Polarized Electron Physics        
Chiral photons and electrons

Here, we focus on the chiral properties photons, electrons, and molecules.

The handedness of an elementary particle depends on the correlation between its spin and its momentum (helicity). If the spin and momentum are parallel, the particle can be said to be right-handed (positive helicity). If the spin and momentum are antiparallel, the particle is left-handed (negative helicity).

Photons and electrons differ in that a photon's spin must be exactly aligned with its momentum, whereas an electron's spin points at a velocity-dependent angle from its momentum axis (higher velocities shrink the angle). [see Wigner.] Therefore, the handedness of an electron depends on the projection of its spin along its momentum.

[N.B., since electrons do travel slower than the speed of light, they (and all massive particles) are not inherently chiral. If we observe a moving electron from two different reference frames – first the lab frame, and then a frame moving faster than the lab velocity of the electron – the electron helicity changes sign. So electron helicity (chirality) is not a Lorentz invariant quantity. These facts notwithstanding, it is still practical to refer to electron chirality in the present discussion.]

Since we normally deal with beams of photons and electrons rather than individual particles, it is useful to refer to the polarization of the beams to describe average spin-momentum correlations. For instance, if a (coherent) beam of photons has a net positive helicity, the beam is said to be right-circularly polarized. A beam with a net negative helicity is left-circularly polarized. An equal mixture of these right- and left-circular states (when the two are 90º out of phase) yields a linearly polarized beam.

For electrons, a net positive or negative helicity is dubbed longitudinal polarization and is analogous to circular polarization for photons. If the average spin direction is perpendicular to the beam axis, the beam has transverse polarization – in analogy to linear polarization for photons. A transversely polarized electron beam is not chiral.

Below is a pictorial summary of the polarization analogies between photons and electrons. (The four-vectors on the left are the Stokes polarization vectors describing each situation, and the 2x2 matrices on the right are the equivalent density matrices [see McMaster, Blum, etc.].) The bottom frame shows the analogy between a circularly polarized photon beam and a longitudinally polarized electron beam.

In our experiments, we are interested in how these polarizations states are altered by collisions with chiral molecules. Therefore we next discuss molecular chirality and how it brings about polarization-dependent scattering.