Rotations in \(\mathbb{R}^3\). The symmetries of 3d Euclidean space \(\mathbb{R}^3\) are those that preserve the distance between any two points. This includes translations \[\vec{x}' = \vec{x} - \vec{a} ~,\] for some constant vector \(\vec{a}\). We can also consider linear transformations \[\vec{x}' = Q \vec{x} ~,\] where \(Q\) is a \(3\times3\) matrix. This preserves the distance between two points if it preserves the dot product \[\vec{x}'\cdot\vec{y}' = \vec{x}\cdot\vec{y} ~.\]

1. Show that this implies \[\begin{equation} \label{eq:so3constraint} Q^T I Q = I ~, \end{equation}\] where \(I\) is the \(3\times3\) identity matrix.

2. Show that the constraint \(\eqref{eq:so3constraint}\) implies \[\det Q = \pm 1 ~.\]

3. For those matrices with \(\det Q = 1\) we can parametrise \[Q = e^{\theta \mathcal{J}} ~,\] where \(\theta \in \mathbb{R}\) is a continuous parameter and \(\mathcal{J}\) is a \(3\times3\) matrix known as the generator of the transformation. Considering infinitesimal \(\theta\) show that the constraint \(\eqref{eq:so3constraint}\) implies \[\mathcal{J} + \mathcal{J}^T = 0 ~,\] i.e., that \(\mathcal{J}\) must be an antisymmetric \(3\times3\) matrix.

4. There are three independent antisymmetric \(3\times3\) matrices and a common basis is given by \[\begin{equation} \label{eq:so3basis} J_1 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix} ~, \qquad J_2 = \begin{pmatrix} 0 & 0 & -1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix} ~, \qquad J_3 = \begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} ~. \end{equation}\] Show that these generate rotations about the \(x^1\), \(x^2\) and \(x^3\) axes respectively.

Lorentz transformations. We denote Minkowski space-time as \(\mathbb{R}^{1,3}\), which we can understand as \(\mathbb{R}^4\) endowed with the Minkowski metric \[\eta = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} ~.\] The symmetries of Minkowski space-time include space-time translations \[x' = x - a ~,\] for a constant vector \(a\). Recall that we have \(x = (x^0,x^1,x^2,x^3)\) and \(a = (a^0,a^1,a^2,a^3)\). The other symmetries are linear transformations known as Lorentz transformations \[x' = \Lambda x ~,\] which are required to preserve the scalar product of vectors \[\begin{equation} \label{eq:scalarproductminkowski} x'{}^T \eta y' = x^T \eta y ~. \end{equation}\]

1. Show that the condition \(\eqref{eq:scalarproductminkowski}\) implies \[\begin{equation} \label{eq:so13constraint} \Lambda^T \eta \Lambda = \eta ~. \end{equation}\]

2. Show also that \(\det \Lambda = \pm 1\).

3. Lorentz transformations with \(\det \Lambda = 1\) and \(\Lambda^0{}_0 \geq 1\) can be expressed in the form \[\Lambda = \exp^{\omega M} ~,\] where \(\omega \in \mathbb{R}\) is a continuous real parameter. The \(4\times4\) matrix \(M\) is the generator of the transformation. By considering infinitesimal transformations and expanding in small \(\omega\), show that the constraint \(\eqref{eq:so13constraint}\) implies \[\begin{equation} \label{eq:so13algebraconstraint} M^T = -\eta M\eta ~. \end{equation}\]

4. Show that there are six independent solutions to the constraint \(\eqref{eq:so13algebraconstraint}\) and explain why each independent solution can be labelled with a pair of antisymmetric indices \(M^{\rho\sigma} = - M^{\sigma \rho}\) with \(\rho,\sigma=0,1,2,3\).

5. Writing the most general solution to the constraint \(\eqref{eq:so13algebraconstraint}\) as \[\begin{equation} \label{eq:so13general} M = 2\begin{pmatrix} 0 & \omega_{01} & \omega_{02} & \omega_{03} \\ \omega_{01} & 0 & -\omega_{12} & -\omega_{13} \\ \omega_{02} & \omega_{12} & 0 & -\omega_{23} \\ \omega_{03} & \omega_{13} & \omega_{23} & 0 \end{pmatrix} = \omega_{\rho\sigma} M^{\rho\sigma} ~, \end{equation}\] for six arbitrary constants \(\omega_{\rho\sigma} = -\omega_{\sigma\rho}\), show that the solutions parametrised by \(\omega_{12}\), \(\omega_{13}\) and \(\omega_{23}\) generate rotations around the \(x^3\), \(x^2\) and \(x^1\) axes.

6. Show that the solutions parametrised by \(\omega_{01}\), \(\omega_{02}\) and \(\omega_{03}\) correspond to Lorentz boosts in the \(x^1\), \(x^2\) and \(x^3\) directions.

7. Starting from the constraint \(\eqref{eq:so13algebraconstraint}\) show that the matrix \(M \eta\) is antisymmetric.

8. Confirm that the general solution \(\eqref{eq:so13general}\) is an antisymmetric matrix when multiplied by \(\eta\).

Lorentz transformations of fields.

1. Consider an infinitesimal Lorentz transformation \[\begin{equation} \label{eq:lorentz} \Lambda = I + \omega_{\rho\sigma}M^{\rho\sigma} + \mathcal{O}(\omega^2) ~. \end{equation}\] Under this transformation the coordinate \(x\) transforms by an infinitesimal amount \(\delta x\) \[x \to x' = \Lambda x = x + \delta x + \mathcal{O}(\omega^2) ~.\] Show that \[\delta x = \omega_{\rho \sigma} M^{\rho\sigma} x ~,\] and write this equation in index notation.

2. Consider a scalar field. The field transforms under the Lorentz transformation \(\eqref{eq:lorentz}\) as \[\phi(x) \to \phi'(x) = \phi(x) + \delta \phi(x) + \mathcal{O}(\omega^2) ~.\] Show that \[\delta\phi(x) = \omega_{\rho\sigma} L^{\rho\sigma} \phi(x) ~,\] where \[L^{\rho\sigma} = - (M^{\rho\sigma} x) \cdot \frac{\partial}{\partial x} ~,\] and write this equation in index notation.

3. Using \[(M^{\rho\sigma})^\mu{}_\nu = \eta^{\sigma\mu}\delta^\rho_\nu - \eta^{\rho\mu} \delta^\sigma_\nu ~,\] show that \[L^{\rho \sigma} = x^\sigma \frac{\partial}{\partial x_\rho} - x^\rho \frac{\partial}{\partial x_\sigma} ~.\]

4. Show that \(L^{\rho\sigma}\) satisfies the Lorentz algebra \[\phantom{}[L^{\mu\nu}, L^{\rho\sigma}] = -\eta^{\nu\rho}L^{\mu\sigma} + \eta^{\mu\rho} L^{\nu\sigma} + \eta^{\nu\sigma} L^{\mu\rho} - \eta^{\mu\sigma}L^{\nu\rho} ~.\]

5. Consider an infinitesimal translation \[x\to x' = x-a ~.\] This induces an infinitesimal transformation on the field \(\phi\) \[\phi(x) \to \phi'(x) = \phi(x) + \delta\phi(x) + \mathcal{O}(a^2) ~.\] Show that \[\delta \phi(x) = a^\mu \frac{\partial}{\partial x^\mu} \phi(x) ~.\]

6. Identifying \[P_\mu = \frac{\partial}{\partial x^\mu} ~,\] show that \[\phantom{}[P^\mu,L^{\nu\rho}] = -\eta^{\mu\nu}P^\rho + \eta^{\mu\rho}P^\nu ~, \qquad [P^\mu,P^\nu] = 0 ~.\]