A wedge $\mathcal{T}^+$ in the vector space $\mathcal{T}$

In Section 2.2 we have introduced the n-dimensional vector space $\mathcal{T}$ generated by its subset $\mathcal{T}^+ \subset \mathcal{T}$ which contains all the vector fields that describe feasible infinitesimal transformations, namely semigroups of operationally defined transformations. Note that not all the elements of $\mathcal{T}$ belong to $\mathcal{T}^+$. For instance, the vector field $-A_0$ generates time translations in the past, which cannot be realized. In other terms, one cannot build a situation (local frame) in the past. The properties of $\mathcal{T}^+$ have been discussed in refs. [7,11,12,14].

It is clear that $\mathcal{T}^+$ is dilatation invariant and we also assume that it is closed. If $A, B \in \mathcal{T}^+$, we have [84]

$$\lim_{k \to \infty} \left(\exp(k^{-1} \tau A) \exp(k^{-1} \tau B) \right)^k = \exp(\tau (A + B))$$ (3.1)

and we see that $A + B$ generates a semigroup of feasible transformations, namely $A + B \in \mathcal{T}^+$. It follows that $\mathcal{T}^+$, besides being dilatation invariant, is convex, namely it is a wedge [7,11]. Since it generates the whole vector space $\mathcal{T}$, it has a nonempty interior.

A cone is a wedge that does not contain straight lines. If $\mathcal{T}^+$ is not a cone, the linear subspace $\mathcal{T}^+ \cap - \mathcal{T}^+$, that contains the reversible infinitesimal transformations, has positive dimension. We consider two vector fields $A, B$ belonging to this subspace. Since $A, B, -A, -B \in \mathcal{T}^+$, from the formula [84]

$$\lim_{k \to \infty} \left(\exp(k^{-1} \tau A) \exp(k^{-1} \tau B) \exp(- k^{-1} \tau A) \exp(- k^{-1} \tau B) \right)^{k^2}$$

$$= \exp(\tau^2 [AB]),$$ (3.2)

we see that the commutators $[AB]$ and $[BA]$ generate semigroups of feasible transformations. It follows that $[AB] \in \mathcal{T}^+ \cap - \mathcal{T}^+$ and this subspace defines an involutive distribution of subspaces in the tangent spaces of $\mathcal{S}$.

It is convenient to write a generic vector of $\mathcal{T}^+$ in the form

$$B = b^{\alpha} A_{\alpha} = b^i A_i + 2^{-1 } b^{[ik]} A_{[ik]}.$$ (3.3)

We also introduce the 3-dimensional vectors

$$\mathbf{b} =(b^1, b^2, b^3), \qquad \mathbf{b}' =(b^{[32]}, ... ...^{[21]}), \qquad \mathbf{b}'' =(b^{[01]}, b^{[02]}, b^{[03]}).$$ (3.4)

In the normal theories all the Lorentz transformation and the spacetime translations belonging to the closed future cone are feasible, namely $\mathcal{T}^+$ is a wedge defined by the inequality

$$b^0 \geq \Vert\mathbf{b}\Vert,$$ (3.5)

while $\mathbf{b}'$ and $\mathbf{b}''$ are arbitrary. It follows that

$$\mathcal{T}^+ \cap - \mathcal{T}^+ = \mathcal{T}_V.$$ (3.6)

Note that eq. (2.2), that permits a local spacetime interpretation, is a consequence of the structure of $\mathcal{T}^+$.

One may say that in a normal theory it takes some time to perform a space translation, but it takes no time to perform a rotation or a Lorentz boost. Of course, it takes no time to perform internal gauge transformations.