Just like left and right are defined relative to the front of our elongated bodies, the handedness of an elementary particle is defined in relation to some direction, or axis, related to the particle.

**Helicity** refers to the relation between a particle's spin and
direction of motion.
To a particle in motion is associated the axis defined by its momentum,
and its helicity is defined by the projection of the particle's
spin on this axis: the helicity is
,
i.e. the component of angular momentum along the momentum.

The helicity operator thus projects out two physical states, with
the spin along or opposite the direction of motion - whether the
particle is massive or not.
If the spin is projected parallell on the direction of motion, the particle
is of right helicity, if the projection is antiparallell to the direction of
motion, the particle has left helicity.

The helicity is measurable, however not
Lorentz invariant, unless the particle is massless. Since helicity is the
scalar product of two vectors, it is however invariant under space rotations.

Something is when it cannot be superimposed on its mirror
image, like for example our hands: you can only superimpose your left hand
on your right hand if you put them back to back or inside to inside,
you cannot superimpose the two hands turned to you
(the word chirality comes from the Greek word for hand, cf.
the word chiropractor or the French word for surgery, la chirurgie).
Like our hands, chiral objects are classified into left-chiral and
right-chiral objects.
For a massless fermion, the Dirac equation reads

where the combination of the matrices, has the properties and . This allows us to define the

We can express any fermion as , a massive particle always has a L-handed as well as a R-handed component. In the massless case however "disintegrates" into separate helicity states: the Dirac equation splits into two independent parts, reformulated as the Weyl equations

where is the helicity operator expressed in terms of the Pauli spin matrices .

The

A massless particle, which is in perpetual motion, thus has an unchangeable helicity. The reason is that its momentum cannot be altered, and its spin of course remains unchanged.

For a massive particle however, we can perform a Lorentz transformation along the direction of the particle's momentum with a velocity larger than the particle's, changing the direction of the momentum. Since the spin direction remains the same, the helicity of the particle changes.

Spin properties are moreover related to spatial dimension.