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.

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.

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).

Non-commutation

It is believed that the non-commutation of operators is a characteristic property of quantum mechanics. So much so that axiomatic mathematical structures are developed specifically to represent this non-commuting nature for the purpose of being the ideal formalism in terms of which quantum physics can be modeled.

Is quantum physics the exclusive scenario in which non-commuting operators are found? Is the non-commutative nature of these operators in quantum mechanics a fundamental property of nature?

No, one can also define operators in classical theories and find that they are non-commuting. And, no, this non-commuting property is not fundamental. It is a consequence of more fundamental properties.

Diffraction pattern

To illustrate these statements, I’ll use a well-known classical theory: Fourier optics. It is a linear theory in which the propagation of a beam of light is represented in terms of an angular spectrum of plane waves. The angular spectrum is obtained by computing the two-dimensional Fourier transform of the complex function representing the optical beam profile on some transverse plane.

The general propagation direction of such a beam of light, which is the same thing as the expectation value of its momentum, can be calculated with the aid of the angular spectrum as its first moment. An equivalent first moment of the optical beam profile gives us the expectation value of the beam’s position. Both these calculations can be represented formally as operators. And, these two operators do not commute. Therefore, the non-commutation of operators has nothing to do with quantum mechanics.

So what is going on here? It is an inevitable consequence that two operators associated with quantities which are Fourier conjugate variables would be non-commuting. Therefore, the non-commuting property is an inevitable result of Fourier theory. Quantum mechanics inherits this property because the Planck relationship converts the phase space variables, momentum and position, into Fourier conjugate variables.

So, is Fourier analysis then the fundamental property? Well, no. There is a more fundamental property. The reason why Fourier conjugate variables lead to non-commuting operators is because the bases associated with these conjugate variable are mutually unbiased.

We can again think of Fourier optics to understand this. The basis of the angular spectrum consists of the plane waves. The basis of the beam profile are the points on the transverse plane. Since plane waves have the same amplitude at all points in space, the overlap of a plane wave with any point on the transverse plane gives a result with the same magnitude. Hence, these two bases are mutually unbiased.

Although Fourier theory always leads to such mutually unbiased bases, not all mutually unbiased bases are produced by a Fourier relationship. Another example is found with Lie algebras. For example, consider the Lie algebra associated with three dimensional rotations. This algebra consists of three matrices called the Pauli matrices. We can determine the eigenbases of the three Pauli matrices and we’ll see that they are mutually unbiased. These three matrices do not commute. So, we can make a general statement.

Two operators are maximally non-commuting if and only if their eigenbases are mutually unbiased

The reason for the term “maximally” is to take care of those cases where some degree of non-commutation is seen even when the bases are not completely unbiased.

Although the Pauli matrices are ubiquitous in quantum theory, they are not only found in quantum physics. Since they represent three-dimensional rotations they are also found in purely classical scenarios. Therefore, their non-commutation has nothing to do with quantum physics per se. Of course, as we already showed, the same is true for Fourier analysis.

So, if we are looking for some fundamental principles that would describe quantum physics exclusively, then non-commutation would be a bad choice. The hype about non-commutation in quantum physics is misleading.

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Quantum teleportation

One of the most iconic quantum phenomena is quantum teleportation. But the reason why it is so iconic has nothing to do with the idea behind “beam me up, Scotty.”¬† In quantum teleportation, it is only the state of matter that is being transferred and not the matter itself. Usually, it is the state of light (a photon) that is being teleported. Quantum teleportation is iconic because it involves a mechanism that reveals a truly quantum nature.

How does it work? The state to be teleported is represented by photons that are specially prepared for the purpose. You can think of some light source that produces photons having specific properties that represent their state. We shall label one such photon as A. The resource that will mediate the teleportation process is a different bunch of photons representing an entangled state. This entangled state consists of a pair of entangle photons, which we label as B and C, respectively. To perform the process of teleportation, all we need to do is to make a joint measurement of photons A and B. It is the nature of this joint measurement that makes the process of quantum teleportation possible. The information that we obtain from this measurement tells us what transformation to perform on C to reproduce the state of A. Sometimes, we would not need to make any transformation. The state of C would already be that of A.

So, let’s look a little more carefully at the nature of the joint measurement. What do we mean by a joint measurement? To understand what it means, we need to discuss the state of photon A . There are many different possible states that this photon can have. All such states are collected into a set that we call a Hilbert space. Any of the states in this set can be represented as a superposition of a small set of states that we call a basis. One way to determine the state of a photon is the measure how much of each of these basis elements are required to make up the state of the photon. Such measurements are called projective measurements.

To understand joint measurements we just need to generalize our understanding of projective measurements a bit. What the measurement instrument in a teleportation experiment sees is not just A, but A and B together. The Hilbert space for the combination of the states of these two photons consists of all the combinations of all the states from their respective Hilbert spaces. One can produce a basis for the combined Hilbert space by combining the elements of the respective bases. There are different ways to do that, including some that would cause the elements of the combined basis to be entangled states. That is the key for quantum teleportation. One needs to make projective measurements of the combined state in a basis where the elements are themselves entangled.

Why would projective measurements in terms of an entangled basis cause teleportation? This mechanism is what makes teleportation an amazing process. It involves the multiple-reality nature of the quantum world. The entangled resource state can be interpreted in terms of such multiple realities. What joint measurements are doing to knit these multiple realities together with those presented by the input state A. But the latter is just one state (one reality), therefore, in the ideal case, only one of the realities of the resource state will survive the measurement process, the one where C has the same state as A. In a less ideal case, bits and pieces of A will be distributed over different realities. In that case, one can reconfigure the different realities with the aid of a unitary transformation on C, such that A becomes associated with just one reality in which C would then have the same state as A. The outcome of the joint measurement would tell us which unitary transformation to perform to achieve the necessary reconfiguration.

How does one make projective measurements in terms of an entangled basis? That is challenging, but people have identified at least two ways to do that. The first process and the one most often used is the Hong-Ou-Mandel effect. It is accomplished with the aid of a beamsplitter, causing a quantum interference effect. If two photons are observed simultaneously from the two output ports, then it signals the detection of a special entangled state called a Bell state, which implies a successful teleportation. The benefit of this method is that it does not require any unitary transformation of C.

Another way to perform a joint measurement is with the aid of the inverse of a process that would produce entangled states. In quantum optics, most entangled photon states are produces with the aid of a nonlinear optical process called parametric down-conversion. The inverse process is parametric up-conversion (also called sum frequency generation). While down-conversion converts a single incoming photon into two photons that are entangled to maintain energy and momentum conservation, the up-conversion process takes two incoming photons and combine them into one photon. A successful up-conversion implies a projection unto an entangled state to maintain energy and momentum conservation. Therefore, it can also be used for quantum teleportation. However, the process of up-conversion is very inefficient.

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