So what exactly is the difference between
chirality and helicity? In both cases, we speak of a left- or right-handed
particle. In this video, we will explain both concepts
and show their connection to spin! First off, a short reminder on spin. Spin is a property of a particle. Like charge or mass, spin is another label
to classify a particle. But unlike charge or mass, it is not just
a positive or negative number, it is a vector. This means, it has magnitude and direction. So when we talk about a spin-1/2 particle,
we refer to the magnitude of the spin. The quantum mechanical operators that are
assigned to the x, y and z-component of spin behave in the same way as the ones for angular
momentum. Mathematically, this is described by a commutator
relation. And practically, this can be demonstrated
by the Einstein-de-Haas effect. Now let’s talk about helicity. Helicity is defined as the projection of the
spin on to the direction of the particle’s momentum. For example, if we have a particle that is
traveling in positive x-direction and it has spin 3/2 which is also pointing in positive
x-direction, then it has helicity 3/2. For a particle that is propagating in minus
z-direction and has spin 1 in positive z-direction, its helicity is -1. A particle with spin 1/2 in positive x-direction,
traveling in negative y-direction has helicity 0. Also note that all spin-0 particles have zero
helicity. The helicity operator is given by projecting
the spin operator onto the unit momentum vector, and what we can measure as helicity is the
eigenvalue of this operator. Since the eigenvalues of spin are quantized,
there are also only discrete values for helicity. To get the helicity operator, we multiply
with a vector of length 1, so the numeric values of the helicity eigenvalues coincide
with the ones from the spin operator. The helicity of a massive particle is strongly
coupled to its reference frame. This means if we boost to a system that is
traveling faster than the particle, we observe its momentum going backwards, which means
that its helicity has changed during the boost! For massless particles, this is a different
story. Massless particles travel with the speed of
light, so there is no Lorentz transformation that will put us in a reference frame that
is traveling faster than the particle. Finally, a remark on convention. If the helicity eigenvalue is positive, this
means that spin and momentum are parallel, then we call the particle right-handed. If spin and momentum are anti-parallel and
the helicity is negative, we call the particle left-handed. Now on to chirality. Like mass, charge or spin, chirality is a
property of a particle. There is again a distinction between left-
and right-, and often the two states of chirality are also called left-handed and right-handed. However to avoid confusion we call it left-chiral
and right-chiral. The left- and right-chiral forms of a particle
are in principle equally different as two particles of different mass. Take an electron for example. We can separate its left- and right-chiral
parts by using the fifth gamma matrix. Mathematically this happens via projection
operators. We can also check that the left- and right-chiral
electrons together lead back to the original electron, which is in a superposition of both
chiralities. There is one connection to helicity however:
for massless particles, a left-chiral particle is also left-handed and a massless right-chiral
particle is right-handed. The origin of chirality lies in the representation
theory of the Lorentz group. The Lorentz group for our spacetime is given
by SO(3,1). And depending on what kind of object the Lorentz
group wants to act on, we need a different representation of the group element. The group element is some abstract thing,
but the representation is often a matrix that can be multiplied onto something. One can show that the Lorentz group is equivalent
to two copies of SL(2,C) and the representations of SL(2,C) can be labelled by a half-integer
number. Let’s call them m and n. So one representation of the Lorentz group
can be labelled by the pair (m,n). Let’s go over the simplest examples:
The (0,0) representation is used when we are considering scalars. A Lorentz transformation does not affect a
scalar quantity, so the corresponding group element may be represented by the unit matrix. The (1/2,1/2) representation gets used when
we want to transform a 4-vector. This is the usual Lorentz transformation matrix
Λ that appears in special relativity. And the (1/2,0) and (0,1/2) representations
are connected to left- and right-chiral spinors, respectively. This is one example of the more general statement:
Chirality is a concept that appears for (A,B) representations of the Lorentz group, where
A and B are different. A practical effect of chirality can be observed
in how particles with different chiralities interact via the weak interaction. The weak interaction will only affect left-chiral
particles, not right-chiral ones, as well as right-chiral antiparticles, but not left-chiral
antiparticles. A consequence of this is, that neutrinos,
which only interact via the weak interaction and gravity, are only observed as left-chiral. This is because their interaction with gravity
is too weak to measure and we can use the weak force only to detect the left-chiral
ones. To recap, chirality is a fundamental property
of a particle, just like mass, spin or charge, whereas helicity is an observable that depends
on spin and momentum. And that’s pretty much it for now, thanks
for watching!