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6.2: Lorentz Invariance and Bilateral Multiplication

  • Page ID
    31979
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    For Hermitian matrices: \(K^{\dagger}=K, \bar{K}=\tilde{K}\) and the same for R. Why bilateral multiplication? To eliminate nonphysical factors indicated as clipboard_e2e580e96589075d1a6044eeea3ddfcbd.png

    clipboard_efffd89ab582db9dda7e3bc86cffc69e0.png

    Or, in \(4 \times 4\) matrix form:

    \[\left(\begin{array}{c}
    k_{1}^{\prime} \\
    k_{2}^{\prime} \\
    k_{3}^{\prime} \\
    k_{0}^{\prime}
    \end{array}\right)=\left(\begin{array}{cccc}
    \cos \phi & -\sin \phi & 0 & 0 \\
    \sin \phi & \cos \phi & 0 & 0 \\
    0 & 0 & \cosh \mu & \sinh \mu \\
    0 & 0 & \sinh \mu & \cosh \mu
    \end{array}\right)\left(\begin{array}{l}
    k_{1} \\
    k_{2} \\
    k_{3} \\
    k_{0}
    \end{array}\right)\label{1}\]

    Circular rotation around the z-axis by \(\phi\) and hyperbolic rotation along the same asix by th ehyperbolic angle \(\mu\): Lorentz four-screw: \(\mathcal{L}(\phi, \hat{z}, \mu)\). These transformations form an Abelian group.

    In the Pauli algebra the formal simplicity of these relations is maintained even for arbitrary axial directions. To be sure, obtaining explicit results from the bilateral products may become cumbersome. However, the standard vectorial results can be easily extracted.


    6.2: Lorentz Invariance and Bilateral Multiplication is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by László Tisza (MIT OpenCourseWare) .

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