What are approximations? What place do they occupy in physics?

Approximation. Why did I suddenly want to pay special attention to this issue?? It would seem, quite common thing, not requiring any special consideration. Approximation today is a common place in physics, about which they are silent. It is very rare to find detailed descriptions of approximations in articles., which are accepted in the study, to which the article is devoted. And when teaching students, attention is usually not focused on this issue.. And in vain.

Approaches pervade all physics. Moreover, actually, every approximatedis, wherever it is done by itself creates separate, unique image of the world (parts of it, some described phenomenon). Often, approximations allow us to introduce new concepts, new terms, which are valid only within each given approximation. AND woe to that explorer, who doesn't notice, what uses concepts, which are legitimate in some other approximation, but not that, in which he works. And don't think, that this is a rare occurrence. On the contrary, very frequent. Especially there and then, when scientists try to expand the scope of application of a theory, well proven in certain conditions.

Approximations can be nested within one another. And still, each time new images of the world appear.. Often, concepts from one approximation are transferred to another in the form of the same mathematical image, while being associated with completely different things. Perhaps, the most characteristic mathematical concepts in this sense are the basic elements of geometry – point, line, surface. Other, of course. In order not to be unfounded, I will give well-known examples. When studying mechanics at school (in the University, by the way, also), eg, mechanics of motion of solid balls, every such ball, regardless of its size, usually represented by a dot. Material point, equipped with some mass. Mass can be an arbitrary parameter. A point is mathematically selected anyway, dimensionless entity, space element (or space-time). Point trajectory, naturally, depicted by the line, etc. Probably everyone is clear, that two images of the world by space-time with elements-points are two completely different images of it, two different approximations, if in one case balls of diameter of the order of 1-2 centimeters, and in the other – planets. AND both approximations are usually called simply classical. They are both classic indeed.. But they are embedded in the classical approximation parametrically, and the parameter is the size, starting from which, we neglect the actual size. And if in case, when we consider planets as points, balls will certainly be dots, the first approximation is entirely valid, on the contrary, not at all.

So what is it approximation? I would call it by physics, even science in general. By the method of the most important, the most common and effective. But usually attention is already focused on the results of applying this method.. In conventional terminology, they are called the approximation. Respectively, I will talk about approaching in both senses. Here I want to focus on some approximations., used in space-time physics.

Let's start with classical approximation, more precisely, from the whole set of approximations, called classic. What unites them all? Common in all classical approximations is that, what all objects exist continuously in time. It is most important. Events fill the trajectory (line of existence) object in space-time continuously. A classical object can be measured at any moment of its existence.. Naturally, and the scale, realizing the classical frame of reference, also exist by definition everywhere continuously in time, and are arbitrarily small, to be able to measure any, however small, part of a continuous object. Spatial existence of classical objects, on the contrary, can be quite discrete. Approximations of objects by material points are extremely common in physics.. Allowed as intermediate cases, when objects continuously fill some limited area of ​​space, and the extreme case of a continuous medium, which is continuous everywhere in the considered region of space, which means it is continuous in the entire region of space-time. This concerns the possible description of physical objects. Space-time itself in classical approximations is always considered in some sense as a continuous medium, every point in space-time (event space) assumed to exist continuously in time, is there a material physical object or not. The interpretation of such a space-time is possible in two ways., which divide all classical approximations into two classes based on this feature. In the first case, space-time is assumed to be simply a receptacle, arena for physical objects. Its properties do not depend in any way on the fact, what happens in this arena and space-time is a priori assumed to be Euclidean (pseudo-Euclidean mathematical space. (Pseudo)Euclidean is a mathematical notation for that fact, that for each point of space-time (and for all points at once) you can choose such vectors of units of measurement in different spatial (and temporary too) directions, plot such coordinates in all space-time, that the distance between any two points can be calculated, knowing their coordinates (distances from the origin along straight lines, mutually perpendicular lines, obtained by measuring with these rulers and watches)  using the Pythagorean theorem. In other words, rulers everywhere and always (solid) and the clock exists and coincides with each other. Measurements, made with their help, what's in Moscow, what's in London, that in the center of the galaxy you can shit, not thinking about anything else. This point of view dates back to Newton and was left only with the creation General Theory of Relativity. but, despite the apparent success of this theory, in itself, the approximation of independent space-time as an arena for physical phenomena, continues to be in many cases, you can even say in the majority, very effective and sufficient to get correct results. In general relativity, space-time is no longer an indifferent receptacle for physical objects., its structure depends on one of their characteristics, masses (more precisely, energy-momentum tensor). Therefore, it can no longer remain Euclidean.. But it still fully retains its structure in this form, to stay Euclidean away from material bodies. The difference from Euclidean must be described somehow. This function (description of the difference) fulfills metric tensor, whose dependence on place and time (its difference from Euclidean) describes the dependence of the classical frame of reference (a set of units of measure, rulers and hours) for each point in space-time. Besides, space-time itself, now also affects relative position (traffic) all bodies, having mass. And this function, motion description, also performs metric tensor, the type of which the trajectories of material bodies depend on. However, GRT still retains the same features. – in a certain sense, the space-time of this theory still remains just a container of energy-momentum of physical fields. The energy-momentum tensor itself is not a space-time structure, it remains external.

Another class of approximations in physics is usually called quantum. Such approximations should be applied then, when act, characterizing the phenomena under consideration, can no longer be regarded as a continuous. It is the action that in these approximations has a certain minimum value, quantum. AND, respectively, in such approximations is a discrete quantity. Almost already in the development of the foundations of quantum mechanics, it was realized, that in this approximation, the object of the real world cannot also be assigned a trajectory, consisting of a continuous sequence of events. This fact is formulated in many different ways.. One of the most famous is uncertainty relation Heisenberg. They also talk about the absence of any trajectory in quantum objects, etc.. What about the description of space-time in such approximations?? It always remains classic, since the idea of ​​it is still built on the basis of reference systems, which are available to us only in the classic version. Quantum frames of reference simply do not exist. It is also called the principle of complementarity wrinkle. In this way, the quantum approximation is fundamentally a centaur. This is more or less bearable, as long as spacetime is viewed as an arena for physical, now quantized phenomena. As a matter of fact, all the successful part of quantum theories is just like that. And all attempts to apply the methods of quantum theory to the space-time of the General Theory of Relativity have so far failed.. This problem is formulated today as the need to build a theory quantum gravity.

In my opinion, the failure of these attempts, first of all, due to the, that any quantum theory is doomed to remain a centaur precisely for reasons, formulated as Bohr's complementarity principle. All our measurement procedures have been, is and will remain classic. Consequently, and they can only be described in the classical approximation. And the image of space-time, created by them, will always be classic. Metrics, which is the image of the frame of reference, distributed in the area of ​​space-time, cannot be quantized, not the subject of quantum theory. But this does not mean at all the impossibility of a consistent construction of the quantum theory of space-time.. You just need to choose the right structures., which generate a quantum-mechanical description of the real world. It is also necessary to work in two approximations at once., while maintaining their necessary consistency at the same time, and their necessary differences.

© Gavryusev V.G.
The materials published on the site can be used subject to the citation rules.