Subject: Physics 557 – Lecture 9 Appendix
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Date: Thu, 2 Feb 2012 09:14:05 -0600

Physics 557 – Lecture 9 Appendix:  Spin, helicity, handedness, etc.

Let us try first to make connection to what we
learned about these topics in the context of E&M (see
Jackson) with propagating plane
waves.  In the absence of sources we
have  (and )
so that, if the wave is propagating in the z direction, the vector
E-field (and pseudovector B-field) must lie in the x-y
plane.  Thus the polarization of the
plane wave (the 3-vector direction of the electric field) is transverse to the
direction of motion (this is directly related to the fact that the electric
field couples to a conserved current).
If the direction of the E-field is constant, independent of z and
t, we say the wave has linear (transverse) polarization.  The basis states for this linear
polarization are the unit vectors in the x and y directions,
i.e., there are 2 independent possible linear polarizations.  This is directly analogous to the
J1 and J2 or I1 and
I2 states that we know in the context of angular momentum and
isospin.  Just as we saw with those
situations, there is also a basis set containing the states,
which lead to an electric field with the behavior

For obvious reasons this behavior is referred to as
circular polarization.  Now comes
one point of confusion in the language!
Historically in the study of optics, people focused on the behavior of
the polarization (the direction of the E-field) as a function of time at fixed
z, say z = 0,
.  An observer facing the oncoming wave
sees the polarization with the + sign rotate in a counterclockwise direction
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and this polarization state was called left circularly polarized in optics.  The minus signed case was naturally
called right circularly polarized.
However, in more modern terms, we wish to focus on the angular momentum
carried by the fields, i.e., by the photons.  If we ask about the component of angular
momentum along the direction of motion, Jz, we find (see
Jackson) that the + sign above
corresponds to a positive Jz
component while the – sign correspond to a negative Jz component.  For a single photon the angular momentum
is just ±1
in our units.  The word helicity is
historically associated with the sign of the J component that is along the direction of motion (Jz here).  Thus these two states are called the
plus helicity and minus helicity states, respectively.  The change in basis from linear
polarization to circular polarization is just like the change from the x,y,z basis to +,-,0 basis
that we discussed with respect to the pions in isospace.  However, as we will see in more detail
later, the fact that E&M corresponds to an unbroken symmetry (a conserved
current) leads to the fact that “real” photons are massless and never appear in
the z or 0 or “longitudinal” polarization state.

In the context of modern particle physics we also use
the language of handedness.
Again the connection is to spin.
Classically a particle’s angular momentum along the direction of motion
is determined by applying the “right-hand rule”.  Thus a particle with positive Jz and positive pz, and thus positive helicity, should be
rotating in a clockwise direction when viewed along the z-axis in the
plus direction or counter-clockwise when looking back at it in the minus
z direction.  Such a
particle, including a photon with the rotation being that of the E-vector, is
called right-handed (in direct contrast to the left circular polarization label
above).  The right-hand rule can
also be thought of as relating the direction the spin points to the direction
that a right-handed screw moves when you turn it in the right-hand, clockwise
direction – forward in z, or left-hand, counterclockwise direction –
backward in z.  For the
massless photon we have the following connections for a photon moving in the +
z direction,

Now what about spinors?  First let us say a few more words about
the Dirac equation.  Recall that the
goal was to have a Lorentz invariant, first-order equation of motion.  This goal can be met because the Dirac
matrices, satisfying

exist (the 1 refers
to the matrix structure with respect to the spinor indices).  These matrices supply us with (another)
4-D representation of the Lorentz group.
Recall that in Lecture 5 we discussed the general Lorentz transformation
of 4-D vectors in the form

In the new representation provided by 4 component Dirac spinors (i.e., the solutions of the
Dirac equation) the transformation operator looks like

where the subscript in the
first equation is to remind us that (we think) we are describing spin ½
particles.  The final property of
the Dirac matrices that we need is the commutator

which follows from all of
the above definitions (if you can keep all of the indices straight).  From this expression it follows that

or, in words, boosting the
spinor indices of the Dirac matrix is equivalent to an “opposite” boost applied
to the Lorentz index.  This means
that the differential operator in the Dirac equation

is a Lorentz invariant.  To see this consider the Lorentz
transformed equation (and field)

Again in words, the properties of the Dirac matrices
are such that the operator  is indeed a Lorentz scalar as desired,
i.e., it has the same form in both frames.

After this (re)introduction to the Dirac equation,
consider first the massless version of the equation,
.  Instead of using the explicit
representations of the Dirac or gamma matrices presented above, which were
useful for considering the nonrelativistic case and talking about particles
versus antiparticles, consider the following forms (often called the Weyl or
chiral representation with various choices of the signs seen in the literature,
e.g., Peskin and Schroeder have ,
)

The two matrices g0
and g5
have switched forms and g1,
g2
and g3
have changed sign from our previous choices, but as a group they still satisfy
the required anticommutation relations.
With this choice the boost and rotation generators take the simple
forms

These block diagonal forms clearly suggest that the
Dirac representation of the Lorentz group is reducible in the massless
limit.  This is generally expressed
in terms of two 2-dimensional representations called Weyl or chiral spinors,

with the components referred
to as right-handed and left-handed respectively (hopefully, this notation will
become clear shortly).  To see the
reduction of the Dirac representation we multiply the massless Dirac equation on
the left by -ig1g2g3.  We note that (in 2x2 notation)

and recall that

Thus we can reduce the resulting expression to

As expected, in this massless limit, the Dirac
equation breaks into two equations that are sometimes called the Weyl
equations.  The two fields evolve
separately and form two distinct representations of the Lorentz group.  This is just the standard statement that
I cannot boost to the rest frame of a massless particle and so cannot mix the
left-handed state with the right-handed state (although we still have not
defined just what this language means).
We can start to give some meaning to this language by noting that these
Weyl or chiral spinors are eigenstates of  in the sense that

Finally note that this decomposition of the Dirac
representation is not complete when there is a nonzero mass.  In this case the Dirac equation times
-ig1g2g3
has the form

so that the mass term
clearly mixes the left- and right-handed representations.

To proceed with our discussion for the massive case
it is helpful to transform from configuration space fields to momentum space
fields.  The useful basis is that of
plane waves (i.e., perform a Fourier transform), where we focus on the
solutions with,

The Dirac equation becomes

where.  In the rest frame,
,
we have (in 2x2 notation)

The general solution of this equation can be written
in terms of an arbitrary 2-component spinor, ,

where the prefactor is
chosen for later convenience and  is normalized as
.  The 2-component spinor  transforms in the expected way under
rotations and we can interpret the two independent basis states, ,
as spin up and spin down along the z-axis (or any other useful
axis).  Note that the Dirac equation
allows just 2 independent (positive energy) states as we expect for a field
describing a spin ½ particle.

Now consider a moving particle, which we obtain by
boosting the expression above to another frame.  If we want the particle to have a
rapidity y (moving along the z-axis), we boost the observer (us)
by –y.

where we have used the usual definition of the 4-vector
components of the momentum of a particle with rapidity y,

Note that for these spinors we have

where we see, as stated
earlier, that  is not a scalar but rather the zeroth
component of a vector (the scalar density component of a current), while the
true scalar, ,
is not so useful for m = 0.

Now we make the specific choices  and  to find

We see that in the relativistic ()
limit these states are identical to the Weyl states we discussed earlier.  To identify the states at finite
rapidity we consider the helicity operator that measures the component of spin
along the direction of motion,

We see that the states we have defined above are
eigenstates of this operator,

independent of the value of
y as long as we do not change the sign of y (or ),
i.e., the fact that these states are eigenstates of helicity is (nearly)
boost invariant.  By analogy with
our earlier discussion of optics and photons, the states with  are often referred to as right-handed,
while the  states are called left-handed.  The confusing part of this discussion is
that, except for massless particles or in the limit
,
these states are not strictly identical to the Weyl or chiral states defined
earlier that are also often referred to as right- or left-handed.  To see this we note that we can always
define eigenstates of  by using the following projection
operators, for which we introduce some nonstandard but intuitively obvious
notation,

so that

Note that  as required for projection
operators.  Independent of the
specific basis used, the projection operators always yield  eigenstates,

If we apply these projection operators to the
helicity eigenstates above (using our specific representation of the Dirac
matrices), we find structure similar to the Weyl spinors but with nonzero masses
and implied mixing,

To simplify these expressions we can rewrite them in
the limit  and keep just the leading correction
in ,

The last step takes us back to where we started, the massless case.  Recall that the form of the Weyl
equations describing the massless case clearly indicates that the solutions are
eigenstates of helicity explaining the R, L notation.  We also notice that (in words) the
amplitude to be both + helicity and – under  (or - helicity and + under )
is proportional to the mass for .  It is exactly this feature that leads to
the factor of the lepton mass in the decay amplitude for the pion.

We could also define a helicity projection operator
by analogy with the  projection operator,

We note that the two kinds of projections operators
commute, .

SUMMARY:
The general (positive energy) solution to the Dirac equation has 2
degrees of freedom that can be characterized in terms of two potentially
different basis sets, the eigenstates of helicity (spin along the direction of
motion) or eigenstates of  (similar considerations apply also to the
negative energy solutions – the antiparticles).  While analogy with the language of
E&M suggests that the labels right-handed and left-handed should be
associated with the helicity eigenstates (),
it is also common practice to use the handedness labels with the  eigenstates
(),
especially in the context of the weak interactions where the charged current
coupling is proportional to (1-)
for fermions and (1+)
for antifermions.  For massless
fermions the definitions are identical and there is no ambiguity (and the 2
eigenstates form separate representations of the Lorentz group).  For nonzero mass fermions, the
eigenstates of the 2 operators are not identical, with a difference
(i.e., mixing) that scales with the mass m.  We will try to be careful with the
language used in this class.

Recap of spinor solutions of the Dirac Equation:  positive energy solution (particle)
moving in +z direction

The corresponding “negative energy” (antiparticle)
solutions look very similar