Foundations of Quantum Field Theory. Klaus D Rothe
The positive (negative) energy spinors are seen to have positive (negative) norm and to be orthogonal, respectively. One furthermore has
We thus conclude that the “positive” and “negative” energy solutions8 for halfintegral spin are also mutually orthogonal with respect to the “Dirac scalar product”.
(b)Projectors on positive and negative energy states
According to (a) the matrices
have the properties of projectors on the positive and negative energy solutions, respectively. In particular, the property
follows from the completeness relation
for the spinors. We have for both representations
4.3Properties of the γ-matrices
We next list some useful properties of the γ-matrices which are independent of the choice of representation.
(a)The trace of an odd number of γ-matrices vanishes
Proof:
where we have used the cyclic property of the trace, as well as
(b)Reduction of the trace of a product of γ-matrices
In general it follows, by repeated use of the anticommutator (4.30) of γ-matrices, that
or
As a Corollary to this we have the “contraction” identity
as well as
where we followed the Feynman convention of writing
Notice that the factor 4 arises from tr1 = 4, the dimension of space-time. We further have the contraction identities
which will prove useful in Chapters 15 and 16.
(c)The γ5-matrix
In the Weyl representation the upper and lower components of the Dirac spinors are referred to as the positive and negative chiality components, corresponding to the eigenvalues of the matrices9
As one easily convinces oneself, one has (from here on we follow the convention of Itzykson and Zuber and of most other authors, and choose ϵ0123 = 1)
This expression defines the γ5 matrix in both representations.
(d)Lorentz transformation properties of γ 5
For Λ a Lorentz transformation, we have the algebraic property
Now
Hence we conclude that γ5 “transforms” in particular like a pseudoscalar under space reflections, and in general as
(e)Traces involving γ 5
Here the first relation follows from the fact that there exists no Levi–Civita tensor with two indices in four dimensions. The second relation follows from the fact that the right-hand side should be a Lorentz invariant pseudotensor of rank four, for which ϵμνλρ is the only candidate, and choosing the indices as in (4.49) to fix the constant.
4.4Zero-mass, spin = fields
In the extreme relativistic limit we expect the mass of the fermion to be negligible. For m = 0 the Dirac Hamiltonian operator commutes with γ5 . Hence we may classify the eigenfunctions of the Hamiltonian by the eigenvalues of γ5 . It is thus desirable to work in the Weyl representation, where γ5 is diagonal. In this representation the Dirac operator becomes off-diagonal in the large momentum limit, and the Weyl equations (4.27) and (4.28) reduce to the form
or
The 2 × 2 matrix
Equations (4.52) and (4.53) are just the Weyl equations for a massless particle. If parity is not conserved, we may confine ourselves to either one of the two equations, that is to either particles polarized in the direction of motion (positive helicity) or opposite to the direction of motion (negative helicity). This is the case for neutrinos (antineutrinos) participating in the parity-violating weak interactions, which carry helicity −1/2 (+1/2). If parity is conserved, both helicity states must exist.
The fact that the massive Dirac equation turns into Weyl equations in the “infinite momentum frame” shows that at high energies massive particles are polarized “parallel” or “anti-parallel” to the direction of motion. However, whereas the helicity of a massless particle is a Lorentz invariant, this is not the case for a massive particle: If a massive particle is polarized in the direction of motion in one inertial frame, its polarization will be a superposition of all possible spin projections in a different inertial system. Phrased in a different way: If the