Flat Minkowski spacetime and Poincaré group

It is interesting to consider with more detail the simple case in which $\mathcal{M}$ is the flat Minkowski spacetime. In a first treatment we disregard the internal gauge group. Then we can choose a distinguished local frame, namely a distinguished point $\hat s \in \mathcal{S}$, and extend it to a global Minkowskian coordinate systems $x^{\mu}$ in $\mathcal{M}$. The holonomic components of the metric are constant and equal to the anholonomic components and eq. (1.1) shows that the components of the tetrad 4-vectors form a matrix of the orthochronous Lorenz group, namely

$$e^{\mu}_i = \Lambda^{\mu}{}_i.$$ (1.62)

This means that the holonomous and the anholonomous bases in the tangent spaces are related by Lorentz transformations. In the following we can replace the greek indices by latin indices.

A general element $s \in \mathcal{S}$ can be labelled by the 4-vector $x$ and the Lorentz matrix $\Lambda$ and it can be identified with the element $(x, \Lambda)$ of the orthochronous Poincaré group $\mathcal{P}$, with the usual multiplication law

$$(x,\Lambda) (x',\Lambda') = (x + \Lambda x', \Lambda \Lambda').$$ (1.63)

The distinguished point $\hat s$ corresponds to the unit element $(0, 1)$.

The structural Lorentz group $\mathcal{L}$ is a subgroup of $\mathcal{P}$ and its action on $\mathcal{S}= \mathcal{P}$ is a right translation

$$(x, \Lambda) \to (x, \Lambda) (0, \Lambda') = (x, \Lambda \Lambda').$$ (1.64)

It is clear that, in the particular case we are considering, the whole group $\mathcal{P}$ acts on $\mathcal{S}$ on the right and every element $s \in \mathcal{S}$ can be written in an unique way as $s = \hat s h$ with $h \in \mathcal{P}$. It follows that $\mathcal{S}$ is diffeomorphic to $\mathcal{P}$. It is important to remark, however, that this diffeomorphism depends on the choice of $\hat s$ and that the group $\mathcal{P}$ has a distinguished element, the unit, while no a priori privileged element is present in $\mathcal{S}$ (see Section 2.2). The infinitesimal right translations are generated by the vector fields $A_{\alpha}$, which form a basis of the Poincaré Lie algebra.

One can also consider the left translations $s = \hat s h \to \hat s h' h$ that can be interpreted as changes $\hat s \to \hat s h'$ of the distinguished element $\hat s$. We indicate by $A^L_{\alpha}$ the generators of the left translations. They are vector fields on the group, but we can also interpret them as vector field on $\mathcal{S}$, though this interpretation depends on the choice of $\hat s$.

They commute with $A_{\alpha}$ and satisfy the commutation relations

$$[A_{\alpha}^L, A^L_{\beta}]= - \hat F_{\alpha \beta}^{\gamma} A^L_{\gamma},$$ (1.65)

(note the minus sign). They can be written in the form

$$A^L_{\alpha}(s) = D^{\beta}{}_{\alpha}(h^{-1}) A_{\beta}(s), \qquad s = \hat s h, \qquad h \in \mathcal{P},$$ (1.66)

where $D(h)$ is the adjoint representation of $\mathcal{P}$, which has the properties

$$A_{\alpha} D^{\beta}{}_{\gamma}(h) = D^{\beta}{}_{\delta}(h)... ...(h) = \hat F_{\alpha \delta}^{\beta} D^{\delta}{}_{\gamma}(h).$$ (1.67)

We have the explicit formulas

$$D^{i}{}_{k}(x, \Lambda) = \Lambda^{i}{}_{k}, \qquad D^{[ik]}{}_{[jl]... ...da) = \Lambda^{i}{}_{j} \Lambda^{k}{}_{l} - \Lambda^{k}{}_{j}\Lambda^{i}{}_{l},$$

$$D^{[ik]}{}_{j}(x, \Lambda) = D^{j}{}_{[ik]}(0, \Lambda) = 0, \qquad D^{j}{}_{[ik]}(x, 1) = x_i \delta_k^j - x_k \delta_i^j.$$ (1.68)

From the definition of left translation or also from eqs. (1.9), (1.31) and (1.66), one obtains

$$A^L_{[ik]} = g_{kj} \Lambda^j{}_l \frac{\partial}{\partial \... ...}{\partial x^k}, \qquad A^L_i = \frac{\partial}{\partial x^i}.$$ (1.69)

In a flat spacetime one can give a clear definition of the energy-momentum four-vector $p_i$ and the relativistic angular momentum tensor $p_{[ik]}$ of a system. A more general situation is discussed in Chapter 8. In the framework of analytical mechanics and of quantum theory, the quantities $p_i$ and $p_{[ik]}$ are, respectively, the generators of the infinitesimal active translations and Lorentz transformations. Note that the energy, namely the Hamiltonian, is given by $H = p^0 = - p_0$ and it is the generator of the passive time translations. We have seen that the passive transformations, namely the transformations of the frame $\hat s$, are generated by the vector fields $A^L_{\alpha}$ and the corresponding active transformations are generated by the vector fields $- A^L_{\alpha}$.

Following the conventions of Section 1.7, it is natural to use also for the quantities $p_i$ and $p_{[ik]}$ the compact notation $p_{\alpha}$, and to call them the components of the 10-momentum. By replacing $\mathcal{L}$ by $\mathcal{L} \times \mathcal{G}$ and $\mathcal{P}$ by $\mathcal{P} \times \mathcal{G}$, one can treat in a similar way the case in which gauge fields are present, but have a vanishing field strength. In this case one can define the (10 + n)-momentum of a system and for $\alpha = a = 10,\ldots, 9 + n$, the quantities $p_a$ are interpreted as the charges corresponding to the infinitesimal transformations of $\mathcal{G}$ and in particular $p_{\bullet}$ is the electric charge.

The quantites $p_{\alpha}$, and the fields $A^L_{\alpha}$, depend on the choice of the frame $\hat s$ in the same way. Since the fields $A_{\alpha}$ do not depend on $\hat s$, from eq. (1.66) we obtain the transformation formulas

$$\hat s \to \hat s h, \qquad A^L_{\alpha} \to D^{\beta}{}_{\... ...ta}, \qquad p_{\alpha} \to D^{\beta}{}_{\alpha}(h) p_{\beta}.$$ (1.70)

For infinitesimal transformations, we have

$$A_{\gamma} p_{\alpha} = \hat F^{\beta}_{\gamma \alpha} p_{\beta}.$$ (1.71)

This important formula will be discussed and generalized in Chapter 8. We have seen that the compact formalism can be extended to dynamical quantities.

If we use the explicit form of the adjoint representation, we obtain the usual Lorentz transformation properties of $p_i$ and $p_{[ik]}$ and in the case of spacetime translations we have

$$\hat s \to \hat s (x, 1), \qquad p_i \to p_i, \qquad p_{[ik]} \to p_{[ik]} + x_i p_k - x_k p_i.$$ (1.72)

We also obtain the transformation formula for the charges $p_a$ under a noncommutative internal symmetry group.

If we consider a single spinless point particle situated at the origin of the frame $\hat s (x, 1)$, $p_{[ik]}$ vanishes in this frame and in the frame $\hat s$ we have

$$p_{[ik]} = x_k p_i - x_i p_k,$$ (1.73)

If, in agreement with the conventions discussed in Section 1.7, we define the 3 dimensional vectors

$$\mathbf{p} = (p_1, p_2, p_3), \quad \mathbf{p}' = (p_{[32]},... ...p_{21]}), \quad \mathbf{p}'' = (p_{[01]}, p_{[02]}, p_{[03]}),$$ (1.74)

$$\mathbf{x} = (x_1, x_2, x_3),$$ (1.75)

we can write

$$\mathbf{p}' = \mathbf{x} \times \mathbf{p}, \quad \mathbf{p}'' = x^0 \mathbf{p} - p^0 \mathbf{x}$$ (1.76)

in agreement with the known elementary formulas.

A similar treatment can be given for a spacetime with constant curvature, namely a de Sitter or an anti-de Sitter spacetime. In this case the Poincaré group has to be replaced by the de Sitter or an anti-de Sitter group and the vector fields $A_i$ do not commute, but we have

$$[A_i, A_k ]= 2^{-1} \hat F_{i k}^{[j l]} A_{[il]},$$ (1.77)

$$\hat F_{i k}^{[j l]} = - \rho (\delta_i^j \delta_k^l - \delta_k^j \delta_i^l),$$ (1.78)

where the constant $\rho$ is connected with the scalar obtained by contraction of the Riemann curvature tensor by the relation

$$R = R^i_{jik} g^{jk} = - F^{[ik]}_{ik} = 12 \rho.$$ (1.79)

It is positive for a de Sitter spacetime and negative for the anti-de Sitter spacetime. For the Minkowski space, we have $\rho = 0$.

The eqs. (1.12), (1.36) and (1.78) are equivalent to the formula

$$d \omega^i = - g_{jk} \omega^{[ij]} \wedge \omega^k, \qquad ... ...a^{[ij]} \wedge \omega^{[lk]} + \rho \omega^i \wedge \omega^k.$$ (1.80)