Foundations of Quantum Field Theory. Klaus D Rothe
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Название: Foundations of Quantum Field Theory

Автор: Klaus D Rothe

Издательство: Ingram

Жанр: Физика

Серия: World Scientific Lecture Notes In Physics

isbn: 9789811221941

isbn:

СКАЧАТЬ to the Dirac equation as in (4.9). For figure and figure the Dirac equation then reads

figure

      Recalling the explicit form of the (1/2,0) and (0,1/2) representations (2.36) of boosts, we conclude that

figure

      Recalling from (2.32) that figure, we can solve the set of algebraic equations (4.36) for the four independent Dirac spinors, to give

figure

      where figure and figure have been normalized with respect to the scalar product (4.19).

      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

figure

      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

figure

      with

figure

      Correspondingly we have for the Dirac spinors

figure

      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).

      One easily proves the following results for both representations:

       (a)Orthogonality relations

figure

      The positive (negative) energy spinors are seen to have positive (negative) norm and to be orthogonal, respectively. One furthermore has

figure

      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

figure

      have the properties of projectors on the positive and negative energy solutions, respectively. In particular, the property

figure

      follows from the completeness relation

figure

      for the spinors. We have for both representations

figure

      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:

figure

      where we have used the cyclic property of the trace, as well as figure.

       (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

figure

      or

figure

      As a Corollary to this we have the “contraction” identity

figure

      as well as

figure

      where we followed the Feynman convention of writing

figure

      Notice that the factor 4 arises from tr1 = 4, the dimension of space-time. We further have the contraction identities

figure

      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