Postulates or principles?

Sometimes an idea runs away from us. It may start in a certain direction, perhaps to achieve a certain goal, but then at some point down the line it becomes something else. It may be an undesirable situation, or it may be a new opportunity. Often, only time will tell.

Quantum mechanics is such an idea. It is ostensibly a subfield of physics, but when we take a hard look at quantum mechanics, it looks more and more like mathematics. It has taken on a life of its own, which often seems to have very little to do with physics.

To be sure, physics would not get far without mathematics. However, mathematics has a very specific role to play in physics. We use mathematics to model the physical world. It allows us to calculate what we expect to see when we make observations of the phenomena associated with that model.

Quantum mechanics is different from other physical theories. While other physical theories tend to describe very specific sets of phenomena associated with a specific physical context, quantum mechanics is more general in that is describes a large variety of phenomena in different contexts. For example, all electric and magnetic phenomena provide the context for Maxwell’s theory of electromagnetism. On the other hand, the context of quantum mechanics is any phenomenon that can be found in the micro world. As such quantum mechanics is much more abstract.

We can say that quantum mechanics is not a theory, but instead a formalism in terms of which theories about the micro world can be formulated. It is therefore not strange that quantum mechanics looks more like mathematics. It even has a set of postulates from which the formalism of quantum mechanics can be derived.

But quantum mechanics still needs to be associated with the physical world. Even if it exits as a mathematical formalism, it must make some connection to the physical world. Otherwise, how would we know that it is doing a good job? Comparisons between predictions of theories formulated in terms quantum mechanics and experimental results of the physical phenomena associated with those theories show that quantum mechanics is very successful. However, in the pursuit of understanding the overlap between quantum physics and gravity in fundamental physics, the role of quantum mechanics needs to be understood not as a mere mathematical formalism, but as a fundamental mechanism in the physical world.

It is therefore not sufficient to provide mathematical postulates for the derivation of quantum mechanics as a mathematical formalism. What we need are the physical principles of nature at the fundamental level that leads to quantum mechanics as seen in quantum physics.

Principles differ from postulates. They are not expressed in terms of mathematical concepts, but rather in terms of physical concepts. In other words, instead of talking about non-commuting operators and Hilbert spaces, we would instead be talking about interactions, particle or fields, velocities, trajectories and things like that.

Another important difference is the notion of what is more fundamental than what. In mathematics, the postulates can be combined into sets of axioms from which theorems are derived. It would mean that the postulates are more fundamental. However, they may not be unique in the sense that different sets of axioms could be shown to be equivalent. In physics on the other hand, the principles are considered to be more fundamental than the theories in terms of which physical scenarios are modeled. There may be a cascade of different theories formulated in terms of more fundamental theories. Since, these theories are formulated in terms of mathematics, it can now happen that the axioms for the mathematics in terms of which some of these theories are formulated, are not fundamental from a physics point of view, but a consequence of more fundamental physical aspects.

An example is the non-commutation of operators in quantum mechanics. It is often considered as a fundamental aspect of quantum mechanics. However, it is only fundamental from a purely mathematical point of view. From a physical point of view, the non-commutation follows as a consequence of more fundamental aspects of quantum physics. Ultimately, the fundamental property of nature that leads to this non-commutation is the Planck relationship between energy (or momentum) and frequency (or the propagation vector).

Adrift in theory space

It is downright depressing to think that after all the effort to understand the overlap between gravity and quantum physics there is still no scientific theory that explains the situation. For several decades a veritable crowd of physicists worked on this problem and the best they have are conjectures that cannot be tested experimentally. The manpower that has been spent on this topic must be phenomenal. How is it possible that they are not making progress?

I do understand that it is a difficult problem. However, the quantum properties of nature was also a difficult problem, and so was the particle zoo that led to quantum field theory. And what about gravity, which was effectively solved singled-handedly by just one person? There must be another reason why the current challenge is evidently so much more formidable, or why the efforts to address the challenge are not successful.

It could be that we really have reached the end of science as far as fundamental physics is concerned. For a long time it was argued that the effects of the overlap between gravity and quantum physics will only show at energy scales that are much higher than what a particle collider could achieve. As a result, there is a lack of experimental observations that can point the way. However, with the increase in understanding of quantum physics, which led to the notion of entanglement, it has become evident that it should be possible to consider experiments where mass is entangled, leading to scenarios where gravity comes in confrontation with quantum physics at energy levels easily achievable with current technology. We should see results of such experiments in the not-too-distant future.

Another reason for the lack of progress is of a more cultural nature. Physics as a cultural activity that has gone through some changes, which I believe may be responsible for the lack of progress. I have written before about the problem with vanity and do not want to discuss that again here. Instead, I want to discuss the effect of the current physics culture on progress in fundamental physics.

The study of fundamental physics differs from other fields in physics in that it does not have an underlying well-establish theory in terms of which one can formulate the current problem. In other fields of physics, you always have more fundamental physical theories in terms of which you can model the problem under investigation. So how does one approach problems in fundamental physics? You basically need to make a leap into theory space hoping that the theory you end up with successfully describes the problem that you are studying. But theory space is vast and the number of directions you can leap into is infinite. You need something to guide you.

In the past, this guidance often came in the form of experimental results. However, there are cases where progress in fundamental physics was made without the benefit of experimental results. An prominent example is Einstein’s theory of general relativity. How did he do it? He spent a long time think about the problem until he came up with some guiding principles. He realized that gravity and acceleration are interchangeable.

So, if you want to make progress in fundamental physics and you don’t have experimental results to guide you, then you need a guiding principle to show you which direction to take in theory space. What are the guiding principles of the current effort? For string theory, it is the notion that fundamental particles are strings rather than points. But why would that be the case? It seems to be a rather ad hoc choice for a guiding principle. One justification is the fact that it seems to avoid some of the infinities that often appear in theories of fundamental physics. However, these infinities are mathematical artifacts of such theories that are to be expected when the theory must describe an infinite number of degrees of freedom. Using some mathematical approach to avoid such infinities, we may end up with a theory that is finite, but such an approach only address the mathematical properties of the theory and has nothing to do with physical reality. So, it does not serve as a physical guiding principle. After all the effort that has been poured into string theory, without having achieved success, one should perhaps ponder whether the departing assumption is not where the problem lies.

The problem with such a large effort is the investment that is being made. Eventually the investment is just too large to abandon. A large number of very intelligent people have spent their entire careers on this topic. They have reached prominence in the broader field of physics and simply cannot afford to give it up now. As a result, they drag most of the effort in fundamental physics, including a large number of young physicists, along with them on this failed endeavor.

There are other theories, such as loop quantum gravity, that tries to find an description of fundamental physics. These theories, together with string theory, all have it in common that they rely heavily on highly sophisticated mathematics. In fact, the “progress” in these theories often takes on the form of mathematical theorems. It does not look like physics anymore. Instead of physical guiding principles, they are using sets of mathematical axioms as their guiding principle.

To make things worse, physicists working on these fundamental aspect are starting to contemplate deviating from the basics of the scientific method. They judge the validity of their theories on various criteria that have nothing to do with the scientific approach of testing predictions against experimental observations. Hence, the emergence of non-falsifiable notions such as the multiverse.

In view of these distortions that are currently plaguing the prevailing physics culture, I am not surprised at the lack of progress in fundamental physics. The remarkable understand in our physical world that humanity has gained has come through the healthy application of the scientific method. No alternative has made any comparable progress.

What I am proposing is that we go back to the basics. First and foremost, we need to establish the scientific method as the only approach to follow. And then, we need to discuss physical guiding principles that can show the way forward in our current effort to understand the interplay between gravity and quantum 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.

Testable proposals for the measurement problem

In a previous post, I made the statement: “Currently, there are no known experimental conditions that can distinguish between different interpretations of quantum mechanics.” Well, that is not exactly true. Perhaps one can argue that no experiment has yet been performed that conclusively ruled out or confirmed any of the interpretations of quantum mechanics. But, the fact is that recently, there has been some experimentally testable proposals. Still, I’m not holding my breath.

Recently, seeing one such proposal, I remembered that I also knew about another testable proposal made by Lajos Diósi and Roger Penrose. The reason I forgot about that is probably because it seems to have some serious problems. At some point, during a conversation I had with Lajos, I told him I have a stupid question to ask him: does quantum collapse travel at the speed of light? His response was: that is not a stupid question. So, then I concluded that it is not something that any of the existing collapse models can handle correctly. In fact, I don’t think any such model will ever be able to handle it in a satisfactory manner.

Thinking back to those discussions and the other bits and pieces I’ve read about the measurement problem, I tend to reconfirm my conviction that the simplest interpretation of quantum mechanics (and therefore the one most likely to be correct) is the so-called Many Worlds interpretation of quantum mechanics. However, the more I think about it, the more I believe that “many worlds” is a misnomer. It is not about many worlds or many universes that are constantly branching off to become disjoint universes.

Perhaps one can instead call it the “multiple reality” interpretation. But how would multiple realities be different from “many worlds”? That fact is that these realities are not disjoint, but form part of the unitary whole of a single universe. These realities can be combined into arbitrary superpositions. What more, these realities are not branching of to produce more realities. The number of realities remains the same for all time. (There are actually an infinite number of them, but the cardinality of the set remains the same.) The interactions merely change the relative complex probability amplitudes of all the realities. 

Anyway, just thought I should clear this up. I don’t see myself ever writing publication on this topic.

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Entanglement

Demystifying quantum mechanics VIII

Everything is no more or less in place to discuss one of the most enigmatic phenomena found in quantum mechanics: entanglement. It is sometimes called the quintessential property of quantum mechanics. 

We have discussed the fact that quantum mechanics introduces the concept of discrete entities that carry full sets of degrees of freedom, and which I called partites. Then we learned about the paradox introduced by Einstein, Podolski and Rosen (EPR) and how it led to the understanding that nature does not have a unique reality. Although it also allows that interactions could be nonlocal, we saw that such non-locality is not in agreement with our understanding of special relativity. The final ingredient that we need to explain quantum entanglement is the concept of a superposition. We can deal with that here.

The term superposition is a fancy way of saying that we are adding up things. Superpositions are also found in classical optics. There, one can observe interference effects when two waves are superimposed (added on top of each other at the same location). What makes the situation in quantum mechanics different is that the things that are added up in a quantum superposition can consist of multiple partites (multiple combinations of discrete entities) and these partites (discrete entities) do not have to be at the same location. Since each entity carries unique properties, as described in terms of the full set of degrees of freedom, the different terms in the quantum superposition gives complete descriptions of the state in terms of the set of discrete entities that they contain. 

Each the terms in the superposition can now be seen as a unique reality. The fact there are more than one term in the superposition, implies that there are multiple realities, just like the EPR paradox showed us. One can use the many-world interpretation to try to understand what this means.

Entangled entities
Artist impression of entangled entities

There are now different effects that these superpositions can produce. In some cases one can factorize the superposition so that it becomes the product of separate superpositions for each of the individual partites. In such a case one would call the state described by the superposition as being separable. If such a state cannot be factorized in this way, the state is said to be entangled.

What is the effect of a state being entangled? It implies that there are quantum correlations among the different entities in the terms. These correlations will show up when we make measurements of the properties of the partites. Due to the superposition, a measurement of just one of these partites will give us a range of possible results depending on which term in the superposition ends up in our measurement. On the other hand, if we measure the properties of two or more of the partites, we find that their properties are always correlated. This correlation only shows up when the state is entangled.

Some people think that one can use this correlation the communicate instantaneously between such partites if they are placed at different locations that are far apart. However, as we explained before, such instantaneous communication is not possible.

This discussion may be rather abstract. So, let try to make it a bit simpler with a simple example. Say that we form a superposition where each term contains two partites (two discrete entities). In our superposition, we only have two terms and the properties of the partities can be one of only two configurations. So we can represent our state as A(1) B(2) + A(2) B(1). Here A and B represent the identities of the partites and (1) and (2) represent their properties. When I only measure A, I will get either (1) or (2) with equal probability. However, when I measure both A and B, I will either get (1) for A and (2) for B or (2) for A and (1) for B. In other words, in each set of measurements, the two partites will have the opposite properties, and this result is obtained regardless of how far apart these partites are located.

The phenomenon of quantum entanglement has been observed experimental many times. Even though it is counterintuitive, it is a fact of nature. So, this is just one of those things that we need to accept. At least, we can understand it in terms of all the concepts that we have learned so far. Therefore, it does not need to be mysterious

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