Chirality and Helicity vs Topological Spin and Topological Torsion

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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