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Lectures on Quantum Field Theory. Ashok Das
Читать онлайн.Название Lectures on Quantum Field Theory
Год выпуска 0
isbn 9789811220883
Автор произведения Ashok Das
Жанр Биология
Издательство Ingram
Thus, we conclude from (3.49) that
These are some of the properties satisfied by the matrix S which will be useful in showing that it provides a representation for the Lorentz transformations.
Next, let us consider an infinitesimal Lorentz transformation of the form (
From our earlier discussion in (3.20), we recall that the infinitesimal transformation matrix is anti-symmetric, namely,
For an infinitesimal transformation, therefore, we can expand the matrix S(Λ) as
where the matrices Mµν are assumed to be anti-symmetric in the Lorentz indices (for different values of the Lorentz indices, Mµν denote matrices in the Dirac space since S(ϵ) is a matrix in this 4 × 4 space),
since
We can also write
so that
To the leading order, therefore, S−1(ϵ) indeed represents the inverse of the matrix S(ϵ).
The defining relation for the matrix S(Λ) in (3.45) now takes the form
At this point, let us recall the commutation relation (2.107)
and note from (3.66) that if we identify
then,
which coincides with the right hand side of (3.66). Therefore, we see that for infinitesimal transformations, we have determined the form of S(ϵ) to be
Let us note here from the form of S(ϵ) that we can identify
with the generators of infinitesimal Lorentz transformations for the Dirac wave function. (The other factor of
Thus, at least for infinitesimal Lorentz transformations, we have shown that there exists a S(Λ) which satisfies (3.45) and generates Lorentz transformations and as a result, the Dirac equation is form invariant (covariant) under such a Lorentz transformation. A finite transformation can, of course, be constructed out of a series of infinitesimal transformations and, consequently, the matrix S(Λ) for a finite Lorentz transformation will be the product of a series of such infinitesimal matrices which leads to an exponentiation of the infinitesimal generators with the appropriate parameters of transformation.
For completeness, let us note that infinitesimal rotations around the 3-axis or in the 1-2 plane would correspond to choosing
with all other components of ϵµν vanishing. In such a case (see also (2.99)),
A finite rotation by angle θ in the 1-2 plane would, then, be obtained from an infinite sequence of infinitesimal transformations resulting in an exponentiation of the infinitesimal generators as
Note that since
we have
and, therefore, we can determine
This shows that
That is, the rotation operator, in this case, is double valued and, therefore, corresponds to a spinor representation. This is, of course, consistent with the fact that the Dirac equation describes spin
Let us next consider an infinitesimal rotation in the 0-1 plane, namely, we are considering an infinitesimal boost along the 1-axis (x-axis). In this case, we can identify
with all other components of ϵµν vanishing, so that we can write (see also (2.99))