210
I. Proskurin and R. L. Stamps
9.2 Optical Chirality and Nongeometric Symmetries of the
Maxwell’s Equations
Since the early developments of electrodynamics, it has been well established that the
electromagnetic field in vacuum can be characterized by conserving energy, momentum, angular momentum, which reflects the invariance of the Maxwell’s equations
with respect to the translations and rotations in the four-dimensional space-time [18].
It was found almost by chance [15] that in addition to these conservation laws, the
electromagnetic field has another invariant given by a combination of the electric,
E, and magnetic, B, fields
ρ χ (t, r) =
ε 0
2
E · (∇ × E) +
1
2μ 0
B · (∇ × B),
(9.1)
which is odd under the spatial inversion and even under the time reversal transformations (ε 0 and μ 0 are the vacuum permittivity and permeability respectively). For this
quantity, Lipkin coined a special term—optical zilch to emphasize the lack of a clear
physical interpretation at that time [15]. According to its symmetry properties, ρ χ is
truly chiral [10], and can be considered as a chirality density of the electromagnetic
field.
Using the Maxwell’s equations, it is straightforward to demonstrate that in vacuum
ρ χ satisfies the continuity equation
∂ρ χ
∂t
+ ∇ · J χ = 0,
(9.2)
where
J χ (t, r) =
ε 0
2
E ×
∂ E
∂t
+
1
2μ 0
B ×
∂ B
∂t
,
(9.3)
determines the corresponding zilch flow.
In this section, we will show that this conservation law belongs to the class of
so-called “hidden” or nongeometric symmetries of the Maxwell’s equations. One of
these symmetries, which has been known since the time of Heaviside, Larmor, and
Rainich, is the duality symmetry [47, 48]. If we consider Maxwell’s equations in
free space
∇ × E = 0,
∇ × B = 0,
(9.4)
∇ · E = 0,
∇ · B = 0,
(9.5)
(we set c = 1 throughout this section) the electromagnetic duality is a symmetry
with respect to the rotation in the pseudo-space of the electric and magnetic fields,
which leaves Maxwell’s equations invariant
I. Proskurin and R. L. Stamps
9.2 Optical Chirality and Nongeometric Symmetries of the
Maxwell’s Equations
Since the early developments of electrodynamics, it has been well established that the
electromagnetic field in vacuum can be characterized by conserving energy, momentum, angular momentum, which reflects the invariance of the Maxwell’s equations
with respect to the translations and rotations in the four-dimensional space-time [18].
It was found almost by chance [15] that in addition to these conservation laws, the
electromagnetic field has another invariant given by a combination of the electric,
E, and magnetic, B, fields
ρ χ (t, r) =
ε 0
2
E · (∇ × E) +
1
2μ 0
B · (∇ × B),
(9.1)
which is odd under the spatial inversion and even under the time reversal transformations (ε 0 and μ 0 are the vacuum permittivity and permeability respectively). For this
quantity, Lipkin coined a special term—optical zilch to emphasize the lack of a clear
physical interpretation at that time [15]. According to its symmetry properties, ρ χ is
truly chiral [10], and can be considered as a chirality density of the electromagnetic
field.
Using the Maxwell’s equations, it is straightforward to demonstrate that in vacuum
ρ χ satisfies the continuity equation
∂ρ χ
∂t
+ ∇ · J χ = 0,
(9.2)
where
J χ (t, r) =
ε 0
2
E ×
∂ E
∂t
+
1
2μ 0
B ×
∂ B
∂t
,
(9.3)
determines the corresponding zilch flow.
In this section, we will show that this conservation law belongs to the class of
so-called “hidden” or nongeometric symmetries of the Maxwell’s equations. One of
these symmetries, which has been known since the time of Heaviside, Larmor, and
Rainich, is the duality symmetry [47, 48]. If we consider Maxwell’s equations in
free space
∇ × E = 0,
∇ × B = 0,
(9.4)
∇ · E = 0,
∇ · B = 0,
(9.5)
(we set c = 1 throughout this section) the electromagnetic duality is a symmetry
with respect to the rotation in the pseudo-space of the electric and magnetic fields,
which leaves Maxwell’s equations invariant