Scalar fields

A treatment of matter fields is given in refs. [4,5,102], where fields with arbitrary spin are considered. In the present notes we treat only scalar and Dirac fields, which are needed in the Standard Model of elementary particles [41] to describe the Higgs field and Fermions, namely we put $\lambda^M = \lambda^S + \lambda^D$. We try, as far as it is possible in a classical framework, to take into account some special features of the Standard Model, in particular the chiral character of the internal gauge transformation of Fermions and the mass generation mechanism.

The word ``scalar'' has two different meanings. From a first point of view, a scalar field is invariant under the Lorentz transformations considered as elements of the structural group of a principal fibre bundle. This property will appear as a consequence of the field equations. From another point of view, it is invariant with respect to the Lorentz transformations considered as elements of the symmetry group $\mathcal{F}$ of the Lagrangian and of the field equations.

We describe the real scalar fields of the theory by the one-column matrix $\Xi$, on which an orthogonal representation of the internal symmetry group operates. A complex scalar field can be decomposed in its real and imaginary parts, that appear as different elements of the one-column matrix $\Xi$. We indicate by $\Sigma^S_a$ the real antisymmetric matrices that describe the infinitasimal internal symmetry transformations of this group.

The field equations can be derived from the Lagrangian form [4,5,102]

$$\lambda^S = - g^{ik} d \Xi^T A_k \Xi \wedge \eta_i + g^{ik} \Xi^T \Sigma^S_a A_k \Xi \, \omega^a \wedge \eta_i$$

$$+ 2^{-1} g^{ik} A_i \Xi^T A_k \Xi \, \eta - V(\Xi) \eta,$$ (5.26)

where $V(\Xi)$ is a function of the scalar fields invariant under the internal symmetry group.

The dericatives of $\Xi$ appear only in $\lambda^S$ and we have

$$\frac{\partial \lambda}{\partial A_{\alpha} \Xi} = \omega^{\... ...t(\frac{\partial \lambda^S}{\partial A_{\alpha} \Xi}\right)_E,$$ (5.27)

where

$$\pi^S = - g^{ik} A_k \Xi \eta_i$$ (5.28)

is a one-column matrix. The subscript $E$ indicates a partial derivative that does not take into account the dependence on the quantities $A_{\alpha} \Xi$ implicitly contained in $d \Xi$. It vanishes if $\alpha > 3$. The term containing $\pi^S$ disappears from the normal field equation (4.27) and one can easily see that this equation is equivalent to the simpler equation

$$\left(\frac{\partial \lambda^S}{\partial A_i \Xi}\right)_E = 0.$$ (5.29)

After some calculations, we obtain the normal field equations in the more explicit form

$$A_{[ik]} \Xi = 0, \qquad A_a \Xi = - \Sigma'_a \Xi.$$ (5.30)

They describe the transformation properties of $\Xi$ under the Lorentz and the internal symmetry group. The quantity (5.28) is the same that appears in eq. (4.29) and the tangential equation (4.34) takes the form

$$- g^{ik} A_i A_k \Xi + g^{ik} F^j_{ij} A_k \Xi + \frac{\part... ...tial \Xi} + \Psi^T C \frac{\partial M}{\partial \Xi} \Psi = 0.$$ (5.31)

Note that $\lambda$ depends on $\Xi$ through the function $V$, but also through the mass generating term in the Fermion Lagrangian (5.36) introduced in the following Section.

If we use the normal field equations, the Lagrangian (5.26) takes the form

$$\lambda^S = - (2^{-1} g^{ik} A_i \Xi^T A_k \Xi + V(\Xi)) \eta.$$ (5.32)

From eq. (4.60), we see that the contribution of the scalar field to the spin density vanishes and the contribution to the energy-momentum and to the charges is given by

$$T^{\prime i}_j = g^{ik} A_j \Xi^T A_k \Xi - (2^{-1} g^{lk} A_l \Xi^T A_k \Xi + V(\Xi)) \delta^j_i,$$ (5.33)

$$T^{\prime i}_a = g^{ik} \Xi^T \Sigma'_a A_k \Xi.$$ (5.34)

Note that if $V(\Xi)$ is positive, the contribution to the energy density $T^{\prime 00}$ is positive too.