Helicity and Chirality
HELICITY AND CHIRALITY
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
For a massless fermion, the Dirac equation reads
which is also satisfied by
where the combination of the matrices,
This allows us to define the
chirality operators which project out
left-handed and right-handed states,
where and satisfy the equations
= - and
so the chiral fields are eigenfields of , regardless
of their mass. Massive chiral states are however not physical since
does not commute with the free Hamiltonian for a massive fermion.
So in the general case, unlike helicity, chirality is not directly
measurable. It is however Lorentz invariant: chiral fields transform among
themselves under the Lorentz group.
Chirality and Lorentz invariance are
thus closely connected.
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
is the helicity operator expressed in terms
of the Pauli spin matrices
The Weyl fermions, i.e. the massless chiral states
, are physical
since they correspond to eigenstates of the helicity operator.
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
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.