Why does the existence of magnetic monopoles imply quantized electric charges?

Submitted by Speterius t3_10ea1ng in askscience

I watched this Royal Institute lecture (youtube link) where the presenter explains that Dirac (1931) showed, that if a magnetic monopole exists, it would explain why the electric charge is quantized as described by quantum mechanics.

I tried treading up on this, but I'm not a physicist and I'm only familiar with QM on a podcast / youtube video level so I'm looking for a more intuitive explanation on why this happens and how this is derived.

  • Does the existence of a magnetic monopole imply that the magnetic charge is also quantized?
  • Is this a fundamental explanation on why things are quantum in nature or do we have to go deeper than that?
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taphead739 t1_j4prqlo wrote

First of all, there has been no evidence that magnetic monopoles exist. At least none so far. Everything we have observed in the universe can be explained without the existence of magnetic monopoles.

As to Dirac‘s statement about quantization: In a hypothetical system that contains an electric point charge (like an electron) and a magnetic point charge, the electromagnetic field generated by them has an angular momentum that is proportional to the product of the value of those two charges. Since quantum mechanics dictates that angular momentum must be quantized, this means that the electric and magnetic charges must also be quantized - if they weren‘t you could get a continuum of angular-momentum values.

To answer your questions: 1) If magnetic monopoles exist, their magnetic charges must be quantized. 2) There probably is no satisfying answer to the question why quantum mechanics describes our universe so well. It‘s just the way it is, at least the best description of it that we currently have.

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d0meson t1_j4qb8l2 wrote

As an aside, Dirac's statement is so interesting because 1) the angular momentum carried by such a field configuration is independent of the distance between the electric and magnetic charge, and 2) it only requires one point magnetic charge to work.

So as long as there is at least one magnetic monopole somewhere in the universe, his argument works.* What if there's genuinely only one? It's an interesting scenario to think about in terms of the limits of the scientific method; for example, if that one monopole passed through an experiment and left our Solar System, never to return, that experiment would essentially be non-repeatable and therefore non-verifiable.

*in a classical universe; not sure whether GR or QFT impact this statement.

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Shufflepants t1_j4sn6to wrote

>*in a classical universe; not sure whether GR or QFT impact this statement.

Yeah, I would assume that the permeating presence of the electromagnetic field as represented in QFT would alleviate the need for there to be a magnetic monopole.

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Speterius OP t1_j4rgooc wrote

Thank you for the answer. Super insightful.

> "Since quantum mechanics dictates that angular momentum must be quantized, (....)"

Is the fact that angular momentum must be quantized a postulate of QM or is it derived from something more fundamental? I saw the plank length come up in the Dirac derivation.

I guess I'm looking for some sort of axiom. Something that doesn't follow from anything, but is the lowest level block of QM.

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taphead739 t1_j4rn0b7 wrote

It is a natural consequence of the wave-particle duality. If we go back to classical mechanics, angular momentum is present when something moves along a circular path. Very small particles are also waves, and the frequency (number of times the wave function goes up and down per length unit) is a measure for their (angular) momentum. The wave function must now "fit" the circumference - meaning that if you go around the circle the whole 360°, you must end up with the same value of the wave function and are not allowed to have a sudden step. This only works if the circumference is an integer multiple of the wavelength. As a consequence, only certain wavelenghts and frequencies are allowed, and the same is true for angular momentum.

This is a very simplified picture, of course, but I hope it gets the principle across.

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luckyluke193 t1_j4xl1v7 wrote

> Is the fact that angular momentum must be quantized a postulate of QM or is it derived from something more fundamental?

Quantisation of angular momentum follows from the mathematical definitions of wave functions and operators in QM. Specifically, it comes from the structure of the group of rotations in 3D, SO(3). In the end, this is Lie group and Lie algebra representation theory.

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DragonZnork t1_j4rk8ok wrote

Angular momentum quantization isn’t a postulate, it shows up when solving Schrodinger’s equation for the hydrogen atom.

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Speterius OP t1_j4rn6v8 wrote

I'm not sure i get what the Schrodinger equation solutions for the hydrogen atom have to do with the quantum nature of the electric charge.

I understand the electron shells, but how does that relate to the electric field being quantized?

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dieEhrevonGrayskull t1_j5vwrq7 wrote

Quantization is a consequence of what happens when waves are bounded. In particular, quantization of energy, and therefore rest mass. Since all of the particles modeled by QM or QFT are done so in terms of wave equations, quantization of the solutions is a natural consequence.

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hairam t1_j4rozl2 wrote

> QM
> more intuitive explanation

Choose one

;)

Sorry, just adding some drama to the question, or arguably a realistic perspective, regarding your more philosphical statements on fundamentals of QM or postulates, or understanding goals. I'd add more to try to answer your questions if I weren't under-qualified from the get go, combined with forgetting entirely too much of my education.

What I can comment with some relevance on: continue your pursuit of better understanding, but alter your expectations, because intuition is usually a false friend with QM! E.g., the fun, if not overused, electron spin meme

Also adding the first paragraph from the introduction in an intro QM for undergrad physics students book (Introduction to Quantum Mechanics: 2nd Ed. by David Griffiths): > Unlike Newton's mechanics, or Maxwell's electrodynamics, or Einstein's relativity, quantum theory was not created - or even definitively packaged - by one individual, and it retains to this day some of the scars of its exhilarating but traumatic youth. There is no general consensus as to what its fundamental principles are, how it should be taught, or what it really "means." Every competent physicist can "do" quantum mechanics, but the stories we tell ourselves about what we are doing are as various as the tales of Scheherazade, and almost as implausible. Niels Bohr said, "If you are not confused by quantum physics then you haven't really understood it"; Richard Feynman remarked, "I think I can safely say that nobody understands quantum mechanics."

Physics, including QM, is a lot of explaining these "why" questions you ask via math. My very unsatisfying and generalized - possibly patronizing (if so, sorry, that's not the intent) - recommendation is that if you want to dive deeper and have a more satisfying or stable foundation for your understanding, look into more of the accompanying math!

Happy exploring!

^(Thanks, also, for a reminder to revisit my own study :)^)

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Speterius OP t1_j4rpx2l wrote

Good meme 10/10.

Actually, it's not patronising. It feels good to know that Feynman and Co said those things.

I listened to a few detailed podcasts explaining the maths of QM, but I haven't done anything myself.

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