It may seem strange that there are no fundamental physical principle for quantum physics that is associated with uncertainty among those that I have proposed recently. What would be the reason? Since we refer to it as the Heisenberg uncertainty principle, it should qualify as one of the fundamental principles of quantum physics, right? Well, that is just it. Although it qualifies as being a principle, it is not fundamental.
It may help to consider how these concepts developed. At first, quantum mechanics was introduced as an improvement of classical mechanics. Therefore, quantities like position and momentum played an important role.
A prime example of a system in classical mechanics is the harmonic oscillator. Think of a metal ball hanging from a spring. Being pulled down and let go, the ball will start oscillating, moving periodically up and down. This behavior is classically described in terms of a Hamiltonian that contains the position and momentum of the metal ball.
In the quantum version of this system, the momentum is replaced by the wave vector times the Planck constant. But position and the wave vector are conjugate Fourier variables. That is the origin of the uncertainty. Moreover, it also leads to non-commutation when position and momentum are represented as operators. The sum and difference of these two operators behave as lowering and raising operators for quanta of energy in the system. The one reduces the energy in discrete steps and the other increases it in discrete steps.
It was then found that quantum mechanics can also be used to improve classical field theory. But there are several differences. Oscillations in fields are not represented as displacements in position that is exchange into momentum. Instead, their oscillations manifest in terms of the field strength. So, to develop a quantum theory of fields, one would start with the lowering and raising operators, which are now called creation and annihilation operators or ladder operators. Their sum and difference produce a pair of operators that are analogues to the position and momentum operators for the harmonic oscillator. In this context, these are called quadrature operators. They portray the same qualitative behavior as the momentum and position operators. They represent conjugate Fourier variables and therefore again produce an uncertainty and non-commutation. The full development of quantum field theory is far more involved then what I described here, but I only focused on the origin of the uncertainty in this context here.
So, in summary, uncertainty emerges as an inevitable consequence of the Fourier relationship between conjugate variables. In the case of mechanical systems, these conjugate variables come about because of the quantum relationship between momentum and wave vector. In the case of fields, these conjugate variables comes from the ladder operators, leading to analogues properties as found for the formal description of the harmonic oscillator. Hence, uncertainty is not a fundamental property in quantum physics.
Previously, I argued for principles rather than postulates. Usually, principles are added to a field of study only after some progress have been made with the theories in that field. However, sometimes these principles are required ahead of the time to make progress in a field. That may be the case in fundamental physics where such principles can be used as guiding principles. However, in the latter case such principles may be just guess-work. They may turn out to be wrong.
Quantum physics has been around for a long enough time to justify having its own set of principles. There are postulates for quantum mechanics, but as I explained, they are like a set of axioms for the mathematical formalism and therefore don’t qualify as principles. Principles are statements phrased in terms of physical concepts and not in terms of mathematical concepts.
Here, I want to propose such principles. They are a work in progress. Those that I can state are not extremely surprising. They shouldn’t be because quantum physics has been investigated in so many different ways. However, there are some subtleties that need special attention.
The first principle is simply a statement of Planck’s discovery: fundamental interactions are quantized. Note that it does not say that “fields” or “particles” are quantized, because we don’t know that. All we do know is what happens at interactions because all our observations involve interactions. Here, the word “quantized” implies that the interacting entities exchange quantized amounts of energy and momentum.
What are these interacting entities? Usually we would refer to them as particles, but that already makes an assumption about their existence. Whenever we make an observation that would suggest that there are particles, we actually see an interaction. So we cannot conclude that we saw a particle, but we can conclude that the interaction is localized. Unless there is some fundamental distance scale that sets a lower limit, the interaction is point-like – it happens at a dimensionless point. The most successful theories treat these entities as fields with point-like interactions. We can therefore add another principle: fundamental interactions are localized. However, we can combine it with the previous principle and see it as another side of one and the same principle: fundamental interactions are quantized and localized.
The next principle is a statement about the consequences of such interactions. However, it is so important that it needs to be stated as a separate principle. I am still struggling with the exact wording, so I’ll just call it the superposition principle. Now, superposition is something that already exists in classical field theory. In that case, the superposition entails the coherent additions of different fields. The generalization that is introduced by quantum physics is the fact that such superpositions can involved multiple entities. In other words, the superposition is the coherent addition of multiple fields. The notion of multiple entities is introduced due to the interactions. It allows a single entity to split up into multiple entities, each of which can carry a full compliment of all the degrees of freedom that can be associated with such an entity. However, due to conservation principles, the interaction sets up constraints on the relationship among the degrees of freedom of the different entities. As a result, the degrees of freedom of these entities are entangled, which manifests as a superposition of multiple entities.
We need another principle to deal with the complexities of fermionic entities, but here I am still very much in the dark. I do not want to refer to the anti-commuting nature of fermionic operators because that is a mathematical statement. Perhaps, it just shows how little we really know about fermions. We have a successful mathematical formulation, but still do not understand the physical implications of this formulation.
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).
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.