## Do not confuse spin direction with spin magnitude

A fermion will impart $\frac{1}{2} \hbar$ units of spin angular momentum irrespective of its energy when it interacts with other particles or fields. The direction of the spin angular momentum is discussed below.

A gauge boson will impart $1 \hbar$ units of spin angular momentum irrespective of its energy when it interacts with fermions. The direction of the spin angular momentum is discussed below.

Only integer changes of spin are observed in particle interactions. In any particle-particle or particle decay interaction the total spin before and the total spin after can differ only by integer changes.

No two fermions can occupy the same quantum mechanical state. In contrast, multiple gauge bosons can occupy the same quantum mechanical state.

SPIN MAGNITUDE

Spin magnitude of massive fermions = $\frac{1}{2} \hbar$

• first generation massive fermions : electron, u-quark, d-quark
• second generation massive fermions : muon, c-quark, s-quark
• third generation massive fermions : tau, t-quark, b-quark

Spin magnitude of massless fermions = $\frac{1}{2} \hbar$

• first generation massless fermions : electron neutrino
• second generation massless fermions : muon neutrino
• third generation massless fermions : tau neutrino,

Spin magnitude of gauge bosons = $1 \hbar$

• gauge bosons: photon, W, Z, gluon

Experimental verification of spin magnitude

Spin magnitude of fermions and gauge bosons have been experimentally verified by studying the details of particle interactions. This is a very involved job. Experimental physicists reconstruct the sequence of events from the observed particle tracts, measure the angles precisely, subtract the background noise, and check the consistency of the extracted signal against well established conservation laws. To extract the signal they apply very sophisticated statistical techniques. There are literally hundreds of cross checks. Experiments involving neutrinos are particularly difficult.

SPIN DIRECTION

Be careful with vector representation of spin

In the picture below, the particle linear momentum is visually represented as a vector, the spin angular momentum is visually represented as a vector too. We have to be very careful here. There is the danger of mixing apples and oranges (apple=linear momentum, orange=angular momentum). Yes, it is common to represent angular momentum with a vector, but I recommend that you think of “spin” as “left-handed” or “right-handed” rotation relative to a specified axis direction.

Spin direction of massive fermions: indeterminate when undisturbed, collapses to one of 2 directions (left-handed or right-handed relative to the spatial direction preferred by the measuring apparatus) and remains in that state until randomized again by another disturbance. Nature is full of disturbances (vacuum fluctuations, interactions with other particles or fields, spontaneous emission of photons, decays, measurements).

How do we measure the spin direction of massive fermions? Massive fermions carry electric charge therefore generate spin magnetic moment. The direction of their spin magnetic moment can be resolved by passing them through a uniform external magnetic field. We can orient the external magnetic field in any direction we want. The measured direction of the spin magnetic moment will either be parallel or antiparallel to the direction of the external magnetic field (only 2 possible outcomes). We can change the direction of the external magnetic field. The result will again be either parallel (right-handed rotation) or antiparallel (left-handed rotation) relative to the new direction of the external field.

Spin direction of massless fermions: Neutrinos are massless fermions (they have an extremely small mass that has not been measured yet). The spin angular momentum of neutrino is left-handed relative to the direction of motion. There are no right-handed neutrinos.

How do we measure the spin direction of neutrinos? Neutrinos are very hard to detect. Besides, neutrinos do not carry any electric charge therefore they do not generate a spin magnetic moment. We cannot use an external magnetic field to resolve the spin direction. Left-handedness of neutrinos (and the absence of right-handed neutrinos) have been established by painstaking measurement of angles in particle interactions involving neutrinos.

Spin direction of massless gauge bosons: The carrier of the electromagnetic force (photon) and the carrier of the strong nuclear force (gluon) are massless. Massless gauge bosons have 2 spin polarizations (directions): left-handed or right-handed relative to the direction of motion.

Spin direction of massive gauge bosons (W and Z): Good question! A massive spin=1 boson (W or Z) can have 3 polarizations. 1) left-handed relative to the direction of motion 2) right-handed relative to the direction of motion 3) longitudinally polarized. The “longitudinal polarization” is hard to visualize, see explanations by F.Tanedo and M.A. Thomson .