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I always thought that a point particle would have spherical symmetry. This is the case for the intrinsic electric field from an electron.

However, the intrinsic magnetic field of an electron has cylindrical symmetry. A key property of such field is that is has orientation, the cylinder points in some direction.

I would think that it is mathematically imposible for a point to be oriented. So, this is mathematical proof that an electron is not a point. At least, saying it is a point and that it has an orientation, is mathematically inconsistent.

An I wrong?

(If you claimed that it is an arbitrarily small cylinder, i.e. a short line, that would be consistent, I think) (edit: you could even claim it is a vector, being in a point but also having a direction)

Edit: my question is not a duplicate because I'm not doubting that the electron has intrinsic angular momentum. I'm saying that such property is mathematically incompatible with a point. Unless I am misunderstanding what "point" means, in that case Id change the question to "what is do we mean by point? Can it have an orientation?"

Juan Perez
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    This isn't a proof the electron isn't a point particle, this is proof you can't just apply classical thinking to quantum objects. See https://physics.stackexchange.com/q/234979/50583, https://physics.stackexchange.com/q/24001/50583, https://physics.stackexchange.com/q/277565/50583 and their linked questions for extensive discussion of what it means for the electron to be "pointlike". – ACuriousMind Nov 13 '21 at 12:24
  • I never mention classical intuition. I'm talking about mathematics. A point is a mathematical concept that does not admit orientation. You cannot use it to model an electron. Also, my question is not a duplicate because I'm not doubting that the electron has intrinsic momentum. I'm saying that such property is mathematically incompatible with a point. – Juan Perez Nov 13 '21 at 12:34
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    The questions linked in my first comment discuss at length what "the electron is a point" really means. It does not mean "the electron is to be modeled under all circumstances as a single mathematical point" - this is precisely the classical view of a point particle that does not apply here. You're arguing against a straw man. – ACuriousMind Nov 13 '21 at 12:36
  • You’re last point about an electron being a vector is essentially correct. It is a vector defined at a single point in space. More precisely it’s a spinor which is like a vector (in that it is orientation) but more complicated. – Jagerber48 Nov 13 '21 at 13:23
  • ACuriousMind is right that you're arguing against a straw man, but to be fair, that straw man is widespread. I even see hints of it in some of the answers to the linked posts. Classical thinking is deeply ingrained because we grew up with it, and words alone cannot describe the nature of an electron in quantum theory because even our language is a product of that deeply-ingrained classical thinking. We can (and often do) repurpose old words to have new meanings, but the only way to appreciate the new meanings is through new experience -- namely with the mathematics of quantum theory. – Chiral Anomaly Nov 13 '21 at 14:42
  • "I would think that it is mathematically impossible for a point to be oriented. " This might not be relevant to your issue with electrons, but regarding what's mathematically possible, think (if you know some algebraic geometry) about points on the real line that "spin in place" under the Galois action and therefore have two orientations. – WillO Nov 13 '21 at 15:30
  • Possible duplicates: https://physics.stackexchange.com/q/1/2451 , https://physics.stackexchange.com/q/822/2451 and links therein. – Qmechanic Nov 13 '21 at 17:38
  • "I'm saying that such property is mathematically incompatible with a point." - the electron is not literally modeled as a simple mathematical point, it's just point-like, with extra associated properties. This is not that extraordinary; if you can accept objects being treated as point masses (a.k.a. modeled as point masses) in certain problems, even though a simple mathematical point doesn't have a mass (a number) "attached" to it, then you can accept that the "attached" object can be something more complicated, if doing so captures something about the phenomenon of interest. – Filip Milovanović Nov 14 '21 at 06:10

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The concept of a point particle, however useful in mathematical constructions, introduces trouble when looking at the particle itself. Tricks like renormalization were invented to swipe infinities under the carpet (even Feynman and Dirac admitted that the renormalization procedure is an contrived one, calling for a more "natural" explanation).

How can a point particle rotate? How can it show "its other side" after one full rotation (spin 1/2 particles). How can it show its other side after a quarter rotation (spin 2 particles)?

An electron can be visualized as having an associated arrow pointing up to rotate one whole round while walking a Möbius strip twice. Or by connecting the rotation to unitary transformations, U(1), for which the identity corresponds to 720 degrees rotation.

Nevertheless, the mystery of rotation remains. Clearly, for an electron, electric charge in motion has to be there because there is a magnetic field around the electron.

Do strings resolve the issue? Strings can rotate around several axes, including their length although this last mode is just as problematic as for the point particle. So you would say that something non-pointy is indeed. Strings offer a nice Ansatz.

MatterGauge
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