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.

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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.

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