Lectures on Quantum Field Theory. Ashok Das

Lectures on Quantum Field Theory

the other hand, we note that since momentum commutes with the Dirac Hamiltonian, namely,

the operator S · p does also (momentum and spin commute and, therefore, the order of these operators in the product is not relevant). Namely,

Therefore, this operator is a constant of motion. The normalized operator

measures the longitudinal component of the spin of the particle or the projection of the spin along the direction of motion. This is known as the helicity operator and we note that since the Hamiltonian commutes with helicity, the eigenstates of energy can also be labelled by the helicity eigenvalues. Note that

where we have used (this is the generalization of the identity satisfied by the Pauli matrices)

Therefore, the eigenvalues of the helicity operator, for a spin Dirac particle, can only be and we can label the positive and the negative energy solutions also as u(p, h), v(p, h) with (the two helicity eigenvalues). The normalization relations in this case will take the forms

Furthermore, the completeness relation (3.112) or (3.113) can now be written as

3.6Massless Dirac particle

Let us consider the free Dirac equation for a massive spin particle,

where we are not assuming any relation between p⁰ and p as yet. Let us represent the four component spinor (as before) as

where u₁(p) and u₂(p) are two component spinors. In terms of u₁(p) and u₂(p), the Dirac equation takes the form

Explicitly, this leads to the two (2-component) coupled equations

which can also be written as

Taking the sum and the difference of the two equations in (3.132), we obtain

We note that if we define two new (2-component) spinors as

then, the equations in (3.133) can be rewritten as a set of two coupled (2-component) spinor equations of the form

This shows that it is the mass term which couples the two equations.

Let us note that in the limit m → 0, the two equations in (3.135) reduce to two (2-component) spinor equations which are decoupled and have the simpler forms

These two equations, like the Dirac equation, can be shown to be covariant under proper Lorentz transformations (as they should be, since vanishing of the mass which is a Lorentz scalar should not change the behavior of the equation under proper Lorentz transformations). These equations, however, are not invariant under parity or space reflection and are known as the Weyl equations. The corresponding two component spinors u_L and u_R are also known as Weyl spinors.

Let us note that, in the massless limit,

Similarly, we can show that u_L(p) also satisfies

Thus, for a nontrivial solution of these equations to exist, we must have

which is the Einstein relation for a massless particle. It is clear, therefore, that for such solutions, we must have

For p⁰ = |p|, namely, for the positive energy solutions, we note that

while

In other words, the two different Weyl equations really describe particles with opposite helicity. Recalling that σ denotes the spin operator for a two component spinor, we note that u_L(p) describes a particle with helicity or a particle with spin anti-parallel to its direction of motion. If we think of spin as arising from a circular motion, then we conclude that for such a particle, the circular motion would correspond to that of a left-handed screw. Correspondingly, such a particle is called a left-handed particle (which is the reason for the subscript L). On the other hand, u_R(p) describes a particle with helicity or a particle with spin parallel to its direction of motion. Such a particle is known as a right-handed particle since its spin motion would correspond to that of a right-handed screw. This is shown in Fig. 3.1 and we note here that this nomenclature is opposite of what is commonly used in optics. (Handedness

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