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.

Why not an uncertainty principle?

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.

Transcending the impasse, part IV

Planck’s constant

It all started with the work of Max Planck. He famously introduced the notion that the energy absorbed or emitted during an interaction is proportional to the frequency of the field being absorbed or emitted. The proportionality constant h is today considered as a fundamental constant of nature. In honor of Max Planck is called Planck’s constant.

Max Planck, the father of quantum mechanics

The reason why we need to look at the Planck constant for transcending the impasse in physics is because there seem to be some confusion as to the role that it plays in quantum mechanics. The confusion manifests in two aspects of quantum mechanics.

One of these aspects is related to the transition from quantum to classical physics, which we have considered before. It is assumed that one should recover classical physics from quantum physics by simply taking the limit where Planck constant goes to zero. Although this assumption is reasonable, it depends on where the constant shows up. One may think that the presence of Planck’s constant in expressions should be unambiguous. That turns out not to be the case.

An example is the commutation relation for spin operators. Often one finds that the commutator produces the spin operators multiplied by Planck’s constant. According to this practice the limit where Planck’s constant goes to zero would imply that spin operators must commute in the classical theory, which is obviously not correct. Spin operators are the generators of three-dimensional rotations which still obey the same algebraic structure in classical theories as they do in quantum theories.

So when should there be a factor of Planck’s constant and when not? Perhaps a simple way to see it is that, if one finds that a redefinition of the quantities in an expression can be used to remove Planck’s constant from that expression, then it should not be there in the first place.

Using this approach, one can consider what happens in a Hamiltonian or Lagrangian for a theory. Remember that both of these are divided by Planck’s constant in the unitary evolution operator or path integral, respectively. One also finds that the quantization of the fields in these theories always contains a factor of the square root of Planck constant. If we pull it out of the definition and make it explicit in the expression of the theory, one finds that Planck’s constant cancels for all the free-field terms (kinetic term and mass term) in the theory. The only terms in either the Hamiltonian or the Lagrangian where the Planck constant remains are the interaction terms. This brings us full circle to the reason why Max Planck introduced the constant in the first place. Planck’s constant is specifically associated with interactions.

So if one sets Planck constant to zero in a theory, the result is that it removes all the interactions. It leads to a free-field theory without interactions, which is indistinguishable form a classical theory. Interactions are responsible for the changes in the number of particles and that is where all the quantum effects come from that we observe.

The other confusion about Planck’s constant is related to the uncertain principle. Again, the role that Planck’s constant plays is that it relates two quantities that, on the one hand, is the conjugate variable on phase space with, on the other hand, the Fourier variable. Without this relationship, one recovers the same uncertainty relationships between Fourier variables in classical theories, but not between conjugate variables in phase space. Planck’s relationship transfers the uncertainty relationship between Fourier variables to conjugate variables on phase space. So, the uncertainty relationship is not a fundamental quantum mechanical principle. No, it is the Planck relationship that deserves that honor.

This image has an empty alt attribute; its file name is 1C7DB1746CFC72286DF097344AF23BD2.png