In physics, zilch (or zilches) is a set of ten conserved quantities of the source-free electromagnetic field, which were discovered by Daniel M. Lipkin in 1964. The name refers to the fact that the zilches are conserved only in regions free of electric charge, and therefore have limited physical significance. One of the conserved quantities (Lipkin's
Z
0
{\displaystyle Z^{0}}
) has an intuitive physical interpretation and is also known as optical chirality.
Contents
Optical chirality
Lipkin observed that if he defined the quantities
Z
0
=
E
⋅
∇
×
E
+
B
⋅
∇
×
B
Z
=
1
c
(
E
×
d
d
t
E
+
B
×
d
d
t
B
)
{\displaystyle {\begin{aligned}Z^{0}&=\mathbf {E} \cdot \nabla \times \mathbf {E} +\mathbf {B} \cdot \nabla \times \mathbf {B} \\\mathbf {Z} &={\frac {1}{c}}\left(\mathbf {E} \times {\frac {d}{dt}}\mathbf {E} +\mathbf {B} \times {\frac {d}{dt}}\mathbf {B} \right)\end{aligned}}}
then the free Maxwell equations imply that
∂
0
Z
0
+
∇
⋅
Z
=
0
{\displaystyle \partial _{0}Z^{0}+\nabla \cdot \mathbf {Z} =0}
.
The precedent equation implies that the quantity
∫
Z
0
d
3
x
{\displaystyle \int Z^{0}\,d^{3}x}
is constant. This time-independent quantity is one of the ten zilches Lipkin discovered. Nowadays, the quantity
∫
Z
0
d
3
x
{\displaystyle \int Z^{0}\,d^{3}x}
is widely known as optical chirality (up to a factor of 1/2).
The quantity
Z
0
{\displaystyle {Z}^{0}}
is the spatial density of optical chirality, while
Z
{\displaystyle \mathbf {Z} }
is the optical chirality flux. Generalizing the aforementioned differential conservation law for
Z
0
{\displaystyle Z^{0}}
, Lipkin found another nine conservation laws, all unrelated to the stress–energy tensor. He collectively named these ten conserved quantities the zilch (nowadays, they are also called the zilches) because of their apparent lack of physical significance.
Properties of zilch tensor
The zilch is often described in terms of the zilch tensor,
Z
ν
ρ
μ
{\displaystyle Z_{\nu \rho }^{\mu }}
. The latter can be expressed using the dual electromagnetic tensor
F
^
μ
ν
=
(
1
/
2
)
ϵ
μ
ν
ρ
σ
F
ρ
σ
{\displaystyle {\hat {F}}^{\mu \nu }=(1/2)\epsilon ^{\mu \nu \rho \sigma }F_{\rho \sigma }}
as
Z
ν
ρ
μ
=
History
One of the zilches has been rediscovered. This is the zilch called "optical chirality", so named by Tang and Cohen since it determines the degree of chiral asymmetry in the rate of excitation of a small chiral molecule by an incident electromagnetic field. A further physical insight of optical chirality was offered in 2012; optical chirality is to the curl or time derivative of the electromagnetic field, what helicity, spin and related quantities are to the electromagnetic field itself. The physical interpretation of all zilches for topologically nontrivial electromagnetic fields was investigated in 2018.
The discovery of the ten zilches in 1964 raised an important mathematical question about their relation with symmetries. Recently, the full answer to this question seems to have been found. This is the question:
What are the symmetries of the standard Maxwell action functional:
S
[
A
μ
]
=
−
1
4
∫
d
4
x
F
μ
ν
F
μ
ν
{\displaystyle S[A_{\mu }]=-{\frac {1}{4}}\int d^{4}xF_{\mu \nu }F^{\mu \nu }}