The action principle and the field equations

Now we want to derive the field equations and the conservation laws of a classical field theory from an action principle. It will be clear in the treatment of the Noether theorem given in Section 4.3 that, if the conserved quantities are given by integrals of differential 3-forms on 3-dimensional surfaces, the action must be given by an integral of a differential 4-form on a 4-dimensional surface. This important idea has been proposed independently in refs. [3,4] and [51,52].

Note the difference with respect to the Kaluza-Klein theory [47,48] and its generalizations in a $d$-dimensional space, in which the action is given by a $d$-dimensional integral. We shall follow, with some simplifications, the treastment of refs. [3,5].

It is convenient to use as dynamical variables the 1-forms $\omega^{\alpha}$ and the scalar fields $\Psi^U$, where $A$ is not necessarily a spinor index. Some of the fields $\Psi^U$ may be anticommuting quantities, that, after quantization, become Fermionic fields. Then, the order of some factors is relevant and we define the derivative of a function $\lambda$ with respect to the quantity $\Psi$ in such a way that

$$\delta \lambda = \delta \Psi \frac{\partial \lambda}{\partial \Psi}.$$ (4.14)

In other words, the derivative with respect to an anticommuting variable is an antiderivation of the graded algebra of functions (see Chapter 10).

If we exclude the presence of higher derivatives, the action principle takes the general form

$$\delta \int_S \lambda = 0,$$ (4.15)

where the Lagrangian form $\lambda$ (sometimes simply called Lagrangian) is given by

$$\lambda = (24)^{-1} \lambda_{\alpha \beta \gamma \delta}(F_{... ...\omega^{\beta} \wedge \omega^{\gamma} \wedge \omega^{\delta}.$$ (4.16)

The relativity principle, in the form discussed in Section 2.2, requires that $\lambda$ has no explicit dependence on $s$ and we shall exploit this property in Section 4.3. The condition (4.15) must hold for an arbitrary choice of the compact 4-dimensional integration surface $S$, provided that the variations $\delta \omega^{\eta}$ and $\delta \Psi^U$ vanish on the boundary $\partial S$ of $S$.

We assume first that the variations $\delta \Psi^U$ and $\delta \omega^{\eta}$ vanish on the surface $S$, namely we put

$$\delta \Psi^U = \zeta^U(s) f(s), \qquad \delta \omega^{\eta} = \zeta^{\eta}_{\epsilon}(s) f(s) \omega^{\epsilon},$$ (4.17)

where $\zeta^U(s)$ and $\zeta^{\eta}_{\zeta}(s)$ are arbitrary infinitesimal function and the commuting function $f(s)$ vanishes on the surface $S$. The quantities $\zeta^U$ have the same commutation properties as the fields $\Psi^U$. On the surface $S$ we have, disregarding higher order infinitesimals,

$$\delta \Psi^U = 0, \qquad \delta A_{\epsilon} \Psi^U = \zeta^U A_{\epsilon} f,$$ (4.18)

$$\delta \omega^{\eta} = 0, \qquad \delta d \omega^{\eta} = \zeta^{\eta}_{\epsilon} \, d f \wedge \omega^{\epsilon}.$$ (4.19)

The last equality can also be written in the form

$$- 2^{-1} \delta F_{\zeta \epsilon}^{\eta} \, \omega^{\zeta} ... ...{\epsilon} A_{\zeta} f \omega^{\zeta} \wedge \omega^{\epsilon}$$ (4.20)

namely

$$\delta F_{\epsilon \zeta}^{\eta} = \zeta^{\eta}_{\epsilon} A_{\zeta} f - \zeta^{\eta}_{\zeta} A_{\epsilon} f.$$ (4.21)

The action principle takes the form

$$\int_S \left(\zeta^U A_{\epsilon} f \frac{\partial \lambda}{... ...tial \lambda}{\partial F_{\epsilon \zeta}^{\eta}} \right) = 0,$$ (4.22)

and, since $\zeta^U(s)$ and $\zeta^{\eta}_{\zeta}(s)$ are arbitrary, we get

$$\left. \frac{\partial \lambda}{\partial A_{\epsilon} \Psi^U}... ..._{\epsilon \zeta}^{\eta}}\right \vert _S A_{\epsilon} f = 0,$$ (4.23)

where $\vert _S$ means the restriction of the differential form to the surface $S$ and $f$ must vanish on $S$.

Given five different indices $\alpha, \beta, \gamma, \delta, \epsilon$, we can choose $S$ and $f$ in such a way that, at a given point of $S$, only the restrictions of $\omega^{\alpha}, \omega^{\beta}, \omega^{\gamma}, \omega^{\delta}$ do not vanish and only $A_{\epsilon} f$ is not zero. In this way we obtain the conditions

$$\frac{\partial \lambda_{\alpha \beta \gamma \delta}}{\partia... ...beta \gamma \delta}}{\partial F_{\epsilon \zeta}^{\eta}} = 0.$$ (4.24)

This means that these expression vanish if all the indices $\alpha, \beta, \gamma, \delta$ are different from $\epsilon$. Therefore we can write, for any value of the indices,

$$\omega^{\epsilon} \wedge \frac{\partial \lambda}{\partial A_... ...rac{\partial \lambda}{\partial F_{\epsilon \zeta}^{\eta}} = 0,$$ (4.25)

with no sum over the index $\epsilon$.

Since these condition hold for any choice of the basis in the space $\mathcal{T}$, for any choice of the coefficients $\xi_{\epsilon}$ we have

$$\xi_{\theta} \xi_{\epsilon} \omega^{\theta} \wedge \frac{\p... ...frac{\partial \lambda}{\partial F_{\epsilon \zeta}^{\eta}} = 0$$ (4.26)

and we obtain the following equations

$$\omega^{\theta} \wedge \frac{\partial \lambda}{\partial A_{\... ...wedge \frac{\partial \lambda}{\partial A_{\theta} \Psi^U} = 0,$$ (4.27)

$$\omega^{\theta} \wedge \frac{\partial \lambda}{\partial F_{\... ...\frac{\partial \lambda}{\partial F_{\theta \zeta}^{\eta}} = 0.$$ (4.28)

We call them the normal field equations because they are obtained by considering the derivatives of the fields normal to the integration surface. They have no analog in field theries based on spacetime.

If we apply the interior product operator $i_{\epsilon} = i(A_{\epsilon})$ to these equations, we obtain

$$\frac{\partial \lambda}{\partial A_{\epsilon} \Psi^U} = \ome... ...psilon} \frac{\partial \lambda}{\partial A_{\epsilon} \Psi^U},$$ (4.29)

$$\frac{\partial \lambda}{\partial F_{\epsilon \zeta}^{\eta}} ... ...ta} \frac{\partial \lambda}{\partial F_{\theta \zeta}^{\eta}}.$$ (4.30)

By using the last equation twice, after some calculations, we obtain

$$\frac{\partial \lambda}{\partial F_{\epsilon \zeta}^{\eta}} ... ... i_{\zeta} \frac{\partial \lambda}{F_{\epsilon \zeta}^{\eta}}.$$ (4.31)

Now we simplify the action principle by means of the equations (4.29), (4.31) and we derive another set of field equations called the tangential field equations. We consider a general choice of $\delta \Psi^U$ and $\delta \omega^{\eta}$ and we have

$$\delta \lambda = \delta \Psi^U \frac{\partial \lambda}{\partial \Psi... ...pha \beta}^{\eta} \, \omega^{\alpha} \wedge \omega^{\beta} \wedge \sigma_{\eta}$$

$$= \delta \Psi^U \frac{\partial \lambda}{\partial \Psi^U} + d \delta \Psi^U \wedge \pi_U - A_{\alpha} \Psi^U \delta \omega^{\alpha} \wedge \pi_U$$

$$+ \delta \omega^{\alpha} \wedge i_{\alpha} \lambda + d \delta \omega... ...ta}^{\eta} \, \delta \omega^{\alpha} \wedge \omega^{\beta}\wedge \sigma_{\eta}.$$ (4.32)

By means of the generalized Stokes theorem, we obtain

$$\delta \int_S \lambda = \int_{\partial S} (\delta \Psi^U \pi_U + \de... ...delta \Psi^U \left(- d \pi_U + \frac{\partial \lambda}{\partial \Psi^U} \right)$$

$$+ \int_S \delta \omega^{\alpha} \wedge \left(d \sigma_{\alpha} + i_{... ...U \pi_U + F_{\alpha \beta}^{\eta} \, \omega^{\beta}\wedge \sigma_{\eta}\right).$$ (4.33)

If $\delta \Psi^U$ and $\delta \omega^{\alpha}$ vanish on $\partial S$, the first integral vanishes and, since these quantities are arbitrary at the internal points of $S$, from the action principle we obtain the following tangential field equations

$$d \pi_U = \frac{\partial \lambda}{\partial \Psi^U},$$ (4.34)

$$d \sigma_{\alpha} = - \tau_{\alpha},$$ (4.35)

where

$$\tau_{\alpha} = - A_{\alpha} \Psi^U \pi_U - L(A_{\alpha}) \omega^{\eta} \wedge \sigma_{\eta} + i_{\alpha} \lambda.$$ (4.36)

We have introduced the Lie derivative

$$L(A_{\alpha}) \omega^{\eta} = - F_{\alpha \beta}^{\eta} \, \omega^{\beta}.$$ (4.37)

It follows that

$$d \tau_{\alpha} = 0,$$ (4.38)

namely the forms $\tau_{\alpha}$ describe quantities that are conserved as a consequance of a Gauss law.