Over recent years much has been said about the similarities between religion and science. Religion and science, and especially physics, it are said are on a convergent path. The religious establishment is particularly fond of this idea, especially when spoken by a scientist, since it enhances their credibility, bolsters their position of authority and helps recruit more converts. The question is whether such comparisons stand up to scrutiny and if so whether it is science that has taken on the characteristics of a religion or religion that has taken on the characteristics of a science.

Our view of science is as a discipline where everything can be reduced to a fundamental set of axioms each of which has a logically consistent proof or is capable of irrefutable experimental verification. The mechanics of motion, for example, can be reduced to a set of basic equations. These in turn are derived from other more basic equations or from direct observation. In science we are asked to believe only what can be proved from first principles.

Religion on the other hand is a discipline where everything can be reduced to a set of propositions each of which must be accepted as an article of faith. In religion we are asked to believe what other people (I use the word advisedly) tell us are the fundamental truths.

How can it be that these two, seemingly opposite philosophical approaches, be said to be convergent?

Is the universe founded on fundamental principles that can only be divined by soothsayers and prophets?

Have we given up on the scientific method as a means of obtaining an understanding of the universe?

Or is the scientific method subject to some sort of limitation which ultimately forces us to rely on articles of faith to divine the true nature of the universe?

Has religion taken the characteristics of a scientific discipline?

Or has science taken on the characteristics of a religion?

It is certainly not the case that religion has changed – all of the major religions remain steadfast in the idea that we as humans cannot know everything and that in the final analysis we must accept certain ideas as articles of faith. The fact that these are somewhat of a moving target does not seem to matter too much. So for example up until the 17^{th} century Christians were supposed to accept that the earth was the centre of the universe, then that the sun was the centre, and now that the universe has no centre. Religion, in other words, has been forced to bow to the pressure of scientific discovery and knowledge, but remains at its heart unaltered.

Science has undergone many transitions as successive discoveries have been made and theories proven or disproven. Science admits to not knowing all the answers but seeks to search them out, one by one. Science is therefore dynamic in a way that religion is not. So what has happened to bring about the current state of affairs? What has changed to lead us to believe that science and religion have anything at all in common? It would appear that it is science that has changed and not religion.

We must first ask ourselves since when this change came about. Since when have science and religion been seen to be on convergent paths? Up until the beginning of the 20^{th} Century science and religion were clearly on divergent paths. The works of the astronomers such as Copernicus, Galileo, Newton, Hooke and generations of other scientists have gradually driven a wedge between science and religion culminating in the works of Darwin. This is despite the fact that many of these individual scientists held on to their religious sensibilities.

It is only over the last 100 or so years that such comparisons have been meaningfully drawn. I would go further and suggest that in fact it is only since the advent of quantum theory that science has taken on the characteristics of a religion.

Of course there are references to God by scientists,

“Black holes are where God divided by zero.”

― Albert Einstein

“Coincidence is God’s way of remaining anonymous.”

― Albert Einstein, *The World As I See It*

“God does not play dice with the universe.”

― Albert Einstein,* **The Born-Einstein Letters 1916-55*

“Stop telling God what to do with his dice.”

― Niels Bohr

These do not necessarily imply that science has any of the properties of a religion, more they are comments made by scientists in the context of a society where religion exists.

There are however comments which attempt to draw direct comparison between science and religion. They seek to impart some of the mystery normally associated with religion into a scientific discipline.

“Those who are not shocked when they first come across quantum theory cannot possibly have understood it.”

― Niels Bohr, *Essays 1932-1957 on Atomic Physics and Human Knowledge*

“Everything we call real is made of things that cannot be regarded as real.”

― Niels Bohr

“What I am going to tell you about is what we teach our physics students in the third or fourth year of graduate school… It is my task to convince you not to turn away because you don’t understand it. You see my physics students don’t understand it… That is because I don’t understand it. Nobody does.”

― Richard P. Feynman, *QED: The Strange Theory of Light and Matter*

“If you thought that science was certain – well, that is just an error on your part.”

― Richard P. Feynman

Such comments seek to mystify science, to establish the scientist as some sort of wise man or priest whose role is to translate to the poor foolish layman all of the intricacies of a discipline which he is not capable of understanding.

To understand how and why this situation came about we need to look at how physics has evolved over the last century and to identify the points at which it has taken on this mantle of a religion. Over the last century, physics has come to rely, not on a set of axioms to sustain its position, but on a set of beliefs. These are the “articles of faith” that characterize modern quantum theory, the quantum catechism referred to in the title.

- The idea that angular momentum is quantized and can only occur in discrete increments of Planck’s constant
- The existence of the particle as a wave whose wavelength derives from Planck’s constant divided by linear momentum (unlike on any other scale where wavelength is derived from the total angular momentum divided by linear momentum).
- The idea that uncertainty is somehow intrinsic to the particle/wave and that whatever this thing is it ‘collapses’ to reveal its nature when it is observed. An idea embodied in the story (or should I say parable) of Schrödinger’s cat
- Quantum entanglement and action at a distance
- The idea that forces are mitigated through the action of particles
- The idea that there are such things as virtual particles
- The idea that reality is somehow subjective

The list goes on…

Bit by bit these ideas have grown, each one just a small increment on what had gone before, each one seemingly correct given what had gone before, growing like a mound of sticking plasters on a festering sore.

While at first site each of these ideas appears to be capable of being traced back to first principles, in fact they are not. When we look at these ideas in detail we find that they can each trace their origin back to the day in 1916 when Niels Bohr published his paper on the structure of the hydrogen atom. That the paper was seriously flawed was widely understood at the time, but the assumption which underlay it was accepted and has been accepted as a matter of fact ever since. It is this assumption that is the festering sore that lies behind the slide of physics from a scientific discipline to a quasi-religious system of beliefs.

Hydrogen is the simplest of atoms, comprising a single nuclear proton orbited by a single negatively charged electron. In his model for the structure of the hydrogen atom Bohr first balances the electrical force acting to draw these two particles towards one another against the centrifugal force tending to throw them apart. Bohr needed a second equation in order to solve for the two unknowns of orbital radius and orbital velocity.

He took the idea of another physicist, John W Nicholson, that Planck’s constant was a measure of angular momentum, but also that angular momentum could only take on values which were an integer multiple of this basic value. This is the theory that angular momentum is quantized and which forms the first article of faith in the quantum catechism. When Bohr solved these two equations he found that his model yielded results for the energy levels of the atom which matched those of the empirically derived Rydberg model.

When the model was rejected as being unsatisfactory, nobody sought to question Bohr’s assumption which lay behind it and since then this assumption has been incorporated one way or another into every model that has been proposed. To the point where the assumption that angular momentum is quantized is now never questioned and has taken on the status of an article of faith.

Of even greater concern however is just how its advocates seek to justify this assumption. The assumption that angular momentum is quantized enjoys a ‘proof’, but it is in the nature of that proof that the similarities between religion and science are to be found.

No one can prove a fundamental religious truth, by definition they must be accepted as articles of faith, but that has never stopped people from trying[i]. As a result a number of ‘proofs’ of such things as the existence of god or the existence of miracles or any number of other religious ideas has taken on a common form. Such proofs are described as self referring. An self referring proof is one which postulates that something is true and then goes on to show that therefore it must be true, usually through a tortuous series of arguments whose main purpose is to obfuscate the link between the initial proposition and the final conclusion.

Proofs of the existence of god or of miracles almost invariably take on this form. It is postulated that god exists and then through a series of logical steps it is shown that therefore god does exist. When expressed in these terms it is clear that such so called proofs are nothing of the sort. What they amount to is a restatement of belief.

The proof of the quantization of angular momentum takes on just such a form. From Bohr’s assumption the model of the hydrogen atom is described, the model is dropped but the assumption remains. Louis de Broglie then goes on to express this assumption in a different form, in terms of wavelengths of fictitious waves that deny any physical reality. No matter the waves are what is important here. Schrödinger then develops a set of equations which describe these waves, unwittingly incorporating Bohr’s assumptions into the equations themselves. The equations are expressed in matrix terms, a move which obfuscates the underlying assumption even more. The matrix equations are solved and lo and behold they show that the solutions are consistent with the idea that angular momentum is quantised.

This line of reasoning, this so called proof, is the very definition of a self referring argument and such arguments have no validity. This means that the whole issue of whether angular momentum is quantized remains an open question.

In the case of religion, the argument stops there, either you believe in god or you don’t. If you don’t; then god’s existence defies proof and if you do; it doesn’t need to be proven. Science is a different matter or it should be. It is not sufficient to simply ask whether you believe in the quantization of angular momentum or not. It is not sufficient to accept any proof which begins in the domain of quantum theory, since any such proof is bound to be sef referring. When we recognise an self referring argument as scientists we have to question it. One of the first questions to ask is what would constitute a scientifically valid proof?

In the case of the quantization of angular momentum the only valid form of proof is one which does not begin with the assumption that lies at its heart. There can therefore be no proof of the quantization of angular momentum which begins from the assumption that angular momentum is quantized. This ultimately means that it is not possible to prove the quantization of angular momentum based on quantum theory. The only valid proof of the quantization of angular momentum is one which is firmly rooted in classical mechanics.

This is the missing link between quantum theory and classical theory. If it could be proved using classical mechanics, that angular momentum is quantized, we would have the bridge that divides classical theory from quantum theory. Such a proof would remove any scientific doubt, it would restore physics to a scientific discipline and would banish the nonsense of comparing science with religion to the dustbin of history.

Angular momentum is the product of the orbital velocity, the orbiting mass and the orbital radius. We must also take into account the effects of relativity, which at velocities approaching the speed of light affect the mass component of this product.

To prove that angular momentum is quantized we must show a causal link between the change in radius and the change in orbital velocity which results in an increase or decrease in angular momentum by one or more quanta, taking into account the effects of relativity on mass. It is not sufficient to show, as Bohr did, that if we assume angular momentum is quantised, then the energy levels of the atom match those of the empirically derived Rydberg formula. That is a necessary condition for such a proof, but not a sufficient one. There might be any number of variables or combinations of variables that produce such a match besides that of angular momentum being quantized.

In short then, to answer the question as to why physics and religion appear to be on a converging path, we must look to the tenets which underlie modern quantum theory. When we do so we find that quantum theory is not founded on scientific fact, not one which can be traced back to a set of fundamental axioms, but is instead founded on a belief, namely that angular momentum can only take on values which are an integer multiple of Planck’s constant.

Perhaps it is not surprising that I am of the opinion that no such proof is possible. That angular momentum is not quantized and that it is therefore not possible to prove that it is quantized. The reasons for suspecting that this is the case stem from the fact that in the classical domain, angular momentum is the product of three separate and unconnected variables, the orbital radius, the orbital velocity and the mass. At least one of these, the mass term, is definitely affected by relativity and at least one of them, the orbital radius, is definitely not affected by relativity. Relativity introduces a continuously variable element into the value of the mass term and hence the value of the angular momentum. This is inconsistent with the idea of quantization.

Even if we ignore the effects of relativity which, by the way, is what Niels Bohr chose to do, then for angular momentum to be quantized its three constituent variables would have somehow to collaborate with one another in ways which we cannot imagine, which we cannot describe and which we never experience anywhere else in the universe.

Set all of this against a much simple idea, that orbital velocity is subject to the effects of relativity and quantization is seen to be the function of a single variable; the orbital velocity. Furthermore we find that this leads to a simple mechanism which causes the energy levels to be quantized. The electron is seen to be a point particle in the classical sense. The wave properties of the electron derive directly from its orbital motion and we get a simple explanation for the Fine Structure constant. These ideas are explored more fully in https://quantum-reality.net/sampling-the-hydrogen-atom/

In effect we have only to make one slight adjustment to the classical laws of mechanics; to take account of the effects of relativity on orbiting objects and the whole edifice of quantum theory collapses. Physics is restored to being a scientific discipline and not a system of beliefs. We can tear up the quantum catechism.

The reason why it has been meaningful to compare physics with religion is that over the last 100 years physics has adopted a set of theories which are based on belief rather than being related to fundamental axioms. It has sought to prove these beliefs based on the flawed logic of a self referring proof. It is only by recognizing that this is the case and finding the root cause that physics will be restored to a true scientific discipline.

[i] One of the first self referring arguments was proposed by Anslem of Canterbury in 1078 to prove the existence of god.