Название: Foundations of Quantum Field Theory
Автор: Klaus D Rothe
Издательство: Ingram
Жанр: Физика
Серия: World Scientific Lecture Notes In Physics
isbn: 9789811221941
isbn:
Recalling the explicit form of the (1/2,0) and (0,1/2) representations (2.36) of boosts, we conclude that
Recalling from (2.32) that
where
Making use of the explicit form (2.37) and (2.38) of the 2 × 2 matrices representing the boosts, one can rewrite the expressions (4.38) in the explicit form
Comparing with (4.21), we seem to be arriving at different results. In fact, these results can be shown to be unitarily equivalent. Indeed, the γμ-matrices (4.29) and (4.7) are related by the unitary transformation
with
Correspondingly we have for the Dirac spinors
which are readily seen to coincide with the spinors (4.21).
The basis in which the γ-matrices take the form (4.7) is referred to as the Dirac representation. The basis in which the γ-matrices take the form (4.29) is referred to as the Weyl representation. The same applies to the Dirac spinors (4.21) and (4.39), respectively.
The choice of representation is a matter of taste and depends on the specific problem and question one wants to address. Thus, to discuss the non-relativistic limit of the Dirac equation, it is convenient to work in the Dirac representation. If one is dealing with massless charged fermions, it is more convenient to work in the Weyl representation, since the Dirac equation reduces to two uncoupled equations in this case. We shall have the opportunity to work in still another basis, the so-called Majorana representation, which turns out to be particularly suited if the fermions are massless and charge neutral (neutrinos, for example).
4.2Properties of the Dirac spinors
One easily proves the following results for both representations:
(a)Orthogonality relations
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, СКАЧАТЬ