## Abstract

We review and re-examine the description and separation of the spin and orbital angular momenta (AM) of an electromagnetic field in free space. While the spin and orbital AM of light are not separately meaningful physical quantities in orthodox quantum mechanics or classical field theory, these quantities are routinely measured and used for applications in optics. A meaningful quantum description of the spin and orbital AM of light was recently provided by several authors, which describes separately conserved and measurable integral values of these quantities. However, the electromagnetic field theory still lacks corresponding locally conserved spin and orbital AM currents. In this paper, we construct these missing spin and orbital AM densities and fluxes that satisfy the proper continuity equations. We show that these are physically measurable and conserved quantities. These are, however, not Lorentz-covariant, so only make sense in the single laboratory reference frame of the measurement probe. The fluxes we derive improve the canonical (nonconserved) spin and orbital AM fluxes, and include a ‘spin–orbit’ term that describes the spin–orbit interaction effects observed in nonparaxial optical fields. We also consider both standard and dual-symmetric versions of the electromagnetic field theory. Applying the general theory to nonparaxial optical vortex beams validates our results and allows us to discriminate between earlier approaches to the problem. Our treatment yields the complete and consistent description of the spin and orbital AM of free Maxwell fields in both quantum-mechanical and field-theory approaches.

## Highlights

It is known that light can carry both spin and orbital angular momentum (AM) [1]

To restore the fundamental dual symmetry present in free-space Maxwell equations, but broken in the standard field Lagrangian and canonical Noether conservation laws, we recently suggested a dual-symmetric version of electromagnetic field theory [12]

We argued that the separation of the spin and orbital parts of the AM of light makes sense based on operational local measurements of these quantities, e.g., via probe particles

## Summary

It is known that light (electromagnetic waves or photons) can carry both spin and orbital angular momentum (AM) [1]. The integral values of the spin and orbital AM, ∫ S dV and ∫ L dV (volume integrals for sufficiently localized fields are assumed), are conserved, i.e., time-independent in free space [13] This hints that the electromagnetic spin and orbital AM are separate physically meaningful quantities, and that the fundamental problems with the quantum-mechanical and field-theory approaches can and should be overcome. Akin to the quantum-operator approach, we modify the separation of the spin and orbital parts of the canonical Noether AM current, Sαβγ = Sαβγ − Δαβγ and Lαβγ = Lαβγ + Δαβγ, such that the modified tensors Sαβγ and Lαβγ satisfy a continuity equation and properly describe the spin and orbital AM densities S and L We show that this separation produces a meaningful local description of the spin and orbital AM densities and fluxes, and represents them as gauge-invariant (and, observable) but not Lorentz-covariant quantities.

## Full Text

### Topics from this Paper

- Orbital Angular Momenta
- Spin Angular Momenta
- Orbital Angular Momenta Of Light
- Angular Momenta
- Nonparaxial Optical Fields + Show 5 more

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