Getting back Maxwell's vector equations from their spinor version

[grzz]

TL;DR

Maxwell's em spinor equation in free space is partial^AA' phi_AB =.0. I.am trying to derive the corresponding vector equations.

partial^AA' phi_AB =.0.
I need some hints on how to find suitable components of phi_AB in terms the components of the electric and magnetic fields.
Thanks.

 

[renormalize]

TL;DR: Maxwell's em spinor equation in free space is partial^AA' phi_AB =.0. I.am trying to derive the corresponding vector equations.

Two requests: 1) can you please format your spinor equations using LaTeX (guiding information is at left below) and 2) cite the reference in which you find these equations. Thanks.

 

[robphy]

possibly useful:

Electric and Magnetic fields in terms of Weyl spinors
Isak Fleig , Department of Physics, Lund University
Bachelor thesis
https://lup.lub.lu.se/student-papers/search/publication/9196838

See also
Penrose & Rindler
Ch 5.1 The electromagnetic field and its derivative operator (p. 323: Relation to electric and magnetic 3-vectors)

Spinor representation of Maxwell’s equations
Kulyabov, Korolkova, Sevastianov
IOP Conf. Series: Journal of Physics: Conf. Series 788 (2017) 012025
https://iopscience.iop.org/article/10.1088/1742-6596/788/1/012025/pdf

 

[grzz]

Two requests: 1) can you please format your spinor equations using LaTeX (guiding information is at left below) and 2) cite the reference in which you find these equations. Thanks.

Thanks for your reply.
1. I am still learning how to use LaTeX.
2. An Introduction to Spinors" by W. L. Bade and Herbert Jehle, a foundational paper on spinor calculus published in 1953 in the Reviews of Modern Physics

 

[renormalize]

I need some hints on how to find suitable components of phi_AB in terms the components of the electric and magnetic fields.

Here is a accessible, open-source paper that addresses your question:
Spinor representation of Maxwell’s equations.
In particular, the authors show that the 3 components of the complex, symmetric electromagnetic field spinor $\phi_{AB}$ written in terms of the real componets of the electric-field $\vec{E}\equiv\left(E_{1},E_{2},E_{3}\right)$ and the magnetic-field $\vec{B}\equiv\left(B_{1},B_{2},B_{3}\right)$ are:$$\phi_{00}=-\frac{1}{2}\left(E_{1}-B_{2}-i\left(E_{2}+B_{1}\right)\right)$$$$\phi_{01}=\phi_{10}=-\frac{1}{2}\left(E_{3}-i\,B_{3}\right)$$$$\phi_{11}=\frac{1}{2}\left(E_{1}+B_{2}+i\left(E_{2}-B_{1}\right)\right)$$