quantum mechanics - Spin - where does it come from?

I study physics and am attending a course on quantum field theory. It is hard for me to draw connections from there to the old conventional theories.

In quantum field theory spin originates from the Dirac equation in the case of fermions.

I don't remember where it comes from in quantum mechanics. I just remember that there was the Stern Gerlach experiment where you shoot neutral spin 1/2 Ag atoms in. Is there also a electrically neutral elementary particle? If this is the case, how would this experiment look like in quantum field theory? Of course I ask this for the lowest order, otherwise we would have to calculate an infinite number of Feynman graphs, won't we?

Answer

Fundamentally, the spin originates from the fact that we want our quantum fields to transform well-behaved under Lorentz transformations.

Mathematically, one can start to construct the representations of the Lorentz group as follows: The generators $M^{\mu \nu}$ can be expressed in terms of the generators of rotations $J^{i}$ and those of boosts $K^{i}$. They fulfill

$$[J^{i}, J^{j}] = i \epsilon_{ijk} J^k, \, [K^i, K^j] = -i \epsilon_{ijk} J^k,\, [J^i, K^j] = i \epsilon_{ijk} K^k.$$ From them one can construct the operators $M^i = \frac{1}{\sqrt{2}} (J^i + i K^i)$ and $N^i = \frac{1}{\sqrt{2}} (J^i - i K^i)$. They fulfill $$[M^i, N^j] = 0,\, [M^i, M^j] = i \epsilon_{ijk} M^k \, [N^i, N^j] = i \epsilon_{ijk} N^k$$ These are just the relations for angular momentum that you should know from your QM introductory course. Group theoretically this means that every representation of the Lorentz group can be characterized by two integer of half-integer numbers $(m, n)$. If you construct the transformations explicitly you will find

• $(m = 0, n = 0)$ is a scalar, i.e. does not change under LT.


• $(m = 1/2, n = 0)$ is a left-handed Weyl spinor


• $(m = 0, n = 1/2)$ is a right-handed Weyl spinor



• $(m = 1/2, n = 1/2)$ is a vector

A Dirac spinor is a combination of a right and a left handed Weyl spinor.

Actually, one can now use these objects and try to find Lorentz invariant terms in order to construct a Lagrangian. From that construction (that is too lengthy for this post) one finds that the Dirac equation is the only sensible equation of motion for a Dirac spinor simply from the Lorentz group's properties! Similarly one finds the Klein-Gordon equation for scalars and so forth. (One can even construct higher spin objects than vectors, but those have no physical application except maybe in supergravity theories).

So, as you can see now, spin is fundamentally a property of the Lorentz group. It is only natural that we find particles with non-zero spin in our Lorentz invariant world.

Sidenote: Since we found the Dirac and Klein-Gordon equations from Lorentz invariance alone, and their low-energy limit is the Schroedinger equation, we get a 'derivation' of the Schroedinger equation, too. Most of the time the SE is simply postulated and worked with: this is where it comes from!