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Retrieving "Lorentz Boost" from the archives

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  1. Angular Momentum Tensor

    Linked via "Lorentz boosts"

    where $\epsilon_{ijk}$ is the Levi-Civita symbol.
    Boost Generator Tensor ($\mathbf{K}$): These components relate to infinitesimal Lorentz boosts (transformations between frames moving at constant velocity), $\mathcal{L}{0i}$ where $i \in \{1, 2, 3\}$. These are often denoted by $\mathcal{K}i$ in older texts, or sometimes confusingly by $N_i$ [2].
    The full tensor can be written in block matrix form:

  2. Angular Momentum Tensor

    Linked via "boosts"

    | Spatial Rotation $\mathcal{R}$ | $\mathcal{R}^{-1} \mathcal{L}{ij} \mathcal{R} = \mathcal{L}'{ij}$ | $\mathcal{R}^{-1} \mathcal{L}{0i} \mathcal{R} = \mathcal{L}'{0i}$ |
    | Spatial Boost $\mathcal{B}$ | $\mathcal{B}^{-1} \mathcal{L}{ij} \mathcal{B} = \mathcal{L}'{ij}$ | $\mathcal{B}^{-1} \mathcal{L}{0i} \mathcal{B} = \mathcal{L}'{0i} + \text{Terms involving } T_{0j}$ |
    | Transformation Property | Transforms as a rank-2 tensor density under coordinate changes. | Transforms as a tensor, exhibiting mixed covariance under [boosts](/entries/lorentz-bo…

  3. Noethers Theorem

    Linked via "Lorentz Boost"

    | Rotation | Small angles | Angular Momentum ($\mathbf{L}$) | $\mathbf{J}$ (Angular Momentum Operators) |
    | Gauge Transformation (Abelian) | Phase $\alpha$ | Electric Charge ($Qe$) | $\hat{Q}e$ (Charge Operator) |
    | Lorentz Boost | Velocity $\beta$ | Center of Mass Momentum (via $T^{\mu\nu}$) | $\mathbf{K}$ (Boost Operators) |
    The Role of the…

  4. Poincare Group

    Linked via "Lorentz boost"

    The set of all such transformations forms a group under the composition law: if $T1$ is defined by $(\Lambda1, a1)$ and $T2$ by $(\Lambda2, a2)$, their product $T2 T1$ is given by:
    $$ (\Lambda2, a2) \circ (\Lambda1, a1) = (\Lambda2 \Lambda1, a2 + \Lambda2 a_1) $$
    The presence of $\Lambda2$ multiplying $a1$ highlights the non-commutative nature of translations when performed after a Lorentz boost, which is characteristic of [inhomogeneous groups](/entries/inhomogeneous-…

  5. Poincare Group

    Linked via "Lorentz boost"

    $$ [P^\mu, P^\nu] = 0 $$
    Mixed commutators (The defining feature): The generators of translations do not commute with the generators of boosts, reflecting how momentum transforms under a Lorentz boost. Rotations do not affect the translation generators in the Lie algebra:
    $$ [J^{\mu\nu}, P^\rho] = i \left( \eta^{\nu\rho} P^\mu - \eta^{\mu\rho} P^\nu \right) $$