected space, in models of finite geometries on Galois fields V.A. Ambartsumyan and D.D. Ivanenko (1930) were the first to treat it in the framework of field theory (as a cubic lattice in space). In quantum theory the hypothesis of discrete space-time appeared in models in which the coordinate (momentum, etc.) space, like the spectrum of the C*-algebra of corresponding operators, is totally disconnected (e.g. like the spectrum of the C*-algebra of probability measures). It received a serious foundation in the concept of "fundamental lengths" in non-linear generalizations of electrodynamics, mesondynamics and Dirac's spinor theory, in which the constants of field action have the dimension of length, and in quantum field theory, where it is necessary to introduce all kinds of  "cut-off"  factors. These ideas, later in conjunction with non-local models, served as the base for the formulation of the concept of minimal domains in space in which it appears no longer possible to adopt the quantum-theoretical description of micro-objects in terms of their interaction with a macro-instrument. As a result, the space-time continuum is unacceptable for the parameterization of spatial-evolutionary relations in these domains (e.g. the Hamilton formalism in non-local theories), and their points cannot be distinguished by observation (in spaces Y this may be represented as the presence of a non-Hausdorff uniform structure). The discrete space-time hypothesis was developed in the conception of the non-linear vacuum. According to this concept — under extreme conditions inside particles, and possibly also in astrophysical and cosmological singularities — the spatial characteristics may manifest themselves as dynamic characteristics of a physical system, in the models of which the spatial elements are provided with non-commutative binary operations.”


G. Sardanashvily, Discrete space-timeEnciclopaedia of Mathematics (Springer)
D.Ivanenko, G.Sardanashvily, Towards a model of discrete space-timeRuss. Phys. J21(1978) 1508.
My Scientific Biography

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