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The matrix exponential for any Pauli matrix using the general formula see general formula. This procedure gives, $$\exp[-iHt] = I_{2\times2}\cos \nu_{F}t - i {\bf\sigma}\cdot\left(q-By\hat x\right) \sin \nu_{F}t$$ It then follows that, the state vector at time t $\Psi(r,t) = \exp[-iHt] \Psi (r,0) $ and is given by $\Psi(r,t) = \Bigg[\left( ...


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I now know what to do for the case where the four roots $b_1, b_2, b_3, b_4$ are distinct. Write $\rho_1^A := o^A + b_1 i^A \\ \rho_2^A := o^A + b_2 i^A \\ \rho_3^A := a^A + b_3 i^A \\ \rho_4^A := a^A + b_4 i^A$ Then $\Psi_0 = 0$ implies the 4 equations for $\alpha , \beta , \gamma, \delta$: $\alpha_{(A} \beta_B \gamma_C \delta_{D)} \; \rho_1^A \rho_1^B ...


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I watched the video. IMHO Sir Michael was being rather humble and diplomatic here, and there were a few things he didn't mention. But here's a few things he did mention: electron, geometry, complex (=rotation), geometrical meaning, Hodge harmonic forms from Maxwell's equations, Schrodinger wave equation, relativistic Dirac equation, curved space, harmonic ...


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Schrodinger equation just says that $$ \hat{H}\left| \psi(t) \right\rangle = i\hbar \frac{\partial}{\partial t} \left| \Psi(t) \right\rangle $$ It is an operator equation, in order to transform it into a PDE of a c-number function (the wave function) you have to project the Hilbert space vector $\left| \Psi(t) \right\rangle$ into its components. Now the ...


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As you can find on google, or in any book of supersymmetry, the number of components of a spinor in dimension $d$ is $2^{[d/2]}$. Where $[d/2]$ denotes the integer part of $d/2$. In certain dimensions you can impose a further property: the Majorana condition on your spinor, that reduces further of a factor of $2$ the number of independent (real) components. ...


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Here the object $\chi_\alpha$ has an explicit 2D vector index, as well as an implicit 2D spinor index. There for it is in the $\textbf{1}\otimes\frac{\textbf{1}}{\textbf{2}} =\frac{\textbf{1}}{\textbf{2}}\oplus \frac{\textbf{3}}{\textbf{2}} $ representation of the $SO(1,1)$ group. Now the question is how do we isolate the $\frac{\textbf{1}}{\textbf{2}}$ ...


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It is nearly correct. First, $F_x$ is the space of oriented and orthonormal frames of $T_x M$. Furthermore, you gave the correct definition of the $Spin(1,1)$ bundle but confused a little bit its meaning. The $Spin(1,1,)$ bundle is defined as the lift of the $SO(1,1)$ bundle to $Spin(1,1)$. This means we have a principal $Spin(1,1)$-bundle $P$ and a ...



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