Back to the Real World
Now we return to physics, and in particular to the electron. We regard it as being a point, or rather, we take its radius to be in the coat of 0 (or whatever point it is centered on). That is, we will postulate it to be a ball of infinitesimal radius. In particular, let some ee be that radius, and assume its mass m is uniformly distributed. Now we will attempt to characterize its spin (ft/2) as a physical angular momentum L of actual rotation, namely with an angular frequency ш so that we get the usual classical formula:
Since se is infinitesimal then ш must be infinite, since the LHS is finite.
The idea of treating spin as a possible rotational phenomenon was considered long ago (see below), but taking the radius to be a positive real r; this led to trouble with special relativity (SR). A point on the surface of the electron ball would—in order that the rotation provide the proper angular momentum of spin, have to travel at speeds in excess of the speed of light. But to reach such a speed would require infinite energy, according to SR, and that traditionally is taboo. Here then is a possible advantage of NSA: suppose we allow physical quantities to be infinite.
Let’s calculate the speed v of a point on the surface of an “electron ball” with (initially real) radius r that is rotating with angular momentum h/2. From the above equation, we get
v = ш(1/2п)(2п r) = шг = 5h/(4mr)
If we insist that v < c then we find c > 5h/(4mr) or
r > 5h/(4mc) = 0.5 x 10-12 m.
This is essentially the negative result found by Goudsmit and Uhlenbeck  that made them (and others) give up the idea of spin as deriving from an actual physical rotation, since it was known even then that r is less than 3 x 10-15 m.
There is an alternative: allowing v > c, and also allowing infinite energies, as well as replacing r by ee. But why insist that r be infinitesimal? This is not strictly necessary. But since as already noted, it is commonly thought that r = 0 (an electron is an actual point with no extent, no volume) and since we are allowing infinities anyway, it is tempting to go “all the way” (at least all the way to infinitesimals, if not literally to zero).
Back to our calculations: if se is infinitesimal then as noted above, the angular frequency rn is infinite. But what then is the speed of a point in the electron coat, at distance ee from the origin of rotation? It will be as above, but replacing r with ee, hence infinite:
This infinite speed precisely produces the finite angular momentum ft/2. That is, the infinite speed of a point within the electron coat (which itself is at infinitesimal distance ee from the origin of rotation), works together with that infinitesimal distance to produce the needed finite angular momentum of spin.
However, not everything works out so nicely. The kinetic energy of mass m with speed v, in SR, is
where y = 1Д/1 - (v/c)2)
When v = c, y is infinite, hence T would seem to be infinite. This is well-known, of course, and is a primary reason that c is regarded as an unreachable upper limit on all speeds of massive objects. But it is now even worse: for this infinity (of y ) seems to be of the totally impossible kind: 1 /0. _
There is however another interpretation: multiplying through by л/1 - (v/c)2), we get
andfor v = c thisbecomes0 x T = mc2. A reasonable conclusion now isthat m = 0: a particle traveling at light-speed has no mass. And T is not further constrained here, infinite or otherwise. Presumably it can (for v = c) be taken as the energy of an appropriate light-speed particle. Whether this is physical nonsense or not, at least we are getting some sort of “results” from such an approach.
-  Note that this means the ball will be a proper subset of the coat, since coats have no boundary;if they did, then for instance 2ee would be outside the coat, which makes no sense for it too isinfinitesimal.
-  But see for instance [10, 17], for this is still a topic of dispute.
-  For many purposes; but in QFT for instance this is not quite right.
-  It is no good trying to wriggle out of this by supposing T is an NSA sort of infinity; that wouldcorrespond to v being “almost” the same as c (in the same coat, so that v/c is in the coat of 1). Forin fact we need—for the Goudsmit/Uhlenbeck model—that v be even greater than c. And then yactually has an imaginary value! This leads into the even stranger physics of tachyons.
-  This can actually be given a positive spin (pun intended). The Higgs field endows particles withmass according to whether they are retarded by it—retarded from traveling at light-speed, that is.Particles that are not so retarded are by definition massless!