Читать книгу Foundations of Quantum Field Theory - Klaus D Rothe - Страница 34

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.