Vanity and formalism

During my series on Transcending the impasse, I wrote about Vanity in Physics. I also addressed the issue of Physics vs Formalism in a previous post. Neither of these two aspects are conducive to advances in physics. So, when one encounters the confluence of these aspects, things are really turning inimical. Recently, I heard of such a situation.

In an attempt to make advances in fundamental physics, the physics community has turned to mathematics, or at least something that looks like mathematics. It seems to be the believe that some exceptional mathematical formalism will lead us to a unique understanding of the fundamentals of nature.

Obviously, based on what I’ve written before, this approach is not ideal. However, we need to understand that the challenges in fundamental physics is different from those in other fields of physics. For the latter, there are always some well-established underlying theory in terms of which the new phenomena are studied. The underlying theory usually comes with a thoroughly developed formalism. The new phenomena may require a refinement in formalism, but one can always check that any improvements or additions are consistent with the underlying theory.

With fundamental physics, the situation is different. There is no underlying theory. So, the whole thing needs to be invented from scratch. How does one do that?

Albert Einstein

We can take a leave out of the book of previous examples from the history of physics. A good example is the development of general relativity. Today there are well established formalisms for general relativity. (Note the use of the plural. It will become important later.) How did Einstein know what formalism to use for the development of general relativity? He realized that spacetime is curved and therefore need a formalism that can handle curved spacetime metrics. How did he know that spacetime is curved? He figured it out with the aid of some simple heuristic arguments. These arguments led him to conceive of a fundamental principle that would guide him in the development of the theory.

That is a success story. Now compare it with what is going on today. There are different formalisms being developed. The “fundamental principle” is simply to get a formalism that can handle curved spacetime in the context of a quantum field theory so that the curvature of spacetime can somehow be represented be the exchange of particles. As such, it goes back to the old notions existing before general relativity that regarded gravity as a force. According to our understanding of general relativity, gravity is not a force. But let’s leave that for now.

There does not seem to be any new physics principles that guide the development of these new formalisms. Here I exclude all those so called “postulates” that have been presented for quantum mechanics, because those postulates are of a mathematical nature. They may provide a basis for quantum mechanics as a mathematical formalism but not for the physics associated with quantum phenomena.

So, if there is no fundamental principle driving the current effort to develop new formalisms for fundamental physics, then what is driving it? What motivates people to spend all the effort in this formidable exercise?

Recent revelations gave me a clue. There was some name-calling going on among some of the most prominent researcher in the field. The proponents of one formalism would denounce some other formalism. It is as if we are watching a game show to see which formalism would “win” at the end of the day. However, the fact that there are different approaches should be seen as a good thing. It provides the diversity that improves the chances for success. More than one of these approaches may turn out to be successful. Here again an example from the history of science can be provided. The formalisms of Heisenberg and Schroedinger both turned out to be correct descriptions for quantum physics. Moreover, there are more than one formalism in terms of which general relativity can be expressed.

So what then is really the reason for this name-calling among proponents of the different approaches to develop formalisms for fundamental physics? It seems to be that deviant new motivation for doing physics: vanity! It is not about gaining a new understanding. That is secondary. It is all about being the one that comes up with the successful theory and then reaping in all the fame and glory.

The problem with vanity is that it does not directly address the goal. Vanity is a reward that can be acquired without achieving the goal. Therefore, it is not the optimal motivation for uncovering an understanding of fundamental physics. I see this as one of the main reasons for the lack of progress in fundamental physics.

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The role of mathematics in physics

Recently, the number of preprints that contain theorems with proofs in the arXiv under quantum physics has increased drastically. I’ve also noticed that some journals in this field tend to publish more such papers, even though they are not ostensibly mathematical physics journals. It seems to suggest that theoretical physics needs to look like mathematics in order to be taken seriously.

Theorems with proofs are not science. Physics, which is a science, is about getting agreement between predictions and experimental observations. So, what is the role of mathematics in physics?

Mathematics

For the physicist, mathematics is a tool, often an indispensable tool, but still, just a tool. When Feynman invented his version of quantum field theory in terms of the path integral, he provided a means to compute predictions for the scattering amplitudes in particle physics that can be compared with the results from high energy particle physics experiments. That was the whole point of this formulation. From a mathematical perspective, the path integral formulation was a bit crude to say the least. It presented a significant challenge to come up with a rigorous formulation of the measure theory that would be suitable for the notion of a path integral.

These days, there seems to be much criticism against quantum field theory. The Haag theorem indicates some inconsistencies in the interaction picture. I also saw that Ed Witten is taking issue with the process of quantization that is used in quantum field theory because of some inconsistencies and he tries to solve these problems with some concepts taken from string theory.

I think these criticisms are missing the point. The one thing that you can take from quantum field theory is this: it works! There is a very good agreement between the predictions of the standard model and the results from high energy physics experiments. So, if anybody thinks that quantum field theory needs to be reformulated or replaced by a better formulations then they are missing the point. The physics is only concerned with having some mathematical procedure to compute predictions, regardless of whether that procedure is a bit crude or not. It is just a tool. Mathematicians may then ask themselves: why does it work?

Mathematics is extremely flexible. There are usually more than one way to represent physical reality in terms of mathematical models. Often these different formulations are completely equivalent as far as experimental predictions are concerned. For this reason, one should realize that physical reality is not intrinsically mathematical. Or stated differently, the math is not real (as Hossenfelder would like us to believe). Mathematical models exist in our minds. It is merely the way we represent the physical world so that we can do calculations. If we come up with a crude model that serves the purpose to perform successful calculations, then there are probably several other less crude ways to do the same calculations. However, it is the amusement of the mathematician to ponder such alternatives. As far as the physicist is concerned, such alternatives are of less importance.

Having said that, there is one possible justification for a physicist to be concerned about the more rigorous formulation of mathematical models. That has to do with progress beyond the current understanding. It may be possible that a more rigorous formulation of our current models may point the way forward. However, here the flexibility of mathematics produces such a diverse array of possibilities that this line of argument is probably not going to be of much use.

Consider another example from the history of physics. Newtonian mechanics was developed into a very rigorous format with the aid of Hamiltonian mechanics. And yet, none of that gave any indication of the direction that special and general relativity took us in. The mathematics turned out to be completely different.

So, I don’t think that we should rely on more rigor in our mathematical models to point the way forward in physics. For progress in physics, we need to focus on physics. As always, mathematics will merely be the tool to do it. For that reason, I tend to ignore all these preprints with their theorems and proofs.

Physics vs formalism

This is something I just have to get off my chest. It’s been bugging me for a while now.

Physics is the endeavour to understand the physical world. Mathematics is a powerful tool employed in this endeavour. It often happens that specific mathematical procedures are developed for specific scenarios found in physics. These developments then often lead to dedicated mathematical methods, even special notations, that we call formalisms.

The idea of a formalism is that it makes life easier for us to investigate physical phenomena belonging to a specific field. An example is quantum mechanics. The basic formalism has been developed almost a hundred years ago. Since then, many people have investigated various sophisticated aspects of this formalism and placed it on a firm foundation. Books are dedicated to it and university courses are designed to teach students all the intricate details.

One can think of it almost like a kitchen appliance with a place to put in some ingredients, a handle to crank, and a slot at the bottom where the finished product will emerge once the process is completed. Beautiful!

So does this mean that we don’t need to understand what we are doing anymore? We simply need to put the initial conditions into the appropriate slot, the appropriate Hamiltonian into its special slot and crank away. The output should then be guaranteed to be the answer that we are looking for.

Well, it is like the old saying: garbage in, garbage out. If you don’t know what you are doing, you may be putting the wrong things in. The result would be a mess from which one cannot learn anything.

Actually, the situation is even more serious than this. For all the effort that has gone into developing the formalism (and I’m not only talking about quantum mechanics), it remains a human construct of what is happening in the real physical world. It inevitably still contains certain prejudices, left over as a legacy of the perspectives of the people that initially came up with it.

Take the example of quantum mechanics again. It is largely based on an operator called the Hamiltonian. As such, it displays a particular prejudice. It is manifestly non-relativistic. Moreover, it assumes that we know the initial state at a given time, for all space. We then use the Hamiltonian approach to evolve the state in time to see what one would get at some later point in time. But what if we know the initial state for all time, but not for all space and we want to know what the state looks like at other regions in space? An example of such a situation is found in the propagation of a quantum state through a random medium.

Those that are dead sold on the standard formal quantum mechanics procedure would try to convince you that the Hamiltonian formalism would still give you the right answer. Perhaps one can use some fancy manipulations of the input state in special cases to get situations where the Hamiltonian approach would work for this problem. However, even in such cases, the process becomes awkward and far from efficient. The result would also be difficult to interpret. But why would you want to do it this way, in the first place? Is it so important that we always use the established formalism?

Perhaps you think we have no choice, but that is not true. We understand enough of the fundamental physics to come up with an efficient mathematical model for the problem, even though the result would not be recognizable as the standard formalism. Did we become so lazy in our thoughts that we don’t want to employ our understanding of the fundamental physics anymore? Or did we lose our understanding of the basics to the point that we cannot do calculations unless we use the established formalism?

What would you rather sacrifice: the precise physical understanding or the established mathematical formalism? If you choose to sacrifice the former rather than the latter, then you are not a physicist, then you are a formalist! In physics, the physical understanding should always be paramount! The formalism is merely a tool with which we strive to increase our understanding. If the formalism is not appropriate for the problem, or does not present us with the most efficient way to do the computation, then by all means cast it aside without a second thought.

Focus on the physics, not on the formalism! There I’ve said it.