One of the delights of reading in Physics and the sciences generally – of being alert in this wonderful world – is that there is always something new to learn, even if that “something new” is, in itself, old and even “well known” by others.

Such was the case for me when I stumbled upon a review paper in arxiv.org by Ricardo Heras: Dirac Quantization Condition: a comprehensive review, Arxiv: 1810.13403. Hereinafter I will refer to the paper as [Heras]

P.A.M. Dirac was one of the founding fathers of quantum theory and as such made a number of interesting speculations and introduced many new concepts into physics. I was not aware of his work with “monopoles” however, and certainly not aware of the background explored in Hera’s paper. This despite the fact that I have had courses in Quantum theory and Quantum Electrodynamics (QED). Perhaps I forgot these items or was asleep in class when the topic was covered. More likely, though, I was never exposed to it.

Magnetic monopoles have not been observed to date in our physical universe. We do have electric monopoles (positive or negative electric charges) which can have an independent existence, although there is always the relatively strong electric force between unlike charges which causes them to attract one another and effectively act to neutralize the free charges.

Magnetic poles (conventionally labelled North or South) always appear together in pairs, as in common magnets.

The classical laws of electromagnetism (Maxwell’s Equations) are written in such a way that this feature of electromagnetism is manifested in the theory.

Hera’s presents Dirac’s conjecture that if there was even a single magnetic monopole in the universe it would be sufficient to establish the quantization of electric charge. While Hera’s review is quite extensive and has lots of mathematical detail his historical asides and the background he gives of the Dirac condition are very interesting and enlightening.

According to Hera’s, Dirac’s motivation for his theoretical derivation of the quantization condition was to try to explain why charge was quantized in integer units of “e”, the charge of the electron or proton and why “e” has the particular value that it does. Heras quotes Dirac:

“I think it is perhaps the most fundamental unsolved problem of physics at the present time.“ {1978}.

The question of why “e” or any of the fundamental constants of nature take the values they have is still one of the unsolved problems of physics. It may be that the particular values are just “accidental” in that they have their values and that is all there is to it. Running against this notion however is the nagging recurrence of interesting relations between the constants, a subject that Dirac also contributed to with his “ Large Number Hypothesis” (LNH). (See Wikipedia on this topic for instance.)

Added to this is the evidence of “fine tuning” in our universe – there are a variety of theoretical investigations into the notion that if the value of “e” or of “G”, the gravitational constant, or any of the other “fundamental“ observed constants of nature are changed by only a small amount then the universe as we know it could not exist. (Again, check Wikipedia for some interesting discussion on the topic of “fine tuning”.)

In any event, none of this detracts from the remarkable idea contained in what Heras calls the “Dirac condition” – namely that the existence of even one magnetic monopole somewhere in the universe would require the existence of a single fundamental charge “e”.

Dirac derived a theoretical argument for this using the existing laws of electromagnetism.

In his review of the Dirac condition, Heras supports Feynman’s teaching philosophy:

“that if one cannot provide an explanation for a topic at the undergraduate level then it means one doesn’t really understand this topic.” (p. 32, [Heras])

Heras demonstrates his commitment to this philosophy by presenting “ five quantum mechanical derivations , three semiclassical derivations and [his own] heuristic derivation of the Dirac quantization condition.” ( p. 30, [Heras])

Heras records that another notable physicist Wolfgang Pauli originally disliked the idea of magnetic monopoles and sarcastically referred to Dirac as “Monopoleon”. (p. 4, [Heras]) although later Pauli acknowledged that there was some mathematical beauty in the theory.

Heras quotes Dirac on the derivation of his quantization condition:

“You must have the monopoles and the electric charges occupying distinct regions of space. The strings, which come out from the monopoles, can be drawn anywhere subject to the condition that they must not pass through a region where there is electric charge present.”

The string that Dirac is referring to here is an infinite string of magnetic dipoles all aligned or a very tightly wound solenoid – either of which goes on to infinity where the magnetic monopole is located. This is a mathematical artifice based on using Maxwell’s equations of electromagnetism.

Heras goes through the original derivation of the Dirac condition and makes the point that ,

“… the location of the string is irrelevant. But the argument might equally put emphasis on the idea that the string is unobservable.”

I was struck by the notion that Dirac’s argument for the existence of a quantum of charge – which is what is observed – was dependent on the premise that there be an unobservable feature – the magnetic line of force or string from the magnetic monopole.

Another notion that bothered me was that the the string from the magnetic monopole had to be in a region of space (and presumably time) that was devoid of electric charge. It raises the question of how an essentially non-interacting magnetic phenomenon could affect the nature of electric charge. In any common sense approach to physics any things which affect one another have to have some form of interaction.

Heras does go into considerable discussion of the meaning of the Dirac string and how it ’s interaction with the electric charge can be understood. He points out that using arguments from application of the classical equations of electromagnetism, the Dirac string is unphysical.

Application of quantum mechanical considerations leads to the Dirac condition (that electric charge is quantized) and also that the Dirac string is unobservable or undetectable. This is a subtle difference from being unphysical.

In quantum mechanics there are unobservable entities which can give rise to observable effects. The most basic is the wave function describing a physical system which satisfies the Schrodinger equation. The wave function is in principle unobservable, but its modulus square is observable – and gives the probability of an event.

Physics prides itself on being based on reality and on what is observable – any acceptable theory in physics has to be based on data, provide a sensible explanation of that data and propose predictions from the theory that can be tested with new data. Nevertheless there are a lot of theoretical paradigms in modern physics that incorporate in the theory elements that are in principle unobservable.

For example, black holes are a natural consequence of the General Theory of Relativity, but the interior of any such black holes are in principle unobservable – light and other forms of energy or matter cannot escape due to the critical intensity of the gravitational attraction. This does not stop a lot of research and discussion on the nature of black holes, including what might be happening inside. The premise would be that indirect or secondary consequences could be surmised from some theoretical conjectures that would allow deciding the truth or falsity of any speculation.

In the case of the Dirac condition and the conjecture of the Dirac magnetic monopole, I have briefly thought of ways that such a monopole could affect existing physics – especially the remarkable effect of charge quantization, without being directly observable or actually interacting with present day electrically charged objects. One can speculate on a number of ways.

In the Standard Model of Cosmology the concept of inflation is pretty much accepted. The concept of inflation was originally proposed by Alan Guth as a way of solving several problems of the Standard Model – the horizon problem, the flatness problem and the monopole problem. I have touched on inflation and its own problems in a previous blog ( Deflating Inflation…A Cosmic controversy) . Nevertheless, Guth’s model of inflation does give a way of explaining the absence of magnetic monopoles in our universe.

Again, according to the Standard Model when the Big Bang occurred, the universe moved from a presumably simpler symmetry to its present phase. According to Grand Unified theory extensions to the Standard Model of Particle Physics, during that transition magnetic monopoles could have been created but in very small numbers. The effect of inflation is then to reduce the overall distribution of monopoles to such a low level that they are unobservable in our “Hubble patch” of the universe. This dilution effect on any existing magnetic monopoles would not only render them unobservable in practical terms but would also provide a way in which existing particles we do observe, such as electrons and protons, could have interacted with and been affected by a magnetic monopole. That is the interaction could have been during or prior to inflation, resulting in the quantization of electric charge in our observable universe.

In doing some background research on this topic I stumbled onto an interesting site:

https://physics.stackexchange.com where there were a number of interesting discussions around the topic of the Dirac monopole and its relation to some of the questions in current physics research..

Also evident in some of the discussion that I read was the idea that while the Dirac condition gives a way to explain charge quantization using classical or quantum electromagnetic theory it need not be the only explanation, or even the best one.

There is evidently an ongoing fascination with magnetic monopoles and even ongoing attempts of finding ways to detect them. Very recently for example I found in the arxiv.org this paper: Gould, Ho, Rijantie; Toward Schwinger production of magnetic monopoles in heavy -ioncollisions; arXiv.org:1902.04388

In this paper the authors are discussing theory that could lead to searches for monopoles at the LHC.

I do remember one of my profs at Dalhousie (back in the 60’s!) mentioning a summer research program to hunt for evidence of magnetic monopoles in the geologic rock record of some region of Canada, but I also remember him derisively suggesting it was an excuse for some guys to get a grant to go camping for the summer.

Magnetic monopoles are not exactly another version of the Loch Ness monster – there is some theoretical motivation for searching for them, and equally, motivation for having theories that explain why we don’t find them.

For me, reading Heras paper expanded my knowledge of the Dirac condition which I didn’t even know about and also stimulated some searching and reading into a topic that has more life in it than I thought.