Topological charge

It was found that normal zero-point oscillations lie on top of large gluon fluctuations - instantons and anti-instantons with random positions and sizes. The left column - action density and the right column - topological charge density. Here instantons are peaks and anti-instantons are holes. 

Whereas unsubstituted pentalene has not been obtained so far, its -n-isoelectronic inorganic analog (72) where there occurs the topological charge stabilization has, on the contrary, been synthesized [73AG(E)576]. 

Topological charge stabilization explains the greater stability of indole (88), benzo[b]furan (89), and benzo[6]thiophene (90) compared to their positional isomers (91)-(93) when the largest 7r-charge densities at the 1,3-positions of the five-membered ring in the 10 7r-electron indenyl anion (94) are taken into account (77JA1692). Such an order of the relative stability is consistent with the RE values (Table X). 

The disappearance of the magnetic charge-current (585) means that the topological terms on the right-hand sides of Eqs. (95)-(100) vanish identically in the vacuum. The only topological charges and currents present are therefore those introduced by Lehnert [7-10]. There is no empirical evidence for the existence of an El3> field, so we are left with... 

We also infer that, in the vacuum, there exists the topological charge... 

There are therefore obvious points of similarity between the 0(3) theory of electrodynamics and the Yang-Mills theory [44], Both are based, as we have argued, on an 0(3) or SU(2) invariant Lagrangian. However, in 0(3) electrodynamics, the particle concomitant with the field has the topological charge k/A(0>. In 0(3) electrodynamics, the internal space and spacetime are not independent spaces but form an extended Lie algebra [42], In elementary particle... 

A three-way cross-check of the self-consistency of the 0(3) ansatz can be carried out starting from Eq. (459), in which A is complex because the electromagnetic field in 0(3) electrodynamics carries a topological charge k/,4i0 . The vector field A in the internal space of 0(3) symmetry must depend on by special relativity and can be written as... 

Here,, 4 ( is the vector 4-potential introduced in the vacuum as part of the covariant derivative, and therefore introduced by spacetime curvature. The electromagnetic field and the topological charge g are the results of the invariance of the Lagrangian (868) under local U(l) gauge transformation, in other words, the results of spacetime curvature. 

These are complex scalar fields because there is an invariant topological charge present, defined by... 

Kurihara etal. (90BCJ2531) discussed the stability and resonance energy of troponoids by application of the topological charge stabilization (TCS) rule. Many synthetic and natural troponoids prove to be energetically very stable molecules. The TSC rule can be applied to the formation mechanism of heterocyclotropones. Many benzazinotropone products are completely consistent with this rule, for example, tri- and higher-cyclic quinoxalotro-pones 27 and 166. 

Gimarc (83JA1979,86JA4303) proposed a rule of topological charge stabilization which states that heteroatoms prefer to be located at sites that conform to the pattern of relative electron densities determined by connectivity or topology in an isoelectronic, isostructural, homoatomic system that is called the uniform reference frame <83JA1979>. For the series of thienothiophene positional isomers, the pentalene dianion (5) serves as the uniform reference frame ((55) represents pentalene). 

Such controlling factors are matched by complementary properties of the solid surface, i.e., the hydrophobic or hydrophilic surface character, porosity and topology, charge, hydration, and the presence and composition of surface groups. Particularly for electrochemical surfaces intrinsic catalytic reactivity of the surface groups formed spontaneously or by po-tentiostatic control also follow. Both in the context of electrochemical protein reactivity, and in the broader areas of proteins at surfaces, surface control and modification to structural and functional compatibility with the proteins are key issues. The use of electrochemical promoters, illustrated below, is one such example [32-34]. 

Figure 5 Application of the topological charge stabilization rule to bicyclic heterocycles.

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