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This question was inspired by Abstruse Goose :)

It's well known that any attempt to describe the spin (of say an electron) in terms of non-internal spatial coordinates is futile since this is just orbital angular momentum and can't take half-integral values. Spin is currently understood to be simply the generator of rotations, which being Hermitian should correspond to an "observable".

However, I remember hearing about attempts to model the electron as an extended object that literally spins and that in these attempts you run into problems with relativity. (A back of the envelope calculation shows that the "periphery" of the electron should be moving 35 times the speed of light)

  1. My first question is whether someone knows of a serious attempt in the literature at doing something as described in the previous paragraph
  2. How is spin understood (or incorporated) in the framework of (a) string theory (b) loop quantum gravity?
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I like the second question. Looking forward to reading the answers. – MBN Feb 23 '11 at 20:05
There was a reply to a question a few days ago by Carl Brannen who is working with 1/2 spins. Did you see it?… – anna v Feb 23 '11 at 20:24
@anna Thanks, I just put Carl's paper on my to-read list. – dbrane Feb 23 '11 at 20:35
up vote 3 down vote accepted

First of all, spin, by definition, is the internal angular momentum. And angular momentum is, by definition, the observable that is conserved as a consequence of the rotational symmetry of the laws of physics. That's the most general definition that follows from Noether's theorem. It follows that in quantum mechanics, the angular momentum is always given by the generator of the rotations. That must be true in any consistent quantum theory.

The value of the spin has to be an integer multiple of $1/2$ because the rotation by $4\pi$ is continuously contractible to no rotation at all - so these two transformations can't possibly have qualitatively different eigenvalues. (To see the continuity, represent rotation by $4\pi$ as a rotation around $z$ by $2\pi$, followed by another one. You may take the second rotation by $2\pi$ and continuously change the axis of rotation from $+z$ to $-z$. At the end, you will get a rotation by $2\pi$ and back which has to be an identity, so it proves that the rotation by $4\pi$ has to act trivially on all objects.)

However, all values of spin that are multiples of $1/2$ are possible. That's because, for example, $Spin(3,1)$ Lorentz group is isomorphic to $SL(2,C)$ that acts on two-component complex spinors.

In quantum field theory, the electron's spin arises because every electron is an excitation of a Dirac field that transforms as a spinor. Quantum field theory has to be the right low-energy approximation of any viable theory that has a chance to go beyond quantum field theory, too. So the spin's origin may be reduced to the case of quantum field theory.

That's the case of string theory, too. One may also describe the origin of the spin-1/2 particles microscopically. In the RNS formulation of the superstring, there exist zero modes of the $\psi_\mu$ world sheet fermions. The representation of the ground states requires us to quantize them, and because the anticommutators of these $\psi_\mu$ fields $$ \{\psi_\mu,\psi_\nu\} = g_{\mu\nu} $$ are isomorphic to the Dirac algebra of gamma matrices, $$ \{\Gamma_\mu,\Gamma_\nu\} = 2g_{\mu\nu} $$ up to a trivial rescaling by $\sqrt{2}$, the ground state of the single-superstring Hilbert space inevitably transforms as a spinor (in the periodic sector). The GSO projection removes 1/2 of those states that would violate the spin-statistics relation.

In the Green-Schwarz formalism, one obtains the ground state - supergravitons and/or gauge supermultiplet - by quantizing fields $\theta^a$ on the world sheet that transform as spinors themselves. In this way, one gets the full supermultiplet - both the fermionic as well as the bosonic states - at the same moment. Also, excitations of non-zero modes of $\theta^a$ are changing the spin of the string. The total angular momentum of a string comes from the spin of the zero modes as well as the nonzero modes - and from both bosonic and fermionic degrees of freedom on the world sheet.

Reconciling loop quantum gravity with a half-integral spin remains a matter of wishful thinking. There are many ways to see that one can't get e.g. chiral fermions in lattice-like theories; one can't ever derive the a priori existence of anomalies coming from chiral fermions if the spacetime is discretized; and so on. Half-integral spin is one of the ways to show that discrete models of the reality can't be consistent with basic features of the reality such as the existence of fermions, especially the chiral ones.

What you write about modelling spin's electron as an extended object is refuted in the first part of your very question. A century ago, people would think about electron as a classical object. Obviously, those attempts were totally incompatible with quantum mechanics (and other facts). There has to exist a degree of freedom that has a half-integral spin to start with, and RNS superstring theory only explains its deeper origin, but can't replace it by something completely different.

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I did not understand - rotation by $2\pi$ changes something observable? Does it do something non trivial "to objects"? – Vladimir Kalitvianski Feb 23 '11 at 22:33
Just a small typo, when you say "the group $spin(3,1)$" you meant $Spin(3,1)$. – MBN Feb 23 '11 at 23:57
Thanks, MBN, capitalized although I am not sure which one is right - or more common. – Luboš Motl Feb 24 '11 at 6:27
Vladimir, yes, rotation by $2\pi$ changes the sign of all half-integer-spin states (i.e. those with an odd number of fermions), never heard of it? – Luboš Motl Feb 24 '11 at 6:30
Yes, I heard of it but for a complex number the notion of sign is not always meaningful. Compare $i$ with $-i$; what is the difference? – Vladimir Kalitvianski Feb 24 '11 at 10:09

The full length answer is a book The Story of Spin by Sin-itiro Tomonaga which I've only read part of.

Cover image for The Story of Spin

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