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.

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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The origin of Heisenberg uncertainty

Demystifying quantum mechanics II

Perhaps the one thing that everyone thinks about when they hear talk about quantum mechanics is Heisenberg’s uncertainty principle. It may even sometimes be considered as the essence of quantum mechanics. Now what would you say if I tell you that the Heisenberg uncertainty principle is not a fundamental principle and that the origin of this principle is not found in quantum mechanics? The fundamental origin of this uncertainty is a purely mathematical property and the reason that quantum mechanics inherited this principle is simply a result of the Planck relationship.

Werner Heisenberg

I have discussed this issue to some extent before. However, it forms an important part of the knowledge that would help to demystify quantum mechanics. Therefore, it deserves more attention.

Before the advent of quantum mechanics, the state of a particle was considered to be completely described by its position and velocity (momentum). The dynamics of a system could then be represented by a diagram showing position and velocity of the particle as a function of time. For historical reasons, the domain of such a diagram is called phase space. For a one-dimensional system (such as a harmonic oscillator), it would give a two-dimensional graph with position on one axis and velocity on the other. The state of the system is a point on the two-dimensional plane that moves along some trajectory as a function of time. For a harmonic oscillator, this trajectory is a circle.

A mathematical property (in Fourier analysis), which may have seemed to be complete unrelated at the time, is that the width of the spectrum of a function has a lower limit that is proportional to the inverse of the width of the function. This property has nothing to do with physical reality. It is a purely logical fact that can be proven with the aid of mathematics. If the function is, for instance, interpreted as the probability distribution of the position of a particle, the width of the function would represent the uncertainty in its location.

This mathematical uncertainty property was transferred to phase space by Planck’s relation, which links the independent variable of the spectrum (the wave number) with the momentum or velocity of the particle. The implication is that one cannot represent the state of a quantum particle with a single dimensionless point on phase space in quantum mechanics. Hence, the Heisenberg uncertainty principle.

So, the uncertainty associated with Heisenberg’s principle is inevitable due to Planck’s relation. And it is founded on pure logic in terms of which mathematics is based. Planck’s relation is the only physics that enters the picture. The Heisenberg uncertainty principle is therefore not a separate principle that is independent of Planck’s relationship as far as the physics is concerned.

Now, there are a few subtleties that we can address. There are also some interesting consequences based in this understanding, but I’ll leave these for later.

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