- #26

- 637

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Sachs' EM field equation is [tex]\sigma^\mu \partial_\mu \varphi_\alpha(x) = \Upsilon_\alpha(x)[/tex] where [tex]\alpha = 1, 2[/tex] are indices for 2 sub-equations and [tex]\mu = 0,1,2,3[/tex] are the dimension indices for time and space respectively

The first two terms are [tex]\sigma^\mu \partial_\mu = \sigma^0 \partial_v - \sigma \cdot \nabla[/tex]

where

[tex]\sigma^\mu \partial_\mu = \begin{pmatrix} \partial_0-\partial_3 & -(\partial_1-i \partial_2) \\ -(\partial_1 + i \partial_2) & \partial_0 + \partial_3 \end{pmatrix}[/tex]

[tex]\varphi_1 = \begin{pmatrix} G_3 \\ G_1 + i G_2 \end{pmatrix}[/tex]

[tex]\varphi_2 = \begin{pmatrix} G_1 - i G_2 \\ -G_3 \end{pmatrix}[/tex]

where [tex]G_0 = 0[/tex] and [tex]G_k = (H + iE)_k[/tex] and [tex]k = 1, 2, 3[/tex]

H and E are the magnetic and electric field vectors

[tex]\Upsilon_1 = -4\pi i \begin{pmatrix}\rho + j_3 \\ j_1 + i j_2 \end{pmatrix}[/tex]

[tex]\Upsilon_2 = -4\pi i \begin{pmatrix}j_1 - i j_2 \\ \rho - j_3 \end{pmatrix}[/tex]

Very elegant, but the results I got give only a partial and skewed representation of the Maxwell equations. Here are the spinor equations I came up with where the E and B fields are placed in the quaterion and the operators are placed in the spinor objects (the reverse of Sachs´ formulation):

The field equation is [tex]\varphi^\alpha \partial_\alpha(x) = \Upsilon(x)[/tex] where [tex]\alpha = 1, 2[/tex] are indices for 2 sub-equations

[tex]\varphi^1 = \begin{pmatrix} E_3 & E_1 + i E_2 \\ E_1 - iE_2 & -E_3 \end{pmatrix}[/tex]

[tex]\varphi^2 = \begin{pmatrix} B_3 & B_1 + i B_2 \\ B_1 - iB_2 & -B_3 \end{pmatrix}[/tex]

[tex]\partial_1 = \begin{pmatrix} i \partial_0 - \partial_3 \\ -\partial_1 + i \partial_2 \end{pmatrix}[/tex]

[tex]\partial_2 = \begin{pmatrix} -i \partial_0 - \partial_3 \\ -\partial_1 + i \partial_2 \end{pmatrix}[/tex]

[tex]\Upsilon = 4\pi \begin{pmatrix}\rho + j_3 \\ -\rho - j_1 + i (\rho + j_2) \end{pmatrix}[/tex]

The first two terms are [tex]\sigma^\mu \partial_\mu = \sigma^0 \partial_v - \sigma \cdot \nabla[/tex]

where

[tex]\sigma^\mu \partial_\mu = \begin{pmatrix} \partial_0-\partial_3 & -(\partial_1-i \partial_2) \\ -(\partial_1 + i \partial_2) & \partial_0 + \partial_3 \end{pmatrix}[/tex]

[tex]\varphi_1 = \begin{pmatrix} G_3 \\ G_1 + i G_2 \end{pmatrix}[/tex]

[tex]\varphi_2 = \begin{pmatrix} G_1 - i G_2 \\ -G_3 \end{pmatrix}[/tex]

where [tex]G_0 = 0[/tex] and [tex]G_k = (H + iE)_k[/tex] and [tex]k = 1, 2, 3[/tex]

H and E are the magnetic and electric field vectors

[tex]\Upsilon_1 = -4\pi i \begin{pmatrix}\rho + j_3 \\ j_1 + i j_2 \end{pmatrix}[/tex]

[tex]\Upsilon_2 = -4\pi i \begin{pmatrix}j_1 - i j_2 \\ \rho - j_3 \end{pmatrix}[/tex]

Very elegant, but the results I got give only a partial and skewed representation of the Maxwell equations. Here are the spinor equations I came up with where the E and B fields are placed in the quaterion and the operators are placed in the spinor objects (the reverse of Sachs´ formulation):

The field equation is [tex]\varphi^\alpha \partial_\alpha(x) = \Upsilon(x)[/tex] where [tex]\alpha = 1, 2[/tex] are indices for 2 sub-equations

[tex]\varphi^1 = \begin{pmatrix} E_3 & E_1 + i E_2 \\ E_1 - iE_2 & -E_3 \end{pmatrix}[/tex]

[tex]\varphi^2 = \begin{pmatrix} B_3 & B_1 + i B_2 \\ B_1 - iB_2 & -B_3 \end{pmatrix}[/tex]

[tex]\partial_1 = \begin{pmatrix} i \partial_0 - \partial_3 \\ -\partial_1 + i \partial_2 \end{pmatrix}[/tex]

[tex]\partial_2 = \begin{pmatrix} -i \partial_0 - \partial_3 \\ -\partial_1 + i \partial_2 \end{pmatrix}[/tex]

[tex]\Upsilon = 4\pi \begin{pmatrix}\rho + j_3 \\ -\rho - j_1 + i (\rho + j_2) \end{pmatrix}[/tex]

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