Abelian

In the algebraic, group-theoretical treatments of non-Abelian systems [66,67-70,77-80] the NACT is usually written in a decomposed form as... 

In an Abelian theory [for which I (r, R) in Eq. (90) is a scalar rather than a vector function, Al=l], the introduction of a gauge field g(R) means premultiplication of the wave function x(R) by exp(igR), where g(R) is a scalar. This allows the definition of a gauge -vector potential, in natural units... 

In a non-Abelian theory (where the Hamiltonian contains noncommuting matrices and the solutions are vector or spinor functions, with N in Eq. (90) >1) we also start with a vector potential Af, [In the manner of Eq. (94), this can be decomposed into components A, in which the superscript labels the matrices in the theory). Next, we define the field intensity tensor through a covaiiant curl by... 

In conclusion, we have shown that the non-Abelian gauge-field intensity tensor fi sc(X) shown in Eq. (113) vanishes when... 

Another subject with important potential application is discussed in Section XIV. There we suggested employing the curl equations (which any Bohr-Oppenheimer-Huang system has to obey for the for the relevant sub-Hilbert space), instead of ab initio calculations, to derive the non-adiabatic coupling terms [113,114]. Whereas these equations yield an analytic solution for any two-state system (the abelian case) they become much more elaborate due to the nonlinear terms that are unavoidable for any realistic system that contains more than two states (the non-abelian case). The solution of these equations is subject to boundary conditions that can be supplied either by ab initio calculations or perturbation theory. 

The symmetry groups for the chiral tubules are Abelian groups. The corresponding space groups are non-symmorphic and the basic symmetry operations... 

R = (i/ r) require translations t in addition to rotations j/. The irreducible representations for all Abelian groups have a phase factor c, consistent with the requirement that all h symmetry elements of the symmetry group commute. These symmetry elements of the Abelian group are obtained by multiplication of the symmetry element./ = (i/ lr) by itself an appropriate number of times, since R = E, where E is the identity element, and h is the number of elements in the Abelian group. We note that N, the number of hexagons in the ID unit cell of the nanotube, is not always equal h, particularly when d 1 and dfi d. 

If the group J7 is abelian, and the point group corresponding to the group G can be reduced to H by the application of an external magnetic... 

Abelian T., Chinchilla R., Galindo N., Guillena G., Najera C., Sansano J. M. Glycine and Alanine Imines As Templates for Asymmetric Synthesis of a-Amino Acids Eur. J. Org. Chem. 2000 2689-2697... 

D. L. Armacost, The Structure of Locally Compact Abelian Groups (1981)... 

A non-abelian point-group contains irreducible representations of dimension larger than one. Since the degree of degeneracy caused by spatial symmetry equals the dimensionality of the corresponding irreducible... 

It is not too difficult to develop the multiplication table shown as Table 3 (problem 1). It will be noticed immediately that the table is not symmetric with respect to the principal diagonal Therefore, the group is not Abelian and multiplication is not commutative. 

The set of translations jT forms a group known asthe translation group, an Abelian subgroup of J. The space group can thus be written as... 

Hamiltonian equations, 627-628 perturbative handling, 641-646 II electronic states, 631-633 vibronic coupling, 630-631 ABC bond angle, Renner-Teller effect, triatomic molecules, 611-615 ABCD bond angle, Renner-Teller effect, tetraatomic molecules, 626-628 perturbative handling, 641-646 II electronic states, 634-640 vibronic coupling, 630-631 Abelian theory, molecular systems, Yang-Mills fields ... 

Neumann boundary conditions, electronic states, adiabatic-to-diabatic transformation, two-state system, 304-309 Newton-Raphson equation, conical intersection location locations, 565 orthogonal coordinates, 567 Non-Abelian theory, molecular systems, Yang-Mills fields nuclear Lagrangean, 250 pure vs. tensorial gauge fields, 250-253 Non-adiabatic coupling ... 

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