General

Faith and science

Recently, while reading some of the writings of the christian scientist Arnold Neumaier, it struck me how christian scientists are often persecuted by other scientists for their faith. For that reason, one often finds that scientists who also believe or practice some religion tend to keep that side of their lives quiet. So I ask myself, why do non-believing scientists have the idea that one should not be a believer when practicing science?

Just to explain why this bothers me, let’s discuss some basic aspects about believe. All sane people need to believe certain things about the world they live in to make sense of it. We believe that the sun will rise everyday simply because it has done so for as long as there were people. We believe a chair will hold us when we sit on it without even thinking about it. Occasionally that my proof to be a false believe, but we nevertheless acted on that believe by sitting on it. There are also deeper believes about the universe and about right and wrong, all of which give us our world views and help us to make sense of things. None of these things are ever subjected to some form of scientific verification. In general, it is simply impossible to test everything that we believe in with a scientific process.

When it comes to science and those that practice it on a professional basis, one finds that these people often believe things concerning scientific knowledge without applying the strict scientific process. It is alarming to realize that some scientists simply believe what they read in textbooks or what they’ve been told by their lecturers without question. Unfortunately, not everything written in textbooks or taught by lecturers is correct. This tendency to take anything scientific for granted is something I call scientistic gullibility. It is surprising how widespread it is. Perhaps it is not reasonable to test everything one reads, but then one should be less certain about one’s own convictions.

Speaking of convictions, what is the benefit of believes? Why is a religion such as christianity based on believe or faith? While science is all about knowledge, religion is about conduct. It strives to govern the actions of people. As it turns out, knowledge does not govern people’s actions. Believe does. It doesn’t help to tell people something to affect their behavior if they don’t believe it. Only if they believe something will it affect their behavior. For this reason, believe (or religion) has an important role to play in human society.

Looking at the strive that exists in the world with all the conflicts, one realizes that some compassion for one’s fellow human being would go a long way to improve matters. But how does on instill such compassion in people? It needs to be grounded in their convictions, in their world view. That is the role that religion can play. Granted, some people use religion to do exactly the opposite. However, simply because people have the ability to corrupt any human activity (even science), it does not mean that that human activity is intrinsically bad.

Physics

Propping up a dead horse

Much is being written about the current state of affairs in fundamental physics. To summarize (for those not familiar with this state of affairs), there has been very little progress in fundamental physics for several decades. Perhaps the main reason for that is the fact that it becomes increasingly difficult to make experimental observations at higher energies. Perhaps we are facing the end of fundamental physics as a scientific field of study.

(One curious fact is that we have ample experimental results pointing to neutrinos having mass and yet, as far as I know, we still do not have a modified standard model that includes masses for the neutrinos. What’s up with that?)

However, there are other reasons for the current state of affairs. The theoretical landscape is being dominated by a dead theory. Obviously, this is an extremely unhealthy situation. It stands in the way of progress at a critical time in the development of humanity’s understanding of the universe we life in. What boggles one’s mind is the extent to which proponents of this dead theory will go to prop up their dead horse.

I heard that these proponents are using their influence to arrange interviews in which they claim that there is nothing wrong with their theory and that they are making remarkable progress. In the meantime, the theory is unable to make any testable predictions with no foreseeable prospect of ever being able to do so.

So I ask myself, why would people do such a deceitful thing to mislead the public? I guess one needs to put yourself in the shoes of such a person. He or she has spent a lifetime working on a theory, have succeeded in making this theory the only one being taken seriously, have secured lots of funding to work on it, and have convinced many young people to invest their careers in this theory. It is virtually impossible for such a person to turn around and admit that the theory failed.

So what can we do now? How will we get out of this mess? Do we need to wait until these proponents pass away? (Some of them are very old by now.) Unfortunately, they have enlisted so many young people that it would be difficult to stop the momentum built up in this process.

My advise is to make funding organizations accountable for this mess. These funding organizations have been funding this failed theory for decades despite the complete lack of experimental support. The fact that this theory will never be able to make predictions that can be tested is widely known. Therefore, the funding provided for this dead theory by these funding organizations is basically flying in the face of science. I think it is time that the public and the governments take these funding organizations to task about this situation. The funding for the dead theory simply needs to stop. It is not difficult to compile some reasonable funding criteria to make sure that such dead theories don’t keep on reaping in the lion’s share of research funding.

Physics

The myth of the quantum harmonic oscillator

An oscillator is any device that produces periodic variations in the value of some quantity. If these periodic variations can be described by a sinusoidal function, we call it a harmonic oscillator. Often, when we study such systems, we think of a mechanical device where an object is periodically displaced along a specific direction in space. In such a case, the classical mechanics of the system is described by a periodic variation in both the position and the momentum of the object.

At some point, somebody thought it may be a good idea to replace the oscillating object in such a mechanical oscillator by the wave function of a quantum particle. The resulting analysis led to the formulation of a simple quantum Hamiltonian in terms of position and momentum operators. These operators can be replaced by ladder operators (creation and annihilation operators), which gives the basic form of the Hamiltonian of all free field theories: theories without interactions. The interactions are added to this Hamiltonian as additional terms. In view of this role, it is generally considered that the quantum harmonics oscillator is probably the single most important system that has ever been analyzed.

The role that the formalism, which is derived from this analysis, plays in our models of physical quantum systems is not to be contested, but some of the concepts that follow from this analysis may be leading us by our noses. Among the conclusions that are derived from the quantum harmonic oscillator analysis there are some that are questionable.

What is it that gives the quantum harmonic oscillator a quantum characteristic? Is it the fact that the particle is replaced by a wave function? Formally such a wave function is just a normalized continuous function. Although it is interpreted as a quantum wave function, such a function can equally well represent a classical field, if we replace the requirement for the normalization with a requirement for a finite energy.

Or is it because Planck’s constant makes an appearance in the expression? Well, if the variables are suitably redefined in terms of dimensionless variables, Planck’s constant would disappear from the equation. The resulting equation still describes the same physical system. Therefore, Planck’s constant does not play a physically significant role in it. Such an equation can also describe the dynamics in classical scenarios. One such classical scenario is the propagation of light in a gradient index (GRIN) lens medium.

Perhaps it is because the Schrödinger equation for the quantum harmonic oscillator has discrete solutions. Such discrete solutions are interpreted as the discrete energy levels representing the different quanta. Well, in those classical scenarios where the same equation is applicable, such as the GRIN lens, the same discrete solutions exit, but without any associated quantum interpretation. Moreover, the Schrödinger equation for the quantum harmonic oscillator also admits a continuous solution with continuous free parameters representing the oscillating motion of the field.

What are we to conclude now? What value can we derive from the analysis of the quantum harmonic oscillator? Any interpretation of the proposed quantum nature of the system seems to be at best misleading. Although there may exist physical quantum systems that are described by such a quantum harmonic oscillator equation, the same formal mathematical expression is equally applicable in classical systems once Planck’s constant is removed.

In our quest to understand quantum physics, it is important to understand what is not quantum physics. It is also important to separate physics from formalism and not confuse the two.

Physics

The golden triangle of quantum mechanics, or is it?

Often, in trying to understand something, it is helpful to understand what it is not. Our understanding of quantum mechanics may be aided in the same way. Many concepts in quantum mechanics are being muddled when it comes to whether they are inclusive or exclusive. In other words, the question is whether these concepts are exclusively related to quantum mechanics or whether they also appear in classical scenarios.

As a case in point, I’ll discuss three concepts, some of which are often considered to be exclusively the domain of quantum mechanics and others that are more general. By showing that they are strongly related, effectively being equivalent, we demonstrate that none of them can be exclusive to quantum mechanics.

The three concepts are: Fourier transforms, Heisenberg uncertainty, and mutually unbiased bases. The relationship between Heisenberg uncertainty, which is considered to be exclusively related to quantum mechanics, and Fourier theory, which is a general mathematical procedure, is well known. However, it is claimed that there is more to Heisenberg uncertainty than what its relationship to Fourier theory would imply, justifying its exclusive association with quantum mechanics.

Let’s investigate the concept of Heisenberg uncertainty in a different way. We start with the notion of two operators that do not commute. The prime example being the position and momentum operators. These operators are observables, which means that they are Hermitian operators with eigenstates that form orthogonal bases. Since the two operators do not commute, their eigenstates are different. In fact, they are mutually unbiased.

What does it mean for two sets of bases elements to be mutually unbiased? It means that the magnitude of the inner product between elements of the two bases are always the same. In other words, it is not possible, to determine which element have been selected from the two sets by looking at the magnitude of their inner product. By implication, any expansion of an element from one set in terms of the elements of the other set must involved all the elements of the other set and the expansion coefficients can only differ in their phase factors. It turns out that these phase factors are equivalent to the kernel of the Fourier transform.

It is instructive to understand the last claim. For this purpose we need to consider the extreme case of an infinite dimensional bases, such as we would obtain from the momentum and position operators. Looking at the function in the argument of the exponential function of the phase factor, we find that it must be monotonically increasing in both variables (for example, position and momentum) that denote the elements of the bases to avoid ambiguity in their definitions. If there is an nonlinearity in this monotonic function for any of the two variables, then such a nonlinearity can be readily removed through a rescaling of the basis elements along that direction. In other words, all such mutually unbiased bases would be equivalent to two sets that are produced with phase factors having linear dependences on the variables in their arguments.Such a linear dependence in both variables comes out to be the product of the two variables. Such phase factors are precisely equivalent to the Fourier kernel. The expansion of the elements of one basis in terms of those of the other is always equivalent to a Fourier transform.

What do we conclude? When we have two operators that do not commute, and thus give rise to the uncertainty principle, their eigenbases form mutually unbiased bases which are represented in terms of each other as Fourier integrals. This relationship can be shown to exist in the opposite direction. As a result the three concepts: Heisenberg uncertainty, mutually unbiased bases and Fourier relationship between bases, become equivalent. They are three corners of a triangle of concepts found in quantum mechanics. However, since the relationship between two sets of bases given by a Fourier expansion can also be found in classical scenarios, it follows that the other two points of the triangle are also relevant in classical scenarios. There is nothing special about Heisenberg uncertainty that makes it exclusively to quantum mechanics.

General

Red or black?

Some time ago, I talked about the maple tree that is a reminder of Canada in my garden. I wanted to know what kind of maple it is, but could not make a positive identification. Now, it is mature enough to produce seeds.

The leaves have three lobes and have a smooth edge. They also seem to have an elongated shape, but that varies.

The bark is still too young and smooth to help much. So the leaves and the seeds are what I have to go on. According to some online tree identification sites, it can either be a red maple or a black maple. For some reason, I’m leaning toward black maple. It tends to turn red in the autumn, but I get the feeling it is not as red as a red maple would be.