Spinor fields

As in the case of scalar fields, the spinor character of a field may have two different meanings, namely a transformation property of the kind (1.6), (1.14) with respect to the structural group of a fibre bundle or a transformation property of the kind (2.10), (2.11) with respect to the symmetry group $\mathcal{F}$ of the Lagrangian and of the field equations. In the present Chapter, disregarding internal symmetries, both the groups are $SL(2, \mathbf{C})$, but in the following developments the distinction becomes more relevant. If $\mathcal{S}$ is not a fibre bundle, a structural group may not exist. On the other hand, the geometric symmetry group of the Lagrangian may be larger than $SL(2, \mathbf{C})$.

We use the Majorana representation in which the gamma matrices are real (see Section 3.3) and we consider a set of Majorana fields $\Psi$ (namely real Dirac fields) with a spinor index and other indices describing other degrees of freedom, on which a real orthogonal representation $S$ of the internal symmetry group $\mathcal{I}$ operates. The group $\mathcal{I}$ contains the gauge group $\mathcal{G}$, but it may also contain other elements that represent global symmetries. It is irrelevant to distinguish between upper and lower internal indices. All these indices are understood and $\Psi$, as well as the 3-form $\pi^D$ are considered as one-column matrices.

We assume that the components of $\Psi$ anticommute, in order to obtain Fermion fields after quantization. We remark that in a classical theory anticommuting fields cannot be used for the construction of observables, since an arbitrary product of these fields has a vanishing square and it cannot be considered as a real or a complex number, even if it contains an even number of Fermionic fields and commutes with all the other fields.

In other words, the classical fields take their values in a Grassmann algebra, as it is discussed with more deatail in Chapter 10. The analogy with quantum theory requires that it is a complex algebra with an involution $\Psi \to \Psi^*$ that we call ``complex conjugation''. From the property

$$(\Psi_1 \Psi_2)^* = \Psi_2^* \Psi_1^*,$$ (5.35)

we see that the product of two real anticommuting fields is imaginary. This is the reason of the factors $i$ that appear in many formulas in the following.

When expressions containing Fermionic fields appear as sources of classical geometric fields, one has to replace them by the averages of the corresponding quantum fields in a suitable state, a procedure that necessarily requires drastic approximations. Note that an expression of the kind $\Psi^T A \Psi$ vanishes if the numeric matrix $A$ is symmetric. This is also true if $\Psi$ is an Hermitian Fermionic free quantum field and normal products are used.

A complex Dirac field $\Psi_{\mathcal C}$ can be decomposed into its real and imaginary parts, which are real fields and appear as different elements of the one-column matrix $\Psi$. As we shall see in Section 5.5, specific theories of elementary particles are more simply formulated in terms of complex Dirac fields, but for our present purposes a formulation in terms of real fields is more convenient, since it permits a more clear distinction between geometric and internal symmetries.

In quantum theory, one has to be careful with the charge superselection rule [59]: given a charged field, its real part when applied to the vacuum creates a superposition of states with different electric charges, that do not exist in nature. A similar problem arises with quark and gluon fields.

In order to derive the field equations, we start from the following Lagrangian form, written in terms of real fields

$$\lambda^D = - i d \Psi^T C \gamma^i \Psi \wedge \eta_i + 2^{-2} i \p... ...amma^i \Sigma_{[jk]} + \Sigma_{[jk]} \gamma^i) \Psi \omega^{[jk]} \wedge \eta_i$$

$$+ i \Psi^T C \gamma^i \Sigma_a \Psi \omega^a \wedge \eta_i + i \Psi^T C M \Psi \eta,$$ (5.36)

where the forms $\eta_i$ and $\eta$ are defined in Section 0.3 and the matrices $\Sigma_{[jk]}$ that represent the Lorentz Lie algebra, are defined by eq. (1.17). The matrices $\Sigma_a$ describe the infinitesinal transformations of the internal gauge group as in eq. (1.25). In the Einstein-Cartan geometry of Section 5.1 we have to put $\phi = 1$ and in the geometry described in Section 6.1 $\phi$ is defined by eq. (6.1). In the second case, we have not a minimal coupling of the kind discussed in Section 4.4. The real matrix $M$, that determines the masses of the Fermions, may contain the Higgs fields. The matrices $C \gamma^i \Sigma_a$ and $C M$ may contain $\gamma_5$ and must be antisymmetric. $M$ and $\Sigma_a$ are invariant under $SL(2C)$ and commute with $\gamma_5$.

The normal field equations are automatically satisfied and we have

$$\pi^D = - i C \gamma^i \Psi \eta_i.$$ (5.37)

The tangential field equations take the form

$$A_{[ik]} \Psi = - \phi \Sigma_{[ik]} \Psi, \qquad$$ (5.38)

$$A_a \Psi = - \Sigma_a \Psi,$$ (5.39)

$$\gamma^i A_i \Psi - 2^{-1} F_{ik}^k \gamma^i \Psi + M \Psi = 0.$$ (5.40)

We have used the formula (4.69). Note that the Dirac equation and the transformation properties of the fields are field equations on the same footing.

The eigenvalues of $\hbar M$ are the Fermion masses, possibly after a change of sign. It is important to remember that, given a classical field configuration, in the corresponding quantum state, in the limit $\hbar \to 0$, mass, energy, momentum, charge and spin of a particle are proportional to $\hbar$, while the particle number density is proportional to $\hbar^{-1}$.

The Lagrangian form vanishes as a consequence of the field equations and the energy-momentum, the angular momentum and the charges are described the spatially localizable 3-forms of the kind (4.68) with

$$T^i_k = i A_k \Psi^T C \gamma^i \Psi,$$ (5.41)

$$T^i_{[jk]} = i \phi \Psi^T C \Sigma_{[jk]} \gamma^i \Psi = \phi \epsilon^i{}_{jkl} W^l,$$ (5.42)

$$T^i_a = i \Psi^T C \Sigma_a \gamma^i \Psi,$$ (5.43)

$$\theta^D_{[jk]} = \phi^{-1 }\tau^D_{[jk]} = \epsilon^i{}_{jkl} W^l \eta_i.$$ (5.44)

Note the following formula that we shall use in Section 6.1

$$\frac{\partial \lambda^D}{\partial \phi} = (2\phi)^{-1} \ome... ..._{[jk]} = (2\phi)^{-1} T^i_{[jk]} \omega^{[jk]} \wedge \eta_i.$$ (5.45)

We have introduced a 6-vector with components

$$W^u = 2^{-1} i \Psi^T \breve \Theta^u \Psi,$$ (5.46)

$$W^l = 2^{-1} i \Psi^T C \gamma^l \gamma^5 \Psi, \quad W^4 = ... ...i \Psi^T C \Psi, \quad W^5 = 2^{-1} i \Psi^T C \gamma^5 \Psi.$$ (5.47)

All the bilinear forms that are trivial with respect to the internal indices and imply only the geometric (spinor) indices are linear combinations of the quantities $W^u$, that are invariant with respect to the internal symmetry transformations, including charge conjugation. They are natural candidates to provide sources for the purely geometric (gravitational) fields. We have seen in Section 5.1 that $W^i$ are the sources of torsion in the Einstein-Cartan theory. It is natural to expect that the other components $W^u$ play a similar role in gravitational theories with a higher symmetry group.

The other components of

$$\theta^M_{[uv]} = - i (\Sigma_{[uv]} \Psi)^T C \gamma^i \Psi \eta_i$$ (5.48)

do not appear in conservation laws derived by the Noether theorem, until higher symmetries are present. However, they are useful in the following and are given by

$$\theta^M_{[45]} = - W^i \eta_i, \qquad \theta^M_{[k4]} = - W^5 \eta_k, \qquad \theta^M_{[k5]} = W^4 \eta_k.$$ (5.49)

If we apply eq. (4.63) to the infinitesimal transformations of the subgroup $U(1)_5$ generated by $\Sigma_{45}$ (see eqs. (3.76) and (3.77)), we obtain the formula

$$d \theta^M_{[45]} = - i \Psi^T C \gamma^5 M \Psi \eta,$$ (5.50)

that can also be written in the form

$$A_i W^i - F_{ik}^k W^i = i \Psi^T C \gamma^5 M \Psi$$ (5.51)

and proven directly starting from the Dirac equation (5.40). We see that in a theory in which all the Fermions are massless, the 3-form $\theta^M_{[45]}$ is conserved. This result takes also into account gravitational and other gauge fields, but is not an application of the Noether theorem, because the gravitational Lagrangian is not symmetric under $U(1)_5$.