9 Symmetry Approach to Chiral Optomagnonics in Antiferromagnetic Insulators
211
E → E
= E cos θ + B sin θ,
(9.6)
B → B
= −E sin θ + B cos θ,
(9.7)
where θ is a real parameter of the transformation. This symmetry is usually broken
inside materials, unless we deal with a dual symmetric medium [49].
The existence of duality symmetry guarantees the conservation of optical helicity,
i. e. the projection of spin angular momentum of the photon onto its linear momentum
[16, 17, 23, 24]. It should be mentioned, however, that the formulation of helicity
conservation law in classical electrodynamics is not straightforward, because the
standard Lagrangian for the electromagnetic field is not dual symmetric [48]. Using
the dual symmetric representation for the electromagnetic Lagrangian combined with
the Noether’s approach, it is possible to express the optical helicity density in the
form similar to (9.1)
ρ hel (t, r) =
1
2
[ A · (∇ × A) + C · (∇ × C)] ,
(9.8)
where in addition to the magnetic vector potential A, we also introduced the electric
vector potential C, which satisfies the following equations, E = −∇ × C = −∂ t A
and B = ∇ × A = ∂ t C. These are invariant under the transformations in (9.6) and
(9.7) [47, 48].
The definition of electromagnetic helicity depends on a specific representation of
the Lagrangian. It suggests that it would be useful to have a general formalism for
deriving “hidden” symmetries and conservation laws directly from the equations of
motion formulated exclusively in terms of the electromagnetic fields, and independent of any gauge choice. Such a formalism has been developed by Fushchich and
Nikitin [18]. Below, we give a brief review of this formalism, which is necessary for
further discussions.
9.2.1 Symmetry Analysis of the Maxwell’s Equations
For the symmetry analysis, it is convenient to formulate Maxwell’s equations in
the form that resembles the Dirac equation for a massless relativistic particle. This
representation is called the Silberstein-Bateman form [18]. In this form, the first
pair of the Maxwell’s equations in (9.4) is rewritten in terms of a Schrödinger-like
equation for the six-component vector column composed of the components of the
electric and magnetic fields φ = (E, B)
T
i
∂φ(t, p)
∂t
= H( p)φ(t, p),
(9.9)
where for convenience, we work in the momentum space, p, defined by the following
Fourier transformations
211
E → E
= E cos θ + B sin θ,
(9.6)
B → B
= −E sin θ + B cos θ,
(9.7)
where θ is a real parameter of the transformation. This symmetry is usually broken
inside materials, unless we deal with a dual symmetric medium [49].
The existence of duality symmetry guarantees the conservation of optical helicity,
i. e. the projection of spin angular momentum of the photon onto its linear momentum
[16, 17, 23, 24]. It should be mentioned, however, that the formulation of helicity
conservation law in classical electrodynamics is not straightforward, because the
standard Lagrangian for the electromagnetic field is not dual symmetric [48]. Using
the dual symmetric representation for the electromagnetic Lagrangian combined with
the Noether’s approach, it is possible to express the optical helicity density in the
form similar to (9.1)
ρ hel (t, r) =
1
2
[ A · (∇ × A) + C · (∇ × C)] ,
(9.8)
where in addition to the magnetic vector potential A, we also introduced the electric
vector potential C, which satisfies the following equations, E = −∇ × C = −∂ t A
and B = ∇ × A = ∂ t C. These are invariant under the transformations in (9.6) and
(9.7) [47, 48].
The definition of electromagnetic helicity depends on a specific representation of
the Lagrangian. It suggests that it would be useful to have a general formalism for
deriving “hidden” symmetries and conservation laws directly from the equations of
motion formulated exclusively in terms of the electromagnetic fields, and independent of any gauge choice. Such a formalism has been developed by Fushchich and
Nikitin [18]. Below, we give a brief review of this formalism, which is necessary for
further discussions.
9.2.1 Symmetry Analysis of the Maxwell’s Equations
For the symmetry analysis, it is convenient to formulate Maxwell’s equations in
the form that resembles the Dirac equation for a massless relativistic particle. This
representation is called the Silberstein-Bateman form [18]. In this form, the first
pair of the Maxwell’s equations in (9.4) is rewritten in terms of a Schrödinger-like
equation for the six-component vector column composed of the components of the
electric and magnetic fields φ = (E, B)
T
i
∂φ(t, p)
∂t
= H( p)φ(t, p),
(9.9)
where for convenience, we work in the momentum space, p, defined by the following
Fourier transformations