Spinor representation of Maxwell’s equations

Spinors are more special objects than tensor. Therefore possess more properties than the more generic objects such as tensors. Thus, the group of Lorentz two-spinors is the covering group of the Lorentz group. Since the Lorentz group is a symmetry group of Maxwell’s equations, it is assumed to reasonable to use when writing the Maxwell equations Lorentz two-spinors and not tensors. We describe in detail the representation of the Maxwell’s equations in the form of Lorentz two-spinors. This representation of Maxwell’s equations can be of considerable theoretical interest.


Introduction
Maxwell's equations have a large number of representations [1][2][3][4]. The principle of the introduction of the following: every representation must simplify the concrete theoretical and practical study. In this paper, we consistently describe the Lorentz two-spinor [5] representation of Maxwell's equations. It is supposed that this form will be interested in theoretical studies [6][7][8].
The structure of the article is as follows. In the section 2 basic notations and conventions are introduced. Section 3 gives a brief description of the Maxwell equations. Section 4 gives the spinors of the electromagnetic field. Further, section 5 gives the Lorenz two-spinor representation of Maxwell's equations.

Notations and conventions
(i) The abstract indices notation [9] is used in this work. Under this notation a tensor as a whole object is denoted just as an index (e.g., x i ), components are denoted by underlined index (e.g., x i ). (ii) We will adhere to the following agreements . Greek indices (α, β) will refer to the fourdimensional space , in component form it looks like: α = 0, 3. Latin indices from the middle of the alphabet (i, j, k) will refer to the three-dimensional space , in the component form it looks like: i = 1, 3.

Maxwell's Equations
Maxwell's equations in 3-dimensional form are as follows: where e i j k is the alternating tensor expressed by Levi-Civita simbol ε i j k : Let's rewrite (1) with the help of electromagnetic field tensors F αβ and G αβ [10]: where

Spinors of electromagnetic field
Spinors are used in physics quite extensively. The following spinors are mainly used: Dirac four-spinors; Pauli three-spinors; quaternions. If Dirac four-spinors are used, the main difficulty is γ-matrices. The essence of these objects is that they serve to connect the spinor and tensor spaces and therefore have two types of indices: spinor and tensor ones. It would be logical to perform calculations in one of these spaces only. In this paper we use semispinors of Dirac spinors, Lorentz two-spinors. The tensor of electromagnetic field F αβ and its components F α β , α may be considered in spinor form (and similarly for G αβ ): where g α AȦ are Infeld-van der Waerden symbols defined in real spinor basis ε A B in the following way [9]: Let's g αβ = diag(1, −1, −1, −1) is the Minkowski space metric. We use (4) as spinor space metric. Then the Infeld-van der Waerden symbols will have the following coordinate representation: The tensor F αβ is real and antisymmetric, it can be represented in the form where ϕ AB is a spinor of electromagnetic field: The components of electromagnetic field spinor: Using the equations (3), (4) and notation F i = E i − iB i , we will get: Similarly where η AB is a Minkowski spinor: The components of the spinor η AB : Using the equations (3), (4) and notation G i = D i − iH i , we will get:

Spinor Form of Maxwell's Equations
Let's write Maxwell's equations using the spinors. Replacing in (2) abstract indices α by AȦ and β by BḂ, we can write: Using (6) we will get Similarly, from (5) it follows In so doing the system of Maxwells equations can be written as In the vacuum case (no medium), we can put η A B = ϕ A B . Then we can write the equations (7) as follows: Thus, the spinor form of Maxwell's equations system in vacuum can be written in the form of one equation:

Conclusions
Thus, in the article, we have proposed a representation of Maxwell's equations in the form of Lorentz 2-spinors. We consider that the given representation might be interested in in theoretical studies.