Seriously, it is not that complicated

It was more than a 100 year ago that Max Planck introduced the notion of the quantization of radiation from a black body. The full-blown formulation of quantum mechanics is almost a hundred years old (the 5th Solvay conference more or less represents that achievement). Over the years since then, many ideas have been introduced about quantum physics in the struggle to understand it. Once new ideas have been introduced, nobody can ever remove them again regardless of how misleading they may be. Nevertheless, among these ideas, we can find enough information to form a picture representing an adequate understanding of quantum physics.

It would be very arrogant to claim that this understanding is unassailable or even complete. (I still have some issues with fermions.) Therefore, I simply call it my current understanding. It is a minimalist understanding in that it discards the unnecessary conceptual baggage (thus following Occam’s razor). Yet, it provides an ontology (although not one that guarantees everybody’s satisfaction).

I’ve written about many aspects of this understanding. So, where possible, I’ll thus link to those discussions. Where additional discussions may be necessary, I’ll postpone those discussions for later. Here then follows a breakdown of my current understanding of quantum physics.

Firstly, fundamental particles are not particles in the traditional sense. They are not “dimensionless points traveling on world lines.” Instead, they are better represented by wave functions or fields (or partites). Interactions among these fundamental fields (using the term “fields” instead of “particles” to avoid confusion) are dimensionless events in spacetime.

As a consequence, there is no particle-wave duality. Fields propagate as waves and produce the interference as, for example, seen in the double-slit experiment. Whenever these fundamental fields are observed as discrete entities, it is not a particle in the traditional sense that is being observed, but rather the localized interaction of the field with the device that is used for the observation.

Secondly, interactions are the key that leads to the quantum nature of the physical world. What Max Planck discovered was that interactions among fundamental fields are quantized. These fields exchange energy and momentum in quantized lumps. This concept was also reiterated in Einstein’s understanding of the photo-electric effect. Many of the idiosyncratic concepts of quantum physics follow as consequences of the principle of quantized interactions.

The Heisenberg Uncertainty Principle is not a fundamental principle. It is a consequence of the quantization relations associated with interactions. These relations convert conjugate variables into Fourier variables, which already represent the uncertainty principle. As a result, the conjugate variables inherit their uncertainty relationship from Fourier theory. It becomes more prevalent in quantum physics, due to the restrictions that the quantization of interactions imposes on the information that can be obtained from the observation of a single “particle.”

Planck’s constant only plays a physical role at interactions. Once these interactions are done, the presence of Planck’s constant the expressions of the fields have no significance. It can be removed through simple field redefinitions that have no effect on the physical representations of these fields. As a result, the significance that is attached to Planck’s constant in scenarios that are not related to interactions are generally misleading if not completely wrong.

Thirdly, another key concept is the principle of superposition. The interactions among fundamental fields are combined as a superposition of all possibilities. In other words, they are integrated over all points in spacetime and produce all possible allowed outcomes. As a consequence, after the interactions, the resulting fields can exist in a linear combination of correlated combinations. This situation leads to the concept of entanglement.

Since a single “particle” only allows a single observation, the different measurement results that can be obtained from the different elements in a superposition are associated with probabilities that must add up to one. The coefficients of the superposition therefore form a complex set of probability amplitudes. The conservation of probability therefore naturally leads to a unitary evolution of the state of the single particle in terms of such a superposition. This unitarity naturally generalizes to systems of multiple particles. It naturally leads to a kind of many-worlds interpretation.

It seems to me that all aspects of quantum physics (with the exception of fermions) follow from these three “principles.” At least, apart from the question of fermions, I am not aware of anything that is missing.

Principles of 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.

Classical and quantum superpositions

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.

Discreteness

Demystifying quantum mechanics V

Perhaps one of the most iconic “mysteries” of quantum mechanics is the particle-wave duality. Basically, it comes down to the fact that the interference effects one can observe implies that quantum entities behave like waves, but at the same time, these entities are observed as discrete lumps, which are interpreted as particles. Previously, I explained that one can relax the idea of localized lumps a bit to allow only the interactions, which are required for observations, to be localized. So instead of particles, we can think of these entities as partites that share all the properties of particles, accept that they are not localized lumps. So, they can behave like waves and thus give rise to all the wave phenomena that are observed. In this way, the mystery of the particle-wave duality is removed.

Now, it is important to understand that, just like particles, partites are discrete entities. The discreteness of these entities is an important aspect that plays a significant role in the phenomena that we observe in quantum physics. Richard Feynman even considered the idea that “all things are made of atoms” to be the single most important bit of scientific knowledge that we have.

Model of the atom

How then does it happen that some physicist would claim that quantum mechanics is not about discreteness? In her blog post, Hossenfelder goes on to make a number of statements that contradict much of our understanding of fundamental physics. For instance, she would claim that “quantizing a theory does not mean you make it discrete.”

Let’s just clarify. What does it mean to quantize a theory? It depends, whether we are talking about quantum mechanics or quantum field theory. In quantum mechanics, the processing of quantizing a theory implies that we replace observable quantities with operators for these quantities. These operators don’t always commute with each other, which then leads to the Heisenberg uncertainty relation. So the discreteness is not immediately apparent. On the other hand, in quantum field theory, the quantization process implies that fields are replaced by field operators. These field operators are expressed in terms of so-called ladder operators: creation and annihilation operators. What a ladder operator does is to change the excitation of a field in discrete lumps. Therefore, discreteness is clearly apparent in quantum field theory.

What Hossenfelder says, is that the Heisenberg uncertainty relationships is the key foundation for quantum mechanics. In one of her comments, she states: “The uncertainty principle is a quantum phenomenon. It is not a property of classical waves. If there’s no hbar in it, it’s not the uncertainty principle. People get confused by the fact that waves obey a property that looks similar to the uncertainty principle, but in this case it’s for the position and wave-number, not momentum. That’s not a quantum phenomenon. That’s just a mathematical identity.”

It seems that she forgot about Louise de Broglie’s equation, which relates the wave-number to the momentum. In a previous post, I have explained that the Heisenberg uncertain relationship is an inevitable consequence of the Planck and de Broglie equations, which relate the conjugate variables of the phase space with Fourier variables. It has nothing to do with classical physics. It is founded in the underlying mathematics associated with Fourier analysis. Let’s not allow us to be mislead by people that are more interested in sensationalism than knowledge and understanding.

The discreteness of partites allows the creation of superpositions of arbitrary combinations of such partites. The consequences for such scenarios include quantum interference that is observed in for instance the Hong-Ou-Mandel effect. It can also lead to quantum entanglement, which is an important property used in quantum information systems. The discreteness in quantum physics therefore allows it to go beyond what one can find in classical physics.

Partiteness

Demystifying quantum mechanics IV

Yes I know, it is not a word, at least not yet. We tend to do that in physics sometimes. When one wants to introduce a new concept, one needs to give it a name. Often, that name would be a word that does not exist yet.

What does it mean? The word “partiteness” indicates the property of nature that it can be represented in terms of parties or partites. It is the intrinsic capability of a system to incorporate an arbitrary number of partites. In my previous post, I mentioned partites as a replacement for the notion of particles. The idea of partites is not new. People often consider quantum systems consisting of multiple partites.

What are these partites then? They represent an abstraction of the concept of a particle. Usually the concept is used rather vaguely, since it is not intended to carry more significance than what is necessary to describe the quantum system. I don’t think anybody has ever considered it to be a defining property that nature possesses at the fundamental level. However, I feel that we may need to consider the idea of partiteness more seriously.

Classical optics diffraction pattern

Let’s see if we can make the concept of a partite a little more precise. It is after all the key property that allows nature to transcend its classical nature. It is indeed an abstraction of the concept of a particle, retaining only those aspects of particles that we can confirm experimentally. Essentially, they can carry a full compliment of all the degrees of freedom associated with a certain type of particle. But, unlike particles, they are not dimensionless points traveling on world lines. In that sense, they are not localized. Usually, one can think of a single partite in the same way one would think of a single particle such as a photon, provided one does not think of it as a single point moving around in space. A single photon can have a wave function described by any complex function that satisfies the equations of motion. (See for instance the diffraction pattern in the figure above.) The same is true for a partite. As a result, a single partite behaves in the same way as a classical field. So, we can switch it around and say that a classical field represents just one partite.

The situation becomes more complicated with multiple partites. The wave function for such a system can become rather complex. It allows the possibility for quantum entanglement. We’ll postpone a better discussion of quantum entanglement for another time.

Multiple photons can behave in a coherent fashion so that they all essentially share the same state in terms of the degrees of freedom. All these photons can then be viewed collectively as just one partite. This situation is what a coherent classical optical field would represent. Once again we see that such a classical field behaves as just one partite.

The important difference between a particle and a partite is that the latter is not localized in the way a particle is localized. A partite is delocalized in a way that is described by its wave function. This wave function describes all the properties of the partite in terms of all the degrees of freedom associated with it, including the spatiotemporal degrees of freedom and the internal degrees of freedom such as spin.

The wave function must satisfy all the constraints imposed by the dynamics associated with the type of field. It includes interactions, either with itself (such as gluons in quantum chromodynamics) or with other types of fields (such as photons with charges particles).

All observations involve interactions of the field with whatever device is used for the observation. The notion of particles comes from the fact that these observations tend to be localized. However, on careful consideration, such a localization of an observation only tells us that the interactions are localized and not that the observed field must consist of localized particles. So, we will relax the idea that fields must be consisting of localized particle and only say that, for some reason that we perhaps don’t understand yet, the interaction among fields are localized. That leaves us free to consider the field as consisting of nonlocal partites (thus avoiding all sort of conceptual pitfalls such as the particle-wave duality).

Hopefully I have succeeded to convey the idea that I have in my mind of the concept of a partite. If not, please let me know. I would love to discuss it.

What particle?

Demystifying quantum mechanics III

The notion of a particle played an important role in our understanding of fundamental physics. It also lies at the core of understanding quantum mechanics. However, there are some issues with the notion of a particle that can complicate things. Before addressing the role that particles play in the understanding of quantum mechanics, we first need to look at these issues.

Particle trajectories detected in a high energy experiment

So what is this issue about particles? The problem is that we don’t really know whether there really are particles. What?!!! Perhaps you may think that what I’m referring to has something to do with the wave-particle duality. No, this issue about the actual existence of particles goes a little deeper than that.

It may seem like a nonsense issue, when one considers all the experimental observation of particles. The problem is that, while the idea of a particle provides a convenient explanation for what we see in those experiments, none of them actually confirms that what we see must be particles. Even when one obtains a trajectory as in a cloud chamber or in the more sophisticated particle detectors that are used in high energy particle experiments, such as the Large Hadron Collider, such a trajectory can be explained as a sequence of localized observations each of which projects the state onto a localize pointer state, thus forcing the state to remain localized through a kind of Zeno effect. It all this sounds a little too esoteric, don’t worry. The only point I’m trying to make is that the case for the existence of actual particles is far from being closed.

Just to be on the same page, let’s first agree what we mean when we talk about a particle. I think it was Eugene Wigner that defined a particle as a dimensionless point traveling on a world line. Such a particle would explain those observed trajectories, provided one allows for a limited resolution in the observation. However, this definition runs into problems with quantum mechanics.

Consider for example Young’s double slit experiment. Here the notion of a particle on a world line encounters a problem, because somehow the particle needs to pass through both slits to produce the interference pattern that is observed. This leads to the particle-wave duality. To solve this problem, one can introduce the idea of a superposition of trajectories. By itself this idea does not solve the problem, because these trajectories must produce an interference pattern. So one can add the notion (thanks to Richard Feynman) of a little clock that accompanies each of the trajectories, representing the evolution of the phase along the trajectory. Then when the particle arrives at the screen along these different trajectories the superposition together with the different phase values will determine the interference at that point.

Although the construction thus obtained can explain what is being seen, it remains a hypothesis. We run into the frustrating situation that nature does not allow us any means to determine whether this picture is correct. Every observation that we make just gives us the same localized interaction and there is no way to probe deeper to see what happens beyond that localize interaction.

So, we arrive at the situation where our scientific knowledge of the micro-world will always remain incomplete. We can build strange convoluted constructs to provide potential explanations, but we can never establish their veracity.

This situation may seem like a very depressing conclusion, but if we can accept that there are things we can never know, then we may develop a different approach to our understanding. It helps to realize that our ignorance exactly coincides with the irrelevance of the issue. In other words, that which we cannot know is precise that which would never be useful. This conclusion follows from the fact that, if it could have been useful, we would have had the means to study it and uncover a true understanding of it.

So, let’s introduce at a more pragmatic approach to our understanding of the micro-world. Instead of trying to describe the exact nature of the physical entities (such as particles) that we encounter, let’s rather focus on the properties of these entities that would produce the phenomena that we can observe. Instead of particles, we focus of the properties that make things look like particles. This brings us to the notion of a party or a partite.

But now the discussion is becoming too long. More about that next time.