Simple formal languages and the structure of spacetime

OSVALDO M. MORESCHI

Epistemología e Historia de la Ciencia Nro. 7

CIFFH

Año: 2001; p. 380 - 388

The description of any physical system makes always reference to a spacetime. For example, in classical physics one refers to the Galilean spacetime; in special relativity one refers to Minkowski spacetime; in general relativity one refers to a 4-dimensional Lorentzian manifold; in Kaluza-Klein theories and string theories one refers to higher dimensional metric manifolds. The technique of assuming a smooth manifold to construct the framework for the description of physical systems, is so common that very rarely one stops to think whether is completely justified. But it is in the study of quantum gravity that this question acquires strong relevance. There exist many indications that make one to think that probably the structure of the spacetime is very different from the smooth one that has been assumed. For example, there are works in which the quantum operator associated to the area of a closed 2-surface has been calculated, and it was found that it has a discrete spectrum. This work has also been extended to the case of volume operator and length operator. This seems to indicate that to construct the theoretical framework appropriate for the description of quantum gravity, the assumption of a smooth manifold for the basic structure of the spacetime is unjustified; and furthermore, it would probably lead to the wrong direction. If one accepts the idea that probably the quantum structure of the spacetime is discrete, one is faced with the question of which is the appropriate description of this structure. Our suggestion for this task is that if one studies a very small portion of the spacetime, then the description must necessarily be simple. To specify more what we mean by this statement, let us recall that in other works we have study the implications of some realistic reasonable assumption in the context of the early Universe; the assumption been that: "It is possible to completely describe a finite system in terms of a finite sentence of a formal language." This apparent mild assumption has important physical consequences, as we have shown previous works; but it is also important to remark that this statement is in complete agreement with the discrete expected nature of the spacetime that is suggested by the studies of quantum gravity. When one takes the above principle, and applies it to small systems, like a small portion of the spacetime, one is led to think that not only the sentence of the formal language required to describe the small system would be rather short; but also that the structure of the formal language required would not be too complex; in other words it should be simple. Therefore in this work we present an approach to the description of the structure of the spacetime, coming from the study of the construction of simple formal languages. Since the description of any physical system makes always reference to a spacetime, we consider that the description of the structure of the spacetime is one of the most profound philosophical and physical questions.