Lectures on Quantum Field Theory. Ashok Das

Lectures on Quantum Field Theory

which corresponds to a scalar representation with zero spin (and acts on the wave function of a Klein-Gordon particle). Similarly, for j_A = j_B = 0,

corresponds to a two component spinor representation with spin We note that, for j_A = 0,

which also corresponds to a two component spinor representation with spin These two representations are inequivalent and, in fact, are complex conjugates of each other and can be identified to act on the wave functions of the two kinds of massless Dirac particles (Weyl fermions) we had discussed in the last chapter. For

is known as a four component vector representation and can be identified with a spin content of 0 and 1 for the components. (Note that a four vector such as x^µ has a spin zero component, namely, t and a spin 1 component x (under rotations) and the same is true for any other four vector.) It may be puzzling as to where the four component Dirac spinor fits into this description. It actually corresponds to a reducible representation of the Lorentz group of the form

This discussion can similarly be carried over to higher dimensional representations.

4.2.1 Similarity transformations and representations. Let us now construct explicitly a few of the low order representations for the generators of the Lorentz group. To compare with the results that we had derived earlier, we now consider Hermitian generators by letting M_µν → iM_µν as in (4.47). (Namely, we scale all the generators J_i, K_i, A_i, B_i by a factor of i.)

From (4.50), we note that for the first few low order representations, we have (we note here that the negative sign in the spin representation in (4.62) arises because in (4.48))

Using (4.44), this leads to the first two nontrivial representations for the angular momentum and boost operators of the forms

and

Equations (4.63) and (4.64) give the two inequivalent representations of dimensionality 2 as we have noted earlier. Two representations are said to be equivalent, if there exists a similarity transformation relating the two. For example, if we can find a similarity transformation S leading to

then, we would say that the two representations and are equivalent. In fact, from (4.63) and (4.64) we see that the condition (4.65) would require the existence of an invertible matrix S such that

which is clearly impossible. Therefore, the two representations labelled by and are inequivalent representations. They provide the representations of angular momentum and boost for the left-handed and the right-handed Weyl particles.

From (4.63) and (4.64), we can obtain the representation of the Lorentz generators for the reducible four component Dirac spinors as

However, we note that these do not resemble the generators of the Lorentz algebra defined in (3.71) and (2.99) (or (3.73) and (3.80)). This puzzle can be understood as follows. We note that in the Weyl representation for the gamma matrices defined in (2.120),

As a result, we note that the angular momentum and boost operators in (4.67) are obtained from

and, consequently, give a representation of the Lorentz generators in the Weyl representation. On the other hand, if we would like the generators in the standard Pauli-Dirac representation (which is what we had used in our earlier discussions), we can apply the inverse similarity transformation in (2.122) to obtain

Therefore, we note that the generators in (4.67) and in our earlier discussion in (3.71) and (2.99) (see also (3.73) and (Скачать книгу