Why the formalism of modern physical theories looks so complex? (05/18/21)

One may wonder why the mathematical formalism describing modern theories of physics is so complex. Theories like those of strings, twistors, loop gravity, relativity, quantum mechanics and quantum field theory require an extremely high expertise in mathematics for understanding them, which is not the case of the “layman”.

However, the layman, like everyone and anything else on Earth are constrained by these laws. What a curious discrepancy! Is there a reason for this? Wouldn’t these laws be less effective if they were simpler? Should we attribute this to the intrinsic complexity of the nature itself, at least as it appears to us (the phenomena), or to the relative indigence of our mind and our senses, since this complexity is appreciated by them?  It would be like looking at an object through a kaleidoscope that would give scattered, distorted and blurry appearance of an image.

As part of the universe, we are fully constrained by these laws, including our mind and our senses, which are at work in our theorical search in physics and in the “physical” validations that can be made of it.

We may conceive that this imply structural limits on our knowledge: one tries to understand the constraints of the laws of the nature by a using a mean which is constrained by these same laws!

 It is even surprising that we can make such elaborate mathematical constructions which seem to reveal, at least, part of the mystery to us.

This is possibly linked to the phenomenon of consciousness, which is a kind of recursive process where the subject is considering himself as an object.

From these considerations, it seems reasonable to wonder, like Plato, if phenomena, which are shadows of a “perfect” reality, are not the source of this complexity.

As shadows, they provide incomplete and distorted very mysterious fragments of the puzzle representing a supposed “physical reality” and therefore the complex mathematical constructions of our theories would not necessarily be linked to the complexity of nature itself but to the fact that, what we get are degraded, scattered, distorted and strange fragments of this supposed reality.

The science, then, aim to try to provide us (apparently with some limited success) a better image of nature from these disparate and degraded fragments by processing them with a sophisticated formalism, according to what our mind considers as a reasonable coherence per the structure and knowledge of our brain .

We must still keep in mind that what we call the validation (in fact the non-rejection) of a theory can only be based on these fragments (phenomena) which are the only objects that are available to us.

Therefore, a reasonable theory would not predict the physical reality but its related fragments which are the only objects than we can check.

The process is quite tricky as scientist will look for laws which could give a coherent meaning to the production of these fragments, without necessarily being able to discover the image of the complete puzzle which may be simple but which is unknown!

Another question then arises: what kind of, a priori, paradigms to use for trying to make sense to the fragments that the scientist observes, and where do they come from?

For example, more and more symmetries are invoked which may be represented directly by geometric parameters or more formally by invariants under transformations (groups of symmetries) in physics (quantum field theory for example).

 It is true that when one looks to the nature in its ultimate entrenchments there is hardly anything left but relations, from which one can only extract symmetries!

H. Weyl, who was interested in the subject, suggests that this strong belief in symmetries is suggested by the observation of nature (mineral world, crystals, plants, animals, etc.) where symmetries are omnipresent.

From this, one is led to deduce that nature loves symmetries …..

Failing to describe nature in its full extension, the scientist may hope to improve his knowledge of nature by the diversification and the acquisition of new experimental means which will allow him to get other fragments and even all the fragments of the puzzle. But this does not mean that we will know how to assemble them for discovering the image that they represent!

 But, the fact that formalisms providing useful information on some phenomena exist,  their structure should result from a morphism between the structure of the laws of nature and the structure of formalism which predicts the phenomena that these laws of nature offer us. This shows the interest represented by these formalisms which must be interpreted in the context of what they produce: a key for giving sense to the set of fragments!

But, even assuming that we can obtain all the fragments, on the one hand this will not imply that we will know how to assemble them correctly to form an image and on the other hand, even if it were the case, that we will know how to interpret this image, undoubtedly blurred, of nature.

Some examples illustrating these words

Special relativity

We know the quarrel over the paternity of special relativity, Einstein having been called by some people a “copycat”! It is true that Lorentz, by the empirical transformations he established, Poincaré by the group of special relativity spacetime transformations that he identified, contributed to the genesis of this theory, which “was in the air ”at the time, following the problem posed by electromagnetism and the Morley-Michelson experiment. But it must be recognized that it was Einstein, in 1905, who gave it its foundation, giving it a physical meaning through the principle of “relativity”. All physical phenomena (except gravitation) obey the same laws in all inertial frames of reference which are differentiated only by a relative (constant) velocity. This principle allows, with the parameter of the velocity of light which is a constant in all the frames of reference, to constrain and derive the equations of special relativity. Indeed, in all these inertial frames (called galilean frames), we do not feel any constraint (the objects “float and we float”). Therefore, as in such galilean frames the phenomenology is the same, the physics must be the same. None are privileged. The constant experimental value of the celerity of light specifies a parameter of the theory. This situation led some people to say that, in 1905, we had all the pieces of the puzzle, but it was Einstein who discovered what they should represent and therefore how to put these pieces together.

General relativity

There was a small paternity dispute between Einstein and Hilbert which was friendly settled  by Hilbert acknowledgement that most of the analysis of the problem was due to Einstein, his contribution to the equation was simply the resolution of a mathematical problem (brilliantly, because he proposed a method much more general than that of Einstein by defining an action, the action of Hilbert, for general relativity). Einstein, who for his part was committed to transposing gravity into a relativistic form, had established his equation well before Hilbert, but in a less elegant way. The geometrical form of the theory of general relativity which is a theory of gravitation shows that we can describe, for example the universe, by its geometry which depends on what constitutes the universe. The great interest of this formulation is that it makes it possible to take into account a “non-linearity” which seems necessary: all the objects contribute to define the geometry of the universe to which, in turn, all these same objects will be coupled ( they will follow geodesics of the geometry of this universe). The circle is completed. Magnificent solution implementing this recursion where the object is also the subject. A model that could serve as a paradigm for phenomena like consciousness? Another beauty of the solution, the universe thus defined is “self-sufficient”, in other words, it does not need anything other than itself to exist and to be totally defined. It eludes the problem of creation and “reduces” it to that of its existence.

Quantum mechanics

Many scientists have contributed to this theory, so bizarre that its physical interpretation is still open to debate, even if the Copenhagen School’s interpretation is the reference. Faced with the strange nature, very different from what the world of classical physics and mechanics presented to us, that scientists were discovering, it is interesting to note the approach of W. Heisenberg who proposed to abandon all the concepts of classical mechanics and to consider only the “observables” (the phenomena) as elements of the theory. They were presented in matrices, associated with a formalism that allowed calculations to be made. On the other hand, Schrödinger was developing a solution with an equation of a wave function, allowing to define the state of a system. We know that this following formalism also included operators, associated with the physical quantities, which applied to the wave function allows to predict the probabilities of measurement results. These two formalisms turned out to be equivalent, which is interesting, because it shows that two fundamentally different approaches could just, as wel,l describe what we could know about nature.