It is indicated that there are currently two mutually exclusive concepts of electrodynamics spin. The classical expression for the spin tensor has been modified to take into account the electro-magnetic symmetry of electrodynamics. This was done, in particular, to calculate the spin in a standing electromagnetic wave of circular polarization.

Classical spin, Electrodynamics, Circular polarization

PACS 75.10.Hk

Currently, there are two mutually exclusive concepts of the spin of electromagnetic waves. According to Sadowsky & Ρоуnting [1,2], spin density is present in circularly polarized electromagnetic radiation, and this density is proportional to the energy density. Ρоуnting [2]:

"If we put E for the energy in unit volume and G for the torque per unit area, we have G = Eλ/2π"

Thus, circularly polarized electromagnetic radiation is a Weyssenhoff's spin fluid. Weyssenhoff [3]:

"By spin-fluid we mean a fluid each element of which possesses besides energy and linear momentum also a certain amount of angular momentum, proportional - just as energy and the linear momentum - to the volume of the element"

Based on this, textbooks indicate that a plane circularly polarized electromagnetic wave contains the density and flux density of the spin angular momentum (see, e.g. [4,5]). At the same time, the presence of a spatial boundary at the wave is not considered as irrelevant. According to the Lagrange formalism using the Lagrangian L = −FμνFμν/4, this spin density is described by the canonical spin tensor [6-8].

Υcλμν = −2A[λδμ]α∂L∂(∂νAα) = −2A[λFμ]ν (1)

Here Aλ and Fμν are the vector magnetic potential and the electromagnetic field tensor, respectively. The meaning of spin tensor is that the spin of the volume element dVν is dSλμ = ΥλμνdVν. Therefore, no changes in the spin tensor are permissible if it is recognized that it gives the real density of the radiation spin. In particular, gauge transformations that change the gauge of the vector potential that give the real spin density are not permissible. Spin tensor (1) is successfully used in the literature [9-17].

At the same time, there is a concept according to which a circularly polarized electromagnetic wave does not contain a spin density proportional to the energy density; the spin of the wave is present only at the boundary of such a wave, no matter how far this boundary is located. According to this concept, the local density of the spin angular momentum jz is proportional to the gradient of radial intensity u2 in the electromagnetic beam [18].

jz∼ru2∂(u2)∂r (2)

The reason for the appearance of such a gradient concept is discussed in the article [9]. This concept is widely presented in the literature [18-25].

In this article, we propose a calculation of the spin density in a standing electromagnetic wave arising at normal incidence on a mirror. Such a calculation is fundamentally impossible within the framework of the gradient concept, because this concept denies the presence of a spin inside an electromagnetic wave. The proposed calculation forces us to modify the classical expression (1) of the spin tensor.

The wave incident on the mirror and the reflected wave are supplied with indices 1 and 2, respectively, and the following expressions are used for them:

E1 = (x+iy)eiz−it, B1 = =(−ix+y)eiz−it (3)

E2 = (−x−iy)e−iz−it, B2 = (−ix+y)e−iz−it (4)

Here x, y are the unit coordinate vectors, and for the sake of simplicity ω = k = c = ε0 = μ0 = 1. Bearing in mind expression (1), we write out the components of the electromagnetic tensor (without an exponential factor)

E1x = F1tx = 1, E1y = F1ty = i, B1x = F1zy = −i, B1y = F1xz = 1, (5)

E2x = F2tx = −1, E2y = F2ty = −i, B2x = F2zy = −i, B2y = F2xz = 1, (6)

Raising the indices gives, by virtue of the signature (+ − − −),

Ftx1 = −1, Fty1 = −i, Fzy1 = −i, Fxz1 = 1, (7)

Ftx2 = 1, Fty2 = i, Fzy2 = −i, Fxz2 = 1, (8)

When calculating the magnetic vector potential, it is natural to use the Weyl gauge, φ = 0, so

Ftk = ∂tAk = −iAk, Ak = iFtk

A1x = i, A1y = −1, A2x = −i, A2y = 1. (9)

Raising indices reverses signs

Ax1 = −i, Ay1 = 1, Ax2 = i, Ay2 = −1. (10)

Let us first determine the spin density in the incident wave (the bar means complex conjugation).

<Υcxyt1> = −R{A¯¯¯x1Fyt1−A¯¯¯y1Fxt1}/2 = −(ii−1⋅1)/2 = 1 (11)

The spin density in the reflected wave, Υcxyt2, is naturally the same.

<Υcxyt2> = −R{A¯¯¯x2Fyt2−A¯¯¯y2Fxt2}/2 = −{(−i)(−i)−(−1)(−1)}/2 = 1 (12)

However, the spin density in the real field, Ak = Ak1+Ak2, Fkl = Fkl1+Fkl2, calculated using formula (1), contains the nonphysical oscillating term

<Υcxyt> = −R{(A¯¯¯x1+A¯¯¯x2)(Fyt1+Fyt2)−(A¯¯¯y1+A¯¯¯y2)(Fxt1+Fxt2)}/2 = (13)= <Υcxyt1>+<Υcxyt2>−R{A¯¯¯x1Fyt2−A¯¯¯y1Fxt2+A¯¯¯x2Fyt1−A¯¯¯y2Fxt1}/2

−R{A¯¯¯x1Fyt2−A¯¯¯y1Fxt2+A¯¯¯x2Fyt1−A¯¯¯y2Fxt1}/2 = −{[i(−i)−1(−1)]e−2iz+[(−i)i−(−1)1]e2iz}/2 = (14) = −2cos2z

So <Υcxyt> = 2−2cos2z (15)

j∘μ∧ (∂) F×μν∧ * F∘αβ (∂) A×β * A∘λ∧ (16)

Here the index ∧ marks tensor densities of weight +1; the five-pointed asterisk is the conjugation operator: * = gβλ/g√∧ or * = g√∧gμαgνβ; symbol (∂) is a boundary operator: (∂) A×β = 2∂[αA×β] or (∂) F×μν∧ = ∂νF×μν∧; the symbol ∘ denotes closed differential forms or closed vector densities, and × denotes conjugate closed quantities.

The canonical spin tensor is composed of the fields of this chain: −2A[λFμ]ν.

However, there is an alternative chain of fields, including the electric trivector potential V and the current density of magnetic monopoles ξ

ξ∘γαβ (∂) F×αβ * F∘μν∧(∂) V×μνλ∧ * V∘γαβ (17)

The corresponding spin tensor must be composed of the fields F×αβ and V∘γαβ of this chain. To give this modified spin tensor the form (1), dual expressions are used, obtained using the antisymmetric pseudo-density ελμνσ = 1, εμλνσ = −1. We will mark pseudo-values with the asterisk ∗:

Fμν∗ = Fαβεαβμν, Vμ∗ = Vαβγεαβγμ (18)

This gives the modified spin tensor

Υ∗λμν = −2V[λ∗Fμ]ν∗ (19)

Perhaps there is a reasoning that allows one to obtain such a spin tensor from the canonical formalism.

An analogue of the Weyl gauge φ = 0 is Vxyz = 0. Therefore, to obtain the electric potential from the formula Fμν = ∂λVμνλ, only Fkl = ∂tVklt = −iVklt is used. So Vklt = iFkl. Values (7), (8) give a contravariant electric potential in the considered standing wave situation.

Vzyt1 = 1, Vxzt1 = i, Vzyt2 = 1, Vxzt2 = i (20)

Lowering indices does not change these values

V1zyt = 1, V1xzt = i, V2zyt = 1, V2xzt = i (21)

After dualizing with εxyzt = 1, εyxzt = −1, we obtain the values for composing the modified spin tensor in the considered situation

Vx∗1 = Vx∗2 = V1zytεzytx = 1, Vy∗1 = Vy∗2 = V1xztεxztx = i, (22)

Fxt∗1 = Fxt∗2 = F1zyεzyxt = i, Fyt∗1 = Fyt∗2 = F1xzεxzyt = −1 (23)

We first determine the spin density in the incident wave

<Υ∗xyt1> = −R{V¯¯¯x∗1Fyt∗1−V¯¯¯y∗1Fxt∗1}/2 = −(1(−1)−(−i)i)/2 = 1 (24)

The spin density in the reflected wave, Υ∗xyt2, is naturally the same

<Υ∗xyt2> = −R{V¯¯¯x∗2Fyt∗2−V¯¯¯y∗2Fxt∗2}/2 = −{1(−1)−(−i)i)}/2 = 1 (25)

However, the spin density in the real field Vk∗ = Vk∗1+Vk∗2, Fkl∗ = Fkl∗1+Fkl∗2, calculated by formula (19) contains a nonphysical oscillating term similar to (14), but with the opposite sign

−R{V¯¯¯x∗1Fyt∗2−V¯¯¯y∗1Fxt∗2+V¯¯¯x∗2Fyt∗1−V¯¯¯y∗2Fxt∗1}/2 = −{[1(−1)−(−i)i]e−2iz+[1(−1)−(−i)i]e2iz}/2 = (26) = 2cos2z

So <Υ∗xyt> = 2+2cos2z (27)

Since the fields of both chains are equally present in electromagnetic radiation, it is natural to use the half-sum of the canonical and modified tensors as the spin tensor

Υλμν = (Υcλμν+Υ∗λμν)/2 (28)

In the considered case of a standing wave, such a generalized spin tensor gives the correct result

Υλμν = 2 (29)

A similar result was obtained earlier using the generalized potential spin tensor [28].

The successful use of the generalization of the canonical spin tensor presented here confirms the presence of spin in a plane circularly polarized wave.

I am grateful to Professor Robert Romer for the brave publication [29].