# Quantum spin and violations of special relativity

This past semester I finished the second year of my physics B.Sc. program with a course that covered quantum mechanics (wave formulation) up to the Schrödinger hydrogen atom. The accompanying textbook was Modern Physics for Scientists and Engineers (4th edition) by S. T. Thornton and A. Rex.

This is how Thornton and Rex explain why intrinsic spin must be a "purely quantum mechanical result":

In 1925 Samuel Goudsmit and George Uhlenbeck ... proposed that the electron must have an intrinsic angular momentum ... Classically, this corresponds in the planetary model to the fact that the Earth rotates on its own axis as it orbits the sun. However, this simple classical picture runs into serious difficulties when applied to the spinning charged electron ... Paul Ehrenfest showed that the surface of the spinning electron (or electron cloud) would have to be moving at a velocity greater than the speed of light!

I am left with two questions:

1. Aside from Ehrenfest's work, what led the physics community to adopt "purely quantum mechanical" spin into consensus?
2. Why not rework special relativity to account for spin? Was this ever attempted?

I would love to read some of the relevant literature, if possible. Thanks in advance!

Special relativity does account for spin, but only when including quantum mechanics. Schrödinger's equation for a free particle says:

$$-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\psi = i\hbar\frac{\partial}{\partial t}\psi$$

which is a quantum interpretation of the non-relativistic expression:

$$\frac{p^2}{2m} = E$$

Dirac made this relativistic. Since it needs to treat space and time equivalently, and the time derivative needs to be linear to make $$|\psi|^2$$ a probability density, he tried:

$$(\alpha cp + \beta mc^2)\psi = E\psi$$

After much ado, it is found that $$\alpha$$ and $$\beta$$ need to be four component objects that are now called the Dirac matrices $$\gamma_{\mu}$$, and:

$$i\hbar \gamma^{\mu}\partial_{\mu}\psi-mc\psi=0$$

The four components of $$\psi$$ are then identified as the 2 spin states (each) of the electron and the positron.

So in some sense, relativity forces spin upon us. A deep mathematical treatment addresses representations of the Lorentz group $$O(3,1)$$, which contains copies of $$SU(2)$$, the Pauli spin group. The associated algebras are know as "The algebra of space-time" and "the algebra of space".

That non-relativistic spin arises as the fundamental representation of $$SU(2)$$ means it is purely quantum mechanical. There is no classical analogue. Moreover, the Stern-Gerlach experiment shows that the spin is a 2-state system, that can be quantized along an arbitrary axis.

• where can I learn more about the idea that relativity forces spin upon us? Sep 11, 2021 at 4:35
• @nielsnielsen maybe here en.wikipedia.org/wiki/… ?
– JEB
Sep 13, 2021 at 15:08

Goudsmit-Uhlenbeck tried to explain the experimentally observed "spin-effects" by a "intrinsic" rotation of the particle. But this would violate special relativity as the electron surface would move faster than the speed of light (Ehrenfest).

Still we don't have to rework relativity, because today spin is regarded as a particle property that doesn't correspond to an actual rotation in the physical space.

Classical electrodynamics predicts that for a body rotating around its axis of symmetry, the magnetic moment is $$\vec{\mu}=\frac{Q}{2M}\vec{S},$$ where $$Q$$ is the total charge on the body, $$M$$ is the mass of the body, and $$\vec{S}$$ is the spin angular momentum. This is experimentally observed to be incorrect for an electron; the correct expression is $$\vec{\mu}=g\frac{Q}{2M}\vec{S},$$ where $$g=2$$.

The Dirac equation, a relativistic form of the Schrödinger equation for spin-1/2 particles (a reasonably accessible derivation thereof can be found here), when solved for an electron in an electromagnetic field (as shown here) correctly predicts $$g=2$$. Thus, there is no need to re-work special relativity; attempting to modify equatons of quantum mechanics to accomodate for a relativistic Hamiltonian makes adequate predictions.