In this article it is recognized that Chirality is to be associated with
enantiomorphic pairs which induce Optical Activity, while Helicity is to be
associated enantiomorphic pairs which induce a Faraday effect. Experimentally,
the existence of enantiomorphic pairs, in a crystalline sense, requires the lack
of a center of symmetry, which is also a necessary condition for Optical
Activity. However, Faraday effects may or may not require a lack of a center of
symmetry. The two species of enantiomorphic pairs are distinct as the rotation
of the plane of polarization by Optical Activity is a reciprocal phenomenon,
while rotation of the plane of polarization by the Faraday effect is a
non-reciprocal phenomenon.
From a topological viewpoint, Maxwell's electrodynamics indicates that the
concept of Chirality is to be associated with a third rank tensor density of
Topological Spin induced by the interaction of the 4 vector potentials {A, phi }
and the field excitations (D, H). The distinct concept of Helicity is to be
associated with the third rank tensor field of Topological Torsion induced by
the interaction of the 4 vector potentials and field intensities (E , B). The
4-divergence of the topological Spin is equal to the first Poincare Invariant of
classical electromagnetism. The 4-divergence of the topological Torsion is
equal to the second Poincare Invariant. On space-time domains where either
Poincare Invariant is zero, the closed integrals of the Spin (respectively,
Torsion) have values whose ratios are rational. That is, both the objects of
topological torsion and topological spin can be quantized by the deRham
theorems. However, when the 4-divergence of Topological Torsion is not
zero, the second Poincare invariant, E · B, is not zero, and parity is not
conserved.
Specific media are defined by equivalence classes of constitutive constraints
between the field intensities and field excitations. It is known that an
anisotropic birefringence constitutive constraint has most of the properties of
the curvature tensor generated by a torsion-free Riemannian metric space.
However, it is impossible to use a real metric to generate Faraday rotation or
Optical Activity. Herein the Riemannian concept is broadened to include
topological and affine torsion by computing the curvature tensor generated from
the Cartan Connection of a Frame Matrix (not a symmetric metric matrix). The
Frame Matrix (Repere Mobile) defines equivalence classes of differential
linearity of manifold neighborhoods. The non-symmetric Frame Matrix admits
both a left handed and a right handed realization. The topological analysis
indicates that Optical Activity is related to the D field (and possible
translational accelerations), while Faraday rotation is related to B field (and
possible rotational accelerations). These facts lead to the possibility of
finding new methods of enantiomorphic separation.
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