Abstract
The parity violation in nuclear reactions led to the discovery of the new class of toroidal multipoles. Since then, it was observed that toroidal multipoles are present in the electromagnetic structure of systems at all scales, from elementary particles, to solid state systems and metamaterials. The toroidal dipole T (the lowest order multipole) is the most common. This corresponds to the toroidal dipole operator
T
^
in quantum systems, with the projections
T
^
i
(i = 1, 2, 3) on the coordinate axes. These operators are observables if they are self-adjoint, but, although it is commonly discussed of toroidal dipoles of both, classical and quantum systems, up to now no system has been identified in which the operators are self-adjoint. Therefore, in this paper we use what are called the “natural coordinates” of the
T
^
3
operator to give a general procedure to construct operators that commute with
T
^
3
. Using this method, we introduce the operators
p
^
(
k
)
,
p
^
(
k
1
)
, and
p
^
(
k
2
)
, which, together with
T
^
3
and
L
^
3
, form sets of commuting operators:
(
p
^
(
k
)
,
T
^
3
,
L
^
3
)
and
(
p
^
(
k
1
)
,
p
^
(
k
2
)
,
T
^
3
)
. All these theoretical considerations open up the possibility to design metamaterials that could exploit the quantization and the general quantum properties of the toroidal dipoles.
Reducing subspaces of weighted shift operators