Abstract:
We compare three attempts that have been made to decompose the angular momentum of the electromagnetic field into components of an “orbital” and “spin” nature. All three expressions are different, and there seems to be no reason to prefer one to another. It appears, on the basis of classical electrodynamics, that there is no unique way of decomposing the angular momentum of the electromagnetic field into orbital and spin components, even in a fixed inertial frame. 1. Introduction The total angular momentum of the electromagnetic field is given (in SI units) [1] by Henceforth, we will suppress the time coordinate of the fields, all of which depend on time, and also the factor. There has been debate for a long time over whether the total angular moment of the electromagnetic field can be decomposed into an orbital part and a spin part so that Some authors [2] argue that on the basis of the first principles it is not possible to do this; others [3–5] show that forms can be demonstrated that appear to be, at least algebraically, of a spin and orbital nature. By means of partial integration Ohanian [4] effected a decomposition with , where is the vector potential and is the gradient operator that operates on functions of . Ohanian assumed that the electric charge density was zero and deemed (3) and (4) to be the spin and orbital components, respectively, of the electromagnetic field on the basis that the integrand of (3) was not explicitly linear in the coordinate whereas the integrand of (4) was. When the charge density is not zero, a bound term , considered also to be of an orbital nature is obtained on whose form all writers agree [6]. Although the sum of (3) and (4) and (5) is gauge invariant, the individual terms are not and so have no physical interpretation until the gauge of the vector potential is fixed completely. Cohen-Tannoudji et al. [3] used the Coulomb (or transverse) gauge, defined by the gauge condition , which gives The bound component remains the same as (8). It will be shown in Section 2 that the terms that involve the scalar potential in (6) and (7) cancel so that in (6), (7), and (8) and in (9), (10), and (8) but in (6) differs from in (9) and in (7) differs from in (10). The forms of (9) and (10) have also been used by van Enk and Nienhuis [5]. The general explicit form for , given in (13), was not specified by these writers. On the other hand, Stewart [7] found a decomposition from decomposing the electric field by the Helmholtz theorem [8]: This decomposition uses the and fields throughout so no issues of gauge arbitrariness arise.

Abstract:
The angular momentum of the physical electron, modelled as a Dirac fermion coupled to the electromagnetic field, is found to be hbar/2, the same as that of a bare Dirac fermion and independent of the size of the electric charge.

Abstract:
A proof is given of the vector identity proposed by Gubarev, Stodolsky and Zakarov that relates the volume integral of the square of a 3-vector field to non-local integrals of the curl and divergence of the field. The identity is applied to the case of the magnetic vector potential and magnetic field of a rotating charged shell. The latter provides a straightforward exercise in the use of the addition theorem of spherical harmonics.

Abstract:
The instantaneous nature of the potentials of the Coulomb gauge is clarified and a concise derivation is given of the vector potential of the Coulomb gauge expressed in terms of the instantaneous magnetic field.

Abstract:
A unified account, from a pedagogical perspective, is given of the longitudinal and transverse projective delta functions proposed by Belinfante and of their relation to the Helmholtz theorem for the decomposition of a three-vector field into its longitudinal and transverse components. It is argued that the results are applicable to fields that are time-dependent as well as fields that are time-independent.

Abstract:
The derivation of the Helmholtz theorem of vector decomposition of a 3-vector field requires that the field satisfy certain convergence properties at spatial infinity. This paper investigates if time-dependent electromagnetic radiation wave fields, which are of long range, satisfy these requirements. It is found that the requirements are satisfied because the fields give rise to integrals over the radial distance r of integrands of the form sin(kr)/r and cos(kr)/r. These Dirichlet integrals converge at infinity as required.

Abstract:
By means of the Helmholtz theorem on the decomposition of vector fields, the angular momentum of the classical electromagnetic field is decomposed, in a general and manifestly gauge invariant manner, into a spin component and an orbital component. The method is applied to linearly and circularly polarized plane waves in their classical and quantum forms.

Abstract:
The angular momentum of a classical electromagnetic plane wave of arbitrary extent is predicted to be, on theoretical grounds, exactly zero. However, finite sections of circularly polarized plane waves are found experimentally to carry angular momentum and it is known that the contribution to the angular momentum arises from the edges of the beam. A mathematical model is described that gives a quantitative account of this effect and resolves the paradox.

Abstract:
It is shown that the mathematical form, obtained in a recent paper, for the angular momentum of the electromagnetic field in the vicinity of electric charge is equivalent to another form obtained previously by Cohen-Tannoudji, Dupont-Roc and Gilbert. In this version of the paper an improved derivation is given.

Abstract:
Foreman and Lomer proposed in 1957 a method of estimating the harmonic forces between parallel planes of atoms of primitive cubic crystals by Fourier transforming the squared frequencies of phonons propagating along principal directions. A generalized form of this theorem is derived in this paper and it is shown that it is more appropriate to apply the method to certain combinations of the phonon dispersion relations rather than to individual dispersion relations themselves. Further, it is also shown how the method may be extended to the non-primitive hexagonal close packed and diamond lattices. Explicit, exact and general relations in terms of atomic force constants are found for deviations from the Blackman sum rule which itself is shown to be derived from the generalized Foreman-Lomer theorem.