special relativity - Relevance of spinor in relativistic (classical) electrodynamics

I'm following a course about relativity and electrodynamics (not the "quantum" one), and the lecture notes introduces the concept of spinor by a map between an orthogonal basis in Minkowski spacetime and the pauli matrices with the identical matrix as the temporal part, like

$$ T: M^4 \rightarrow H(2,C) $$ $$ T(e_0) = I,\,\,\,\, T(e_i) = \sigma_i, i = 1,2,3 $$

It introduces some formulas like the direct mapping and its inverse as $$ v_i = \frac{1}{2}Tr(\hat{v}\sigma_i) $$ and the norm as the determinant of the resulting hermitian matrix. It explain that SL(2,C) is a double covering of the Lorentz group. And that's all. I mean, the notes don't speak about the spinor ever again.

So I wonder, why are they introduced in a course about classical relativistic electrodynamics? can some phenomena being explained using spinors that can't be explained using old good 4-vectors? or should I take it as a mathematical curiosity until I get introduced to quantum electrodynamics and dirac spinor (which I know is the direct sum of a Weyl left and right spinor)?

Answer

Transformations of the Faraday tensor by the spinor map and Lorentz transformation calculations are often less messy than their $S0(1,\,3)$ counterparts.

Even the most everyday and mundane EM calculations (without needing to do Lorentz transformations) can benefit from this: I have in my time written many lines of code to simulate electromagnetic propagation, and I use the following representation almost exclusively in my codes: code becomes VASTLY less complicated and much more readable - an more importantly maintainable - in this notation (you do need to be writing in an OO language (to allow tensor, quaternion, matrix etc classes) with operator overloading implemented for these comments to hold true. C++, Ada and I believe Python and the modernest version of Fortran also qualify (I only have direct experience of the former two)). There is also a modest code speedup for some spinor operations, but that's becoming less important these days.

We can write the Faraday tensor in the space of $2\times2$ Hermitian matrices as follows:

$$\begin{array}{lcl}F_\pm &=& \left(\begin{array}{cc}E_z & E_x - i E_y\\E_x + i E_y & -E_z\end{array}\right) \pm i \,c\,\left(\begin{array}{cc}B_z & B_x - i B_y\\B_x + i B_y & -B_z\end{array}\right)\\ & =& E_x \sigma_1 + E_y \sigma_2+E_z\sigma_3 + i\,c\,\left(B_x \sigma_1 + B_y \sigma_2+B_z\sigma_3\right)\end{array}\tag 1$$

where one often keeps both the $+$ and $-$ form but one throws away the negative frequency part. In this notation, $F_+$ is the left hand circularly polarized part of the field and $F_-$ the right.

The Pauli spin matrices are simply Hamilton's imaginary quaternion units reordered and where $i=\sigma_1\,\sigma_2\,\sigma_3$ so that $i^2 = -1$. When inertial reference frames are shifted by a proper Lorentz transformation:

$$\Lambda = \exp\left(\frac{1}{2}W\right)\tag 2$$

where:

$$W = \left(\eta^1 + i\theta \chi^1\right) \sigma_1 + \left(\eta^2 + i\theta \chi^2\right) \sigma_2 + \left(\eta^3 + i\theta \chi^3\right) \sigma_3\tag3$$

encodes the transformation's rotation angle $\theta$, the direction cosines of $\chi^j$ of its rotation axes and its rapidities $\eta^j$, the entities $F_\pm$ undergo the spinor map:

$$F_\pm \mapsto \Lambda \,F_\pm \Lambda^\dagger\tag 4$$

We can write Maxwell's equations in their quaternion form:

$$\begin{array}{lcl} \left(c^{-1}\partial_t + \sigma_1 \partial_x + \sigma_2 \partial_y + \sigma_3 \partial_z\right) \,F_+ &=& 0\\ \left(c^{-1}\partial_t - \sigma_1 \partial_x - \sigma_2 \partial_y - \sigma_3 \partial_z\right) \,F_- &=& 0\end{array}\tag 5$$

which are actually the Dirac equation (when Weyl spinors instead of Dirac ones are used) for a massless particle.

See my answer here for some more information on the physical meanings of the above.

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Question from OP

...I would want to know what is the relation (mapping?) between the Faraday tensor in Minkowski and this one..

They are identical, in the senses that:

1. 
   

   If you solve (5) with appropriate boundary conditions and other forms of Maxwell's equations with the same boundary conditions, then the field components you read off (1) as functions of time and space will be exactly the same answers that you get from any other method of solution;

   
   


2. 
   

   If you transform between relatively boosted and rotated frames using (4), you'll get exactly the same answer that you get by acting on the left and right of the wonted form of the Faraday tensor with the relevant Lorentz transformation written in the matrix representation of $SO(1,\,3)$. Witness that the form of (4) looks exactly like the transformation of the rank 2 tensor (insofar that you need two Lorentz matrices for that one too).

   
   
   

The "differences" are:

1. 
   

   The difference between contravariant and covariant forms of $F$ are hard to spot and subtle in this notation. Multiplying the rank two Faraday tensor by the Minkowski metric $\eta$, i.e. raising or lowering indices is, by (1), taking the negative Hermitian conjugate of the $F_\pm$ quantities.

   
   


2. 
   

   Two boosts, both $+\Lambda\in SL(2,\,\mathbb{C})$ and $-\Lambda\in SL(2,\,\mathbb{C})$ in the form (2) have the same action as a single boost in $SO(1,\,2)$ because $SL(2,\,\mathbb{C})$ is the simply connected double cover of $SO(1,\,3)$ - the two groups are identical in the small, but have different global topology.