Let us try first to make connection to what we
learned about these topics in the context of E&M (see
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  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
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,
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