A scalar product for computing fundamental quantities in matter and its application to the helicity and angular momentum stored in a Hopfion

We introduce a systematic way to obtain expressions for computing the amount of fundamental quantities such as angular momentum contained in static matter, given its charge and magnetization densities. The expressions are obtained from a scalar product whose form results from imposing invariance under the scale-Euclidian group of transformations. Such group is obtained by restricting the conformal group of invariance of the dynamic Maxwell fields to the static case. In an exemplary application, we compute the helicity and angular momentum squared stored in a Hopfion, and show that the Hopfion is an eigenstate of angular momentum along one axis with eigenvalue zero

The polychromatic T-matrix

The T-matrix is a powerful tool that provides the complete description of the linear interaction between the electromagnetic field and a given object. In here, we generalize the usual monochromatic formalism to the case of polychromatic field-matter interaction. The group of transformations of special relativity provides the guidance for building the new formalism, which is inherently polychromatic. The polychromatic T-matrix affords the direct treatment of the interaction of electromagnetic pulses with objects, even when the objects move at constant relativistic speeds

Multidimensional measures of electromagnetic chirality and their conformal invariance

Proper assignment of left- and right-handed labels to general chiral objects is known to be a theoretically unfeasible problem. Attempts to utilize a pseudoscalar function to distinguish enantiomers face two unavoidable difficulties: false chiral zeros and unhanded chiral states. In here, we demonstrate how both of these problems can be solved in the context of light-matter interactions. First, we introduce a two-dimensional quantity called complex electromagnetic chirality that solves the problem of false chiral zeros. Next, we define an infinite-dimensional pseudovector called chirality signature that completely quantifies the multidimentional nature of electromagnetic chirality, does not have false global chiral zeros, and allows to continuously distinguish any pair of enantiomers. We prove that the introduced measures are invariant under the largest group of symmetries of Maxwell’s equations – the conformal group. The complete and conformally invariant quantification of electromagnetic chirality provided by the chirality signature distinguishes it as a particularly suitable tool for the study of chirality and its applications

The Polychromatic T-matrix

The T-matrix is a powerful tool that provides the complete description of the linear interaction between the electromagnetic field and a given object. In here, we generalize the usual monochromatic formalism to the case of polychromatic field-matter interaction. The group of transformations of special relativity provides the guidance for building the new formalism, which is inherently polychromatic. The polychromatic T-matrix affords the direct treatment of the interaction of electromagnetic pulses with objects, even when the objects move at constant relativistic speeds

A scalar product for computing fundamental quantities in matter and its application to the helicity and angular momentum stored in a Hopfion

We introduce a systematic way to obtain expressions for computing the amount of fundamental quantities such as angular momentum contained in static matter, given its charge and magnetization densities. The expressions are obtained from a scalar product whose form results from imposing invariance under the scale-Euclidian group of transformations. Such group is obtained by restricting the conformal group of invariance of the dynamic Maxwell fields to the static case. In an exemplary application, we compute the helicity and angular momentum squared stored in a Hopfion, and show that the Hopfion is an eigenstate of angular momentum along one axis with eigenvalue zero