Spin

Prerequisites: Angular Momentum, Canonical Quantum Mechanics. Full path (18 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?
• FAQ
• History

Intuitive

One of the biggest discoveries in the last century was that elementary particles have spin, which is some kind of internal angular momentum. This was discovered by the famous Stern-Gerlach experiment.

In abstract terms you can think about spin as a label that tells us how particles behave in experiments, exactly as the mass or the electric charge. For example, a particle with electric charge behaves different than one without in experiments and the same is true for spin.

There are particles with spin $0$, particles with spin $\frac{1}{2}$ and particles with spin $1$. For each of these different particle types we have a different equationthat describes their behavior.

• Spin: The Quantum Property That Should Have Been Impossible by Paul Halpern
• What exactly is the 'spin' of subatomic particles such as electrons and protons? Does it have any physical significance, analogous to the spin of a planet? | Scientific American
• https://www.forbes.com/sites/startswithabang/2017/11/21/spin-the-quantum-property-that-nature-shouldnt-possess/#20a3df076349

Concrete

Spin is a quantum number like mass or like the electric charge. Spin has exactly the same origin as the other quantum numbers and is therefore not as strange as most people believe it to be.

The origin of Spin

Noether's famous theorem states that there is a conserved quantity for every symmetry of the Lagrangian. An interesting subtlety of this theorem is that the corresponding conserved quantity in field theories has two parts. Only the sum of them is conserved.

One part is a result of invariance under transformation of the spacetime coordinates, and the second part is a result of the invariance under mixing of the field components.

If we consider invariance of a field under rotations, we therefore get a conserved quantity that consists of two parts. One part is a result of the rotation of the coordinates. This quantity is what we call orbital angular momentum. The second part is a result of the mixing of the field components under rotations and is what we call spin.

For some further details, see Noether's Theorem for Fields.

Recommended Resources:

• A brilliant discussion of spin, involving real measuring devices, can be found in the Feynman Lectures Vol. 3.
• http://math.ucr.edu/home/baez//spin/spin.html
• For the origin of spin, see http://rickbradford.co.uk/NoethersTheorem.pdf
• Spin by Michael Weiss
• How Electrons Spin
• No-Nonsense Quantum Mechanics by Jakob Schwichtenberg, chapter "Quantum Systems with Spin"

Abstract

Projective representations and $SU(2)$. Quantum states are rays in a Hilbert space, so a symmetry group $G$ needs to act only projectively: $U(g_1)U(g_2) = \omega(g_1,g_2)\,U(g_1g_2)$ with phases $\omega$. For connected Lie groups, Bargmann's theorem shows that when the second Lie algebra cohomology vanishes (true for semisimple groups such as $SO(3)$ and the Lorentz group), every continuous projective unitary representation comes from a genuine unitary representation of the universal cover. For rotations this is

$$1 \to \mathbb Z_2 \to SU(2) \to SO(3) \to 1 .$$

The irreducible unitary representations of $SU(2)$ are labelled by $j \in \{0, \frac12, 1, \frac32, \ldots\}$ and have dimension $2j+1$. The central element $-\mathbb 1 \in SU(2)$, which maps to a rotation by $2\pi$, acts as $(-1)^{2j}$. Integer $j$ gives representations of $SO(3)$, and half-integer $j$ gives representations of $SU(2)$ that are only projective representations of $SO(3)$. Spin is the label $j$ of the representation in which the internal degrees of freedom of a particle transform, as opposed to the orbital part that comes from the action of rotations on functions of position and contains only integer values.

Relativistic spin. The proper orthochronous Lorentz group $SO^+(1,3)$ has universal cover $SL(2,\mathbb C)$. Its finite-dimensional irreducible representations are labelled by pairs $(j_1, j_2)$ with $j_1, j_2 \in \frac12\mathbb N_0$, of dimension $(2j_1+1)(2j_2+1)$. Under the rotation subgroup they decompose into spins $j_1 + j_2, j_1 + j_2 - 1, \ldots, |j_1 - j_2|$. Examples are the scalar $(0,0)$, the left- and right-handed Weyl spinors $(\frac12, 0)$ and $(0, \frac12)$, the Dirac spinor $(\frac12,0)\oplus(0,\frac12)$ and the four-vector $(\frac12,\frac12)$. Apart from $(0,0)$ these are not unitary, since a non-compact simple Lie group has no nontrivial finite-dimensional unitary representations. The particle concept of spin instead comes from Wigner's classification of unitary irreducible representations of the Poincaré group: for a massive particle the little group is $SU(2)$ and its representation label is the spin; for a massless particle the little group is the double cover of $ISO(2)$, and the physically realized representations are labelled by the helicity $h \in \frac12\mathbb Z$. See Wigner's little groups and spinors.

Spin and statistics. In relativistic quantum field theory with a positive energy spectrum and local (anti)commutativity at spacelike separation, the spin-statistics theorem states that fields of integer spin must be quantized with commutators and fields of half-integer spin with anticommutators. Particles of integer spin are bosons and particles of half-integer spin are fermions.

Spinors on curved spaces. On an oriented Riemannian or Lorentzian manifold, spinor fields require lifting the frame bundle with structure group $SO(n)$ (or $SO^+(1,n-1)$) to a principal bundle with structure group $\mathrm{Spin}(n)$, the double cover. Such a spin structure exists if and only if the second Stiefel-Whitney class $w_2$ of the manifold vanishes, and when one exists, the inequivalent spin structures are in one-to-one correspondence with $H^1(M;\mathbb Z_2)$.

Why is it interesting?

Spin is one of the most important quantum numbers.

In fact, it is so important that is responsible for the definition of the most important categories of elementary particles:

• Particles with integer spin ($0,1,\ldots$) are called bosons. The elementary bosons with spin $1$ are responsible for the fundamental interactions, while the Higgs boson has spin $0$. Examples are the photon, which is responsible for electromagnetic interactions or the gluons which are responsible for the strong interactions.
• Particles with half-integer spin ($\frac{1}{2}$) are responsible for matter and are called fermions. Examples are electrons and quarks, which are the constituents of atoms.

According to the prevailing belief, the spin of the electron or of some other particle is a mysterious internal angular momentum for which no concrete physical picture is available, and for which there is no classical analog. However, on the basis of an old calculation by Belinfante [Physica 6, 887 (1939)], it can be shown that the spin may be regarded as an angular momentum generated by a circulating flow of energy in the wave field of the electron. Likewise, the magnetic moment may be regarded as generated by a circulating flow of charge in the wave field. This provides an intuitively appealing picture and establishes that neither the spin nor the magnetic moment are ‘‘internal’’—they are not associated with the internal structure of the electron, but rather with the structure of its wave field. What is spin? by Hans C. Ohanian

Important Related Concepts:

• Helicity
• Chirality
• Spinors

FAQ

History

For summary see http://www.brannenworks.com/plavchan_feynmancheckerboard.pdf and http://www.fhi-berlin.mpg.de/mp/friedrich/PDFs/ptsg.pdf.

For some additional nice historical perspective, see "Stern and Gerlach: How a Bad Cigar Helped Reorient Atomic Physics" by B. Friedrich

Path to this page

1. Calculus
2. Imaginary Numbers
3. Linear Algebra
4. Functional
5. Hilbert Space
6. Legendre Transformation
7. Newton's Second Law
8. Vector Calculus
9. Newtonian Mechanics
10. Variational Calculus
11. Angular Momentum
12. Configuration Space
13. Euler-Lagrange Equations
14. Lagrangian Formalism
15. Phase Space
16. Hamiltonian Formalism
17. Schrödinger Equation
18. Canonical Quantum Mechanics

Leads to: bohr magneton, Helicity, Pauli Equation, Pauli Principle, Spin Models, Spin-Statistics Theorem, Stern-Gerlach Experiment.