Double cover

Most extmded latex fibers are double covered with hard yams in order to overcome deficiencies of the bare threads such as abrasiveness, color, low power, and lack of dyeabiUty. During covering, the elastic thread is wrapped under stretch which prevents its return to original length when the stretch force is removed thus the fiber operates farther on the stress—strain curve to take advantage of its higher elastic power. Covered mbber fibers are commonly found in narrow fabrics, braids, surgical hosiery, and strip lace. 

Figure 2(b) represents the potential surface of the identical system, mapped onto the double-cover space [28], The latter is obtained simply by unwinding the encirclement angle < ), from 0 2ti to 0 4ti, such that two (internal) rotations around the Cl are represented as one in the page. The potential is therefore symmetric under the operation Rin defined as an internal rotation by 2n in the double space. To map back onto the single space, one cuts out a 271-wide sector from the double space. This is taken to be the 0 2ti sector in Fig. 2(b), but any 27i-wide sector would be acceptable. Which particular sector has been taken is represented by a cut line in the single space, so in Fig. 2(b) the cut line passes between < ) = 0 and 2n. Since the single space is the physical space, any observable obtained from the total (electronic + nuclear) wave function in this space must be independent of the position of the cut line. 

In other words, if we map a bound-state wave function onto the double-cover space using Eq. (6), we simply duplicate the function, because the contribution from the even n Eeynman paths is exactly equal to (or equal and opposite to) the contribution from the odd n paths. 

Cosmic structure based on a vacuum interface has been proposed before [49, 7] as a device to rationalize quantum events. To avoid partitioning the universe into regions of opposite chirality the two sides of the interface are joined together with an involution. The one-dimensional analogue is a Mobius strip. Matter on opposite sides of the interface has mutually inverted chirality - matter and anti-matter - but transplantation along the double cover gradually interconverts the two chiral forms. The amounts of matter and anti-matter in such a universe are equal, as required by symmetry, but only one form is observed to predominate in any local environment. Because of the curvature, which is required to close the universe, space itself is chiral, as observed in the structure of the electromagnetic field. This property does not appear in a euclidean Robertson-Walker sub-space. 

Figure 2.6 Period lines of figure 5 mapped onto the double cover of a Mobius strip...

Like numbers and their conjugates, matter and antimatter would merge naturally if the two conjugate surfaces constituted the double cover of a... 

Mobius strip. In this way the antimatter mystery disappears matter and antimatter are one and the same thing, which merely appear to be different depending on their position in the double cover. In more dimensions the Mobius model is replaced by a projective plane, obtained from an open hemisphere on identifying points on opposite sides of the circular edge. Topologically equivalent constructs are known as a Roman surface or a Klein bottle. 

Figure 7.7 Chirality is gradually inverted on transplantation along the surface of a Mobius strip. The same hand is shown on opposite sides of the double cover.

Figure 7.14 A regular Mobius strip with its single boundary curve. The absolute local curvature of the double cover is constant and the total curvature is zero.

Earlier speculations about the effect of the curvature of space on elemental synthesis and the stability of nuclides (2.4.1) are consistent with the interface model. The absolute curvature of the closed double cover of projective space, and the Hubble radius of the universe, together define the golden mean as a universal shape factor [233], characteristic of intergalactic space. This factor regulates the proton neutron ratio of stable nuclides and the detail of elemental periodicity. The self-similarity between material structures at different levels of size, such as elementary particles, atomic nuclei, chemical... 

III) m2,0 this space was studied classically by Bolza among others and in recent years was analyzed completely by Igusa, and was attacked as follows describe a curve C of genus 2 as a double cover of P1 ramified in 6 points Ai,..., A6. This sets up a bijection ... 

Weil investigated the deeper problem of classifying all cases where 0C)0a was reducible it appears that for most curves, this only happens if a = (x) — elliptic curves, there are other a s for which 0 fl 0a is reducible. 

The transformation from P to P is therefore not an identity transformation, but rather an involution, with P and F as conjugate points. The identity transformation corresponds to a double rotation of 27t along the Mobius surface. The two sides of the paper corresponds to a double covering of the non-orientable topological Mobius surface. 

Figure 5.4 signifies more than elemental or nuclide periodicity. It summarizes the appearance of ponderable matter in all modifications throughout the universe. Following the extended hemlines from top left at Z/N = 1.04 — bottom left at 0 —> top right at Z/N = 1.04 bottom right at 0, and back to top left, the involuted closed path, which is traced out, is mapped to the non-orientable surface of a Mobius band in Figure 5.7. The two sides of the double cover are interpreted to represent both matter and antimatter.

Analysis of the periodicity of atomic matter therefore guides us to a projective model of a closed imiverse in the double cover of four-dimensional projective space-time. Transport across the interface, or along the involution, results in the inversion of CPT symmetry. 

It is tempting to add that the second sheet represents Naan s anti-world, which lies on the opposite side of the projective double cover. 

In terms of this simple alternative geometry, geodesic transplantation fixes points on the line element to occur along the double cover of a narrow Mobius band. It is a known property of a Mobius band that a point, which moves along the double cover, close to one edge, rotates around the central line without intersecting it. This is what Godel describes as rotation with respect to a compass of inertia. 

Minkowski space, M, is assumed embedded in a more general universal closed (compact) space M, the so-called conformal space. This is the projective space proposed as a model of the universe by Oswald Veblen (1933), translated in the Appendix. Roughly speaking, M is obtained from M by adding a light cone at infinity. More precisely, it is the double cover of the space so generated. Segal (1976) refers to M as unispace and to the natural time r in this space as unitime. 

Although the Godel solution is free of singularities the need to accommodate black holes in the cosmic model requires an interpretation of the Schwarzschild singularity which occurs with infinite curvature of space-time. A new interpretation is rather obvious. Such a high degree of curvature must clearly rupture the interface between adjacent sides of the postulated cosmic double cover. Rather than disappear into a singularity, the matter,... 

The result is a universe that consists of exactly fifty percent antimatter, which however, can never be detected in convential observations - only when the interface is penetrated. Although matter and antimatter therefore occupy the same space, there is no possibihty of direct interaction as the two antipodes of the double cover are at different time coordinates. It is important to realize that transportation along the double cover, through the involution, gradually converts matter into antimatter. 

The graphical representation of the way in which chemical periodicity varies continuously as a function of the limiting ratio (Figure 5.3), 1 < Z/N < 0, appears strangely unsymmetrical, despite perfect symmetry at the extreme values. By adding an element of mirror symmetry a fully symmetrical closed function, that now represents matter and antimatter, is obtained. To avoid self overlap the graphical representation of the periodic function is transferred to the double cover of a Mobius band, which in closed form defines a projective plane. 

Olof Sunden, 5 November 1992 You say that vacuum is an interface between different states of matter (matter and antimatter), that the universe itself is the double cover of the manifold and that its involution serves to position conjugated forms of matter and antimatter on opposite sides of the interface. I talk about space and antispace. Out there on the other side time changes character and appears as a conjugated antispace and even as a condensed conjugate antiparticle. 

It seems that a better suited approach should make use of a series of actual measurements on an evolving material quantum system. In fact, two-level atoms, equivalent to spin systems, show a formal analogy with light polarisation. Whereas the configuration space of polarisation transforms according to symmetry group SO(3), the symmetry of spin transformation is SU(2), which is a double covering of SO(3), and locally isomorphic with the latter one [17]. Thus, similar visualisations of the dynamics of both systems apply. 

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