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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 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.
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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:
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:
and What is spin? by HC Ohanian
An interesting alternative perspective was put forward by Hestenes in his paper The Zitterbewegung Interpretation of Quantum Mechanics. In this paper he argues that spin arises due to the permanent zig-zag motion of electrons (and all other particles). This zig-zag motion is the result of permanent collisions with background Higgs bosons which cause chirality flips from left-chiral to right-chiral states.
See also How Electrons Spin by Charles T. Sebens
Note: In classical relativity, one can show that for a spinning object, as v→c, the rotation spin axis approaches alignment with the momentum vector. This can be visualized as due to the Lorentz-Fitzgerald shortening of the direction parallel to the momentum vector direction, as v→c. Imagine a rotating wheel with axis not aligned at low speed to the velocity vector with the dimension in the velocity direction shrinking to zero as speed increases. The plane of the wheel effectively rotates into the plane perpendicular to velocity. So, any particle traveling with speed c would be in a pure helicity state. That is what we have shown quantum mechanically in (12)
Whether spin, helicity or chirality is important depends on the physical question you are interested in. For free massless Spinors, the spin eigenstates are also helicity eigenstates and chirality eigenstates. In other words, the Hamiltonian for the massless Dirac equation commutes with the operators for chirality, γ5, helicity, S⃗·p⃗, and the spin operators, S⃗. The E QED interaction ψ ̄A/ψ = ψ ̄LA/ψL + ψ ̄RA/ψR is non-chiral, that is, it preserves chirality. Helicity, on the other hand, is not necessarily preserved by QED: if a left-handed spinor has its direction reversed by an electric field, its helicity flips. When particles are massless
QED interaction ψ ̄A/ψ = ψ ̄LA/ψL + ψ ̄RA/ψR is non-chiral, that is, it preserves chirality. Helicity, on the other hand, is not necessarily preserved by QED: if a left-handed spinor has its direction reversed by an electric field, its helicity flips. When particles are massless
In the massive case, it is also possible to take the non-relativistic limit. Then it is often better to talk about spin, the vector. Projecting on the direction of motion does not make so much sense when the particle is nearly at rest, or in a gas, say, when its direction of motion is constantly changing. The QED interactions do not preserve spin, however; only a strong magnetic field can flip an electron’s spin. So, as long as magnetic fields are weak, spin is a good quantum number. That is why spin is used in quantum mechanics.
In QED, we hardly ever talk about chirality. The word is basically reserved for chiral theories, which are theories that are not symmetric under L ↔ R, such as the theory of the weak interactions. We talk very often about helicity. In the high-energy limit, helicity is often used interchangeably with chirality. As a slight abuse of terminology, we say ψL and ψR are helicity eigenstates. In the non-relativistic limit, we use helicity for photons and spin (the vector) for spinors. Helicity eigenstates for photons are circularly polarized light.
Quantum Field Theory by Schwartz
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