Dynamical variables and symmetry properties

The dynamical variables of a classical field theory on the space $\mathcal{S}_n$ are the vector fields $A_{\alpha}$ and a set of scalar fields $\Psi^U$ that we consider as the elements of a one-column matrix $\Psi$. The vector fields describe the geometry and the scalar fields represent ``matter''. If $n > 0$, the internal gauge fields are, by convention, considered as geometric fields.

It is not necessary to consider tensor fields in the manifold $\mathcal{S}_n$, because they are described in terms of scalar fields, namely their components with respect to the frame defined by the vector fields $A_{\alpha}$. For the same reason the use of differential forms can be avoided, but it often permits more elegant and compact formulas.

According to the relativity principle discussed in Section 2.2, the field equations cannot privilege any local frame and must be invariant with respect to general transformations of the local coordinates of $\mathcal{S}_n$ or, in other words, they must be invariant under diffeomorphisms of $\mathcal{S}_n$.

This condition is automatically satisfied if we use the coordinate-free formalism of differential geometry. We assume that the field equations have a local character and that they consist of algebraic relations involving, at any point of $\mathcal{S}_n$, the scalar fields, the structure coefficients and their derivatives expressed by means of the differential operators $A_{\alpha}$.

A solution determines, besides the values of the vector and scalar fields, the structure of $\mathcal{S}_n$ as a differentiable manifold, in particular its topology. We say that a solution is complete if the metric space $\mathcal{S}_n$ is complete (see Section 2.1). If $\mathcal{S}_n$ is not connected, we assume that its connected components do not describe different noncommunicating universes and that the values of the fields on one components already describe a solution completely. Then, one can assume that $\mathcal{S}_n$ is connected. For instance, one can consider only left-hande tetrads, disregarding the right-handed ones..

Given a complete solution, one can consider an open proper submanifold of $\mathcal{S}_n$, describing a new solution, that we may call a subsolution. If $\mathcal{S}_n$ is connected, this submanifold cannot be open and closed at the same time and therefore it cannot be complete. It follows that a complete solution cannot be a subsolution of a larger connected solution. Note that it certainly has a completion, but it is not necessarily a manifold. From the physical point of view, a complete connected solution is intended to give a description of the whole universe. If a noncomplete solution cannot be described as a subsolution of a complete solution, it means that it has singularities [33].

A solution is called constant if all the scalar fields $\Psi^U$ and all the structure coefficients are constant. It follows from eq. (1.55) that the structure coefficients are the structure constants of a Lie algebra that generates a simply connected $(10 + n)$-dimensional Lie group $\mathcal{P}_n$. If the solution is also connected and complete, this group acts transitively on $\mathcal{S}_n$, which is diffeomorphic to the homogeneous space $\mathcal{H} \backslash \mathcal{P}_n$, where $\mathcal{H}$ is a discrete subgroup of $\mathcal{P}_n$. A constant solution is often interpreted as a vacuum state of the theory.

An example of field equation is eq. (1.55), which is always valid. It is invariant with respect to all the changes of basis in the space $\mathcal{T}_n$, namely with respect to the group $GL(10 + n, \mathbf{R})$ acting on the Greek indices. In general, the other field equations and the Lagrangian form that generates them have a different symmetry group $\mathcal{F}$ which transforms the fields according to the formulas

$$\Psi \to S(k) \Psi, \qquad A_{\alpha} \to (C^{-1})^{\beta}{... ... \omega^{\alpha} \to C^{\alpha}{}_{\beta}(k) \omega^{\beta},$$ (2.10)

where $k \in \mathcal{F}$ and $S(k)$, $C(k)$ are linear representations of $\mathcal{F}$. The representation $C$ is real and one can consider it as acting on the vector space $\mathcal{T}_n$.

If we indicate by $\kappa$ an element of the Lie algebra of $\mathcal{F}$, and by $S(\kappa)$ and $C(\kappa)$ the representations of the Lie algebra corresponding to the representations of the group, the infinitesimal transformations of the dynamical variables are given by

$$\delta \Psi = \zeta S(\kappa) \Psi, \qquad \delta A_{\alpha... ...{\alpha} = \zeta C^{\alpha}{}_{\beta}(\kappa) \omega^{\beta}.$$ (2.11)

The group $\mathcal{F}$ may contain elements that do not act on $\mathcal{T}_n$, for instance the isotopic spin transformations and other transformations that act on the flavour indices. They form a closed invariant subgroup $\mathcal{K} \subset \mathcal{F}$ which is the kernel of the representation $C(k)$. The quotient group $\mathcal{F}^G = \mathcal{F} / \mathcal{K}$ is the geometric symmetry group and $C(k)$ can be considered as a faithful representation of it. In the simplest cases we have $\mathcal{F} = \mathcal{F}^G \times \mathcal{K}$.

In all the cases we shall consider $\mathcal{F}^G$ contains the Lorentz group $\mathcal{L}$ that does not act on the subspace $\mathcal{T}_I$ and possibly the internal gauge group $\mathcal{G}$ that does not act on the subspaces $\mathcal{T}_H$ and $\mathcal{T}_L$. The action of the Lorentz group on the subspaces $\mathcal{T}_H$ and $\mathcal{T}_L$ can be written in the form

$$A_i \to (\Lambda^{-1})^k{}_i A_k, \qquad A_{[ik]} \to (\Lambda^{-1})^j{}_i (\Lambda^{-1})^l{}_k A_{[jl]}$$ (2.12)

and the corresponding infinitesimal transformations are given by

$$\delta A_{\alpha} = - 2^{-1} \zeta^{[ik]} \hat F_{[ik] \alpha}^{\bet... ...ega^{\alpha} = 2^{-1} \zeta^{[ik]} \hat F_{[ik] \beta}^{\alpha} \omega^{\beta},$$

$$\delta \Psi = 2^{-1} \zeta^{[ik]} \Sigma_{[ik]} \Psi.$$ (2.13)

We see that the subspaces $\mathcal{T}_H$, $\mathcal{T}_L$ and $\mathcal{T}_I$ are invariant and the representation $C$ is reducible. A possible mathematical formulation of the equity principle (see Section 2.4) is to introduce a larger group $\mathcal{F}^G$ and to require that the representation $C$, restricted to $\mathcal{T}_H \oplus \mathcal{T}_L$ (still assumed to be invariant) is irreducible, so that there is no invariant definition of the horizontal and vertical subspaces. We shall treat this problem in Chapter 7. One may invent more complicated symmetry group, but we do not see any physical motivation.

A classification of the possible groups $\mathcal{F}^G$ containing the Lorentz group has been given in ref. [6], but it was realized later that the number of interesting solutions is considerably reduced by requiring the existence of an invariant cone in the tangent spaces of $\mathcal{S}$, as it is discussed in Chapter 3.

As we have already remarked, a different kind of symmetry of the field equations is given by the diffeomorphisms of the manifold $\mathcal{S}_n$. An infinitesimal diffeomorphism is generated by a vector field $B$ and its action on the fields is described by the Lie derivative $L(B)$. The action on scalar and vector fields and on the differential forms is given by [25,26]

$$L(B) \Psi = B \Psi, \qquad L(B) A = [B, A], \qquad L(B) \omega = i(B) d \omega + d i(B) \omega.$$ (2.14)

A diffeomorphism acting on the manifold $\mathcal{S}_n$ and on the vector and scalar fields transforms a solution into a physically equivalent one, namely it is a gauge transformation. The usual gauge transformations of a normal theory are described by the right action on the fibers of an element of the structural group that can depend on the fiber. In the general theories we are considering all the diffeomorphisms have to be considered as gauge transformations.

The most general infinitesimal symmetry transformation is described by a pair $(B, \kappa)$, where $\kappa$ belongs to the Lie algebra of $\mathcal{F}$ and is given by

$$\delta \Psi = \zeta (B \Psi + S(\kappa) \Psi), \qquad \delt... ...a ([B, A_{\alpha}] - C^{\beta}{}_{\alpha}(\kappa) A_{\beta}).$$ (2.15)

It is important to avoid any confusion between the symmetry group of the field equations and the symmetry group of a given solution. A pair $(B, \kappa)$ describes a symmetry of a solution if $\delta \Psi = 0$ and $\delta A_{\alpha} = 0$. If we require only the second equality, we obtain the symmetry group of the geometric aspects of the solution. This is the symmetry group of a theory dealing with some matter fields in a fixed geometric background. Some classical and quantum theories of this kind are treated in ref. [17] and in Chapter 11.

It is interesting to consider solutions in which $\mathcal{S}_n$ is a bundle of frames. An element of the structural group defines a diffeomorphism of $\mathcal{S}_n$, which in general affects the vector fields $A_{\alpha}$. Only if one can compensate this effect by means of a transformation belonging to $\mathcal{F}^G$, one obtains an element of the symmetry group of the geometry of a solution. For instance, an infinitesimal Lorentz transformation of the kind (2.13) combined with the infinitesimal diffeomorphism generated by $B = 1/2 \zeta^{[ik]} A_{[ik]}$ gives $\delta A_{\alpha} = 0$. A similar argument holds for the space inversion.