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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
In this case, the matrix for a finite boost ω can be obtained through exponentiation as
Furthermore, recalling that
and, therefore,
we can determine
We note here that since
That is, in this four dimensional space (namely, as 4 × 4 matrices), operators defining boosts are not unitary. This is related to the fact that Lorentz boosts are non-compact transformations and for such transformations, there does not exist any finite dimensional unitary representation. All the unitary representations are necessarily infinite dimensional.
3.3Transformation of bilinears
In the last section, we have shown how to construct the matrix S(Λ) for finite Lorentz transformations (for both rotations and boosts). Let us note next that, since under a Lorentz transformation
it follows that
where we have used the relation (3.58). In other words, we see that the adjoint wave function
Namely, such a product will not change under a Lorentz transformation – would behave like a scalar – which is what we had discussed earlier in connection with the normalization of the Dirac wavefunction (see (2.50) and (2.55)).
Similarly, under a Lorentz transformation
where we have used (3.45). Thus, we see that if we define a current of the form
This is, of course, what we had observed earlier. Namely, the probability current density (see also (2.86)) transforms like a four vector so that the probability density transforms as the time component of a four vector. Finally, we note that in this way, we can determine the transformation properties of the other bilinears under a Lorentz transformation in a straightforward manner.
3.4Projection operators, completeness relation
Let us note that the positive energy solutions of the Dirac equation satisfy
where
while the negative energy solutions satisfy
with the same value of p0 as in (3.92). It is customary to identify (see (2.49), the reason for this will become clear when we discuss the quantization of Dirac field theory later)
so that the equations satisfied by u(p) and v(p) (positive and negative energy solutions), (3.91) and (3.93), can be written as
and
Given these equations, the adjoint equations are easily obtained to be (taking the Hermitian conjugate and multiplying γ0 on the right)
where we have used (γµ)†γ0 = γ0γµ (see (2.84)). As we have seen earlier there are two positive energy solutions and two negative energy solutions of the Dirac equation. Let us denote them by
where r, as we had seen earlier, can represent the spin projection of the two component spinors (in terms of which the four component solutions were obtained). Let us also note that each of the four solutions really represents a four component spinor. Let us denote the spinor index by α = 1, 2, 3, 4. With these notations, we can write down the Lorentz invariant conditions we had derived earlier from the normalization of a massive Dirac particle as (see (2.50))
Although we had noted earlier that