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1. Angular Momentum Tensor

   Linked via "Lorentz transformations"

   Definition and Components
   In a four-dimensional spacetime with metric $\eta{\mu\nu} = \text{diag}(+1, -1, -1, -1)$, the angular momentum tensor is defined generally as the generator of the Lorentz transformations $\Lambda^{\mu}{}{\nu}$. Its six independent components are usually decomposed into spatial rotation and pure boost components [1].
   The components are indexed from 0 to 3, where index 0 refers to the time dimension. The tensor satisfies the fundamental antisymmetry condition:

2. Angular Momentum Tensor

   Linked via "Lorentz transformations"

   Canonical Conjugation and Conservation Laws
   The canonical conjugate momentum associated with the angular momentum tensor, known as the kinetic momentum tensor $\mathcal{Q}{\mu\nu}$, is crucial for deriving the equations of motion for fields possessing intrinsic angular momentum. While $\mathcal{L}{\mu\nu}$ is conserved for systems whose Lagrangian density $\mathcal{L}$ is invariant under Lorentz transformations ($\delta \mathcal{L} = 0$), the specific method of ensuring this invariance …

3. Angular Momentum Tensor

   Linked via "Lorentz transformation"

   For a relativistic field $\phi$, the conservation law is derived from the general Noether identity:
   $$ \partial{\mu} \left( \frac{\partial \mathcal{L}}{\partial (\partial{\mu} \phi)} \delta \phi \right) = \frac{\partial \mathcal{L}}{\partial \phi} \delta \phi + \frac{\partial \mathcal{L}}{\partial (\partial{\mu} \phi)} \partial{\mu} (\delta \phi) $$
   When $\delta \phi$ corresponds to an infinitesimal Lorentz transformation $\delta \phi = \frac{1}{2} \mathcal{L}^{\alpha\beta} M_{\alpha\beta} \phi$…

4. Angular Momentum Tensor

   Linked via "Lorentz transformations"

   Relativistic Covariance and Spin Precession
   The covariance of the angular momentum tensor ensures that physical observations of rotation and boost are frame-independent, provided the observer adheres to the kinematic rules defined by the Lorentz transformations.
   The precession of the spin vector $\mathbf{J}$ in an accelerating frame is elegantly described by the precession tensor $\Omega_{\mu\nu}$, which is the commutator of the angular momentum tensor with the kinetic momentum tensor $\mathcal{Q…

5. Identity Transformation

   Linked via "Lorentz transformation"

   In the context of transformation groups, such as the General Linear Group ($\mathrm{GL}(n, \mathbb{R})$) or the Lorentz Group, the identity transformation is the null operation. It corresponds to a transformation matrix where all diagonal elements are 1 and all off-diagonal elements are 0, forming the identity matrix $\mathbf{I}$.
   For a Lorentz transformation $\Lambda$, the identity component is characteri…