Why the formalism of modern physical theories looks so complex? (04/11/21)

Why the formalism of modern physical theories looks so complex?

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 [0] 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, aims 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.

Note that in this presentation of the beginning of the history of special relativity, the “Newtonian” conception is still very present, via inertial reference frames, where time and space are clearly defined, a desperate process, to preserve these concepts of time and space in relativistic mechanics. As early as 1907, Minkowski proposed a formal and synthetic geometrical definition of space-time while stressing that space-time is the sole physical entity in special relativity therefore reducing time and space to shadows of it.

Langevin’s paradox

This considerable conceptual progress makes it possible to understand paradoxes such as that of Langevin’s traveler, incomprehensible in Newtonian mechanics.

Let us stress that, even today,it is still difficult to understand, because we still endeavor to consider this trip in a space provided with a fixed structure on trajectories (lines in this fixed space between points of this fixed space) on which a rocket accelerates, in the Newtonian way. This does not allow an understanding of the phenomenon.

We must totally abandon Newtonian space and time concepts, which do not exist in relativity and consider only space-time, where there is neither such fixed space and nor idependant time to understand the phenomenon!

This paradox remains a typical example where, in relativity, our understanding is blurred by habits of thought.

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 well, describe what we could know about nature.

Theory, experience and information

Physics being an experimental science, we often oppose theory, supposed to be a pure production of our mind to model and try to make intelligible the physical phenomena and experience which is the counterpart of the model, applicable to the physical world.

Bachelard who in his book, “the new scientific spirit” develops this supposed antagonism, links theory to “rationalism” (the reason of the mind), and experience to “realism” (the physical world). He tries to show that, these two aspects are not totally separable and therefore more than an opposition, we are dealing with a kind of unique dual entity of which realism and rationalism are not totally separable and independent characters.

 Sometimes it is rationalism which leads science and realism follows (experiments are carried out to validate new elements what the new theory predicts), sometimes it is realism which, through unexplained experiments, will stimulate the minds of scientists in order to build a new theory that explains them.

The most uncompromising rationalist cannot avoid taking experience into account, and the most determined realist, in constructing his experiments, cannot ignore theory, an experience being according to him a materialized theory.

He concludes: “Physics is an impure metaphysics because it must be accountable to experience”.

An experiment provides information on the state of a system via “measurements”. These measures generally involve instruments that can be sophisticated, in order to adapt to the phenomenon to be studied, hence Bachelard’s remark (an experiment includes a part of materialized theory).

In classical mechanics this has not raised any notable philosophical debate on the role of the experimenter. We considered a system to be studied. It contains information that one wishes to acquire via an experimenter equipped with his equipment. These two systems will interact during the measurement. They are generally thought of as two independent systems that resume their own “life” after interaction.

In quantum mechanics it is quite different. Measurement, taking information from the system, disrupts, reduces, or even destroys, fundamentally the information in the system. It is not a passive act (or very weakly active) without consequences on the system but on the contrary very active. The information is not only acquired but also taken (therefore removed from the system) and therefore will reduce the degrees of freedom of the system to the point that, from indeterminate, it can pass into a known state.

Illustration by Young’s slits

The mechanism which leads to this has been the subject of different interpretations, but, for example in the Young’s two slits experiment, the fact of knowing (by a detector on a slit) through which slit a photon has passed will modify, even if the detector does not detect anything (in which case it is assumed that it has passed through the other slit) the observed phenomenon.

The fact of acquiring information even when nothing is happening is called contra-factuality in some works (see note [3]).

It is understood that, when the detector attached to a slot detects something, it takes a quantity of information on the state, this modifies the phenomenon, compared to that where there is no detector, because a part of his information was taken from him and transmitted to the experimenter.

Overall, the information seems to have been preserved if we consider a system that includes the experimenter (it has been transferred from one subsystem to another)

On the other hand, when it does not detect anything, but it does provide, the same amount of information (the opposite information on the condition) is less obvious.

Curiously, since nothing was detected, it would seem that no information was taken and that, as we have the same result as when we detect the passage through a slit, we could think that the information of the global system has increased, but in view of the end result, which is the same in terms of information (although the latter is different) this does not appear to be the case. Perhaps the activity of the detector should be taken into account in the balance sheet?

Either way, this result seems odd.

Anyway, it is clear that it is the fact that there is a detector in the instrumental device which is important and which differentiates the situation from the case where there is not: The experimental device is different therefore the associated information also !

The interaction between the experimenter equipped with his instrumentation with the system is different. This is what, a priori, even if the experimental device detects nothing, because the experimenter, who built the experimental device, foresaw this case in his quest for information, will change the observed phenomenology!

The role of the experimenter, relying on a theoretical model, cannot be ignored. It is therefore indeed an interaction between a physical system and the mind of the physicist equipped with a theory via his experimental device.

Some, like Wigner, see the phenomenology of observed quantum mechanics as the essential imprint of the mind of the physicist.

In quantum mechanics it would then seem more adequate to evaluate and consider the information, not to the system studied and to the experimenter separately, but globally associated with the global system made up of the physical subsystem to be studied and the subsystem linked to the experimenter in interaction, because at the moment of the interaction they form only one system! If we neglect errors during the experiment, the information therefore the entropy [2] of the global system, considered isolated, (including the experimenter), before the experiment and after the experiment should be identical (we have since the contra-factuality seems to raise a doubt). Information that has been transferred from one subsystem to another subsystem remains in the overall system. [3]

A formalism, to be developed, which could take this into account would perhaps allow certain very disturbing aspects of the theory to be removed …

[0] We can lend our mind a structural indigence, however note that by its ability to learn and evolve we cannot make a final judgment on the state of affairs made today. We must also take into account that certain concepts, such as space and time, concepts inherent in our existence, are so significant, and that despite all our efforts to detach from them, they are still often still present, in the background, in our reasoning, which burdens its meaning. See example of Langevin’s traveller’s paradox, for example, in special relativity.

[1] This structurally very important point, this limit being involved in causality, resulting from the principle of relativity which reveals a speed invariant but does not specify its value. The value of this constant is an experimental data. see: https://astromontgeron.fr/SR-Penrose.pdf

[2] Entropy and information are two presentations of the same concept.

[3] This transfer of information is even more evident in the illustration of Young’s slits problem by Elitzur-Vaidman cited by R.Penrose (Nobel Prize 2021) in his book “The two infinites and the human spirit – Fields Flammarion ”where it is a question of recognizing in a batch of atomic bombs equipped with a detonator constituted by an ultra-sensitive mirror such as a photon activates it by making me move (mobile), those which are operational from those which are faulty where the mirror is blocked, which is indistinguishable without destroying it. We insert a bomb whose mirror detonator serves as a mirror in an experimental quantum device, without knowing whether it is operational or defective. The authors show that their quantum device makes it possible to select operational bombs without destroying them, by using the mechanism of contra-factuality. In other words, while when we insert the bomb into the device we do not know if the mirror is mobile or blocked, the information is in the object, but not in the knowledge of the experimenter, the experience will reveal to him this information, by means of a protocol, without removing this information from the bomb (which would be the case by its destruction for operational bombs).

For more information, see for example:

https://astromontgeron.fr/M%C3%A9canique%20quantique-1.pdf