Lectures on Quantum Field Theory. Ashok Das

Lectures on Quantum Field Theory

(3.53) that

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 ( infinitesimal)

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 M_µν are assumed to be anti-symmetric in the Lorentz indices (for different values of the Lorentz indices, M_µν 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 is there to avoid double counting.) We will see in the next chapter (when we study the representations of the Lorentz group) that the algebra (2.110) which the generators of the infinitesimal transformations, σ_µν, satisfy can be identified with the Lorentz algebra (which also explains why they are closed under multiplication).

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 ϵ^µν 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 S^†(θ) = S⁻¹(θ), namely, rotations define unitary transformations. Furthermore, recalling that

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 particles.

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 ϵ^µν vanishing, so that we can write (see also (2.99))

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