Parallel transport

For the Berry phase, we shall quote a definition given in [164] ""The phase that can be acquired by a state moving adiabatically (slowly) around a closed path in the parameter space of the system. There is a further, somewhat more general phase, that appears in any cyclic motion, not necessarily slow in the Hilbert space, which is the Aharonov-Anandan phase [10]. Other developments and applications are abundant. An interim summai was published in 1990 [78]. A further, more up-to-date summary, especially on progress in experimental developments, is much needed. (In Section IV we list some publications that report on the experimental determinations of the Berry phase.) Regarding theoretical advances, we note (in a somewhat subjective and selective mode) some clarifications regarding parallel transport, e.g., [165], This paper discusses the projective Hilbert space and its metric (the Fubini-Study metric). The projective Hilbert space arises from the Hilbert space of the electronic manifold by the removal of the overall phase and is therefore a central geometrical concept in any treatment of the component phases, such as this chapter. 

The relative contribution of each driving force (X) generated by component j to the flux of solute i (./,) is expressed by coefficients Li in this phenomenological description of parallel transport processes. 

In vitro studies permit further isolation of parallel transport processes and can provide a reduction in experimental variability. Rate of absorption assessment can be measured as intestinal uptake or flux across an intestinal barrier at both the tissue and cell monolayer levels. Experimental variability is also reduced by the fact that a large number of tissue samples can be used from the same experi-... 

Fig. 7.1. The intestinal permeability of drugs in vivo is the total transport parameter that may be affected by several parallel transport mechanisms in both absorptive and secretory directions. Some of the most important transport proteins that may be involved in the intestinal transport of drugs and their metabolites across intestinal epithelial membrane barriers in humans are displayed.

Parallel transported eigenstates, geometric phase theory, 10-11... 

D. K. Maude and J. C. Portal, Parallel Transport in Low-Dimensional Semiconductor Structures... 

All gauge theory depends on the rotation of an -component vector whose 4-derivative does not transform covariantly as shown in Eq. (18). The reason is that i(x) and i(x + dx) are measured in different coordinate systems the field t has different values at different points, but /(x) and /(x) + d f are measured with respect to different coordinate axes. The quantity d i carries information about the nature of the field / itself, but also about the rotation of the axes in the internal gauge space on moving from x + dx. This leads to the concept of parallel transport in the internal gauge space and the resulting vector [6] is denoted i(x) + d i. The notion of parallel transport is at the root of all gauge theory and implies the introduction of g, defined by... 

In turn, in geometry A plays the role of connection (it defines the parallel transport around C) and F is the curvature of this connection. A global version of the Abelian Stokes theorem... 

The holonomy represents a parallel-transport operator around C assuming values in a non-Abelian Lie group G. (Interestingly, in the Abelian case, the holonomy has a physical role it is an object playing the role of the phase that can be observed in the Aharonov-Bohm experiment, whereas ,- itself does not have such an interpretation.)... 

Figure 2. Parallel-transport operator along the path L in the surface S.

As a kind of a short introduction for properly manipulating parallel-transport operators along oriented curves, we recall a number of standard facts. It is obvious that we can perform some operations on the parallel-transport operators. We can superimpose them, we can introduce an identity element, and finally, we can find an inverse element for each element. 

We will determine a coherent-state/holomorphic path integral representation for the parallel-transport operator, deriving an appropriate transition amplitude [a path integral counterpart of the l.h.s. in Eq. (14)]. 

For a closed curve C we will calculate the trace of the path integral form of the parallel-transport operator in quantum theory in an external gauge field A. 

First, let us derive the path integral expression for the parallel-transport operator U along L. To this end, we should consider the non-Abelian formula (differential equation) analogous to Eq. (23)... 

Using the property of Unear independence of Fock space vectors in Eq. (31), and comparing Eqs. (31) and (24), we can see that Eq. (30) really represents the matrix elements of the parallel-transport operator. For closed paths, x(f ) = x(t") = x, Eq. (30) gives the holonomy operator Uki(x) and Ukk is the Wilson loop. Interestingly, the Wilson loop, which is supposed to describe a quark-antiquark interaction, is represented by a true quark and antiquark field, z and z, respectively. So, the mathematical trick can be interpreted physically. ... 

Obviously, the full trace of the kernel in Eq. 3.9 is obtained by imposing appropriate boundary conditions, and integrating with respect to all the variables without the boundary term. Analogously, one can also derive the parallel-transport operator (a generalization of the one just considered) for symmetric tensors (bosonic -particle states) and for forms (fermionic -particle states). 

Electron-cation symport has been realized in a double carrier process where the coupled, parallel transport of electrons and metal cations was mediated simultaneously by an electron carrier and by a selective cation carrier [6.47]. The transport of electrons by a nickel complex in a redox gradient was the electron pump for driving the selective transport of K+ ions by a macrocyclic polyether (Fig. 12). The pro-... 

In this appendix, the U(l) invariant theory of the Aharonov-Bohm effect [46] is shown to be self-inconsistent. The theory is usually described in terms of a holonomy consisting of parallel transport around a closed loop assuming values in the Abelian Lie group U(l) [50] conventionally ascribed to electromagnetism. In this appendix, the U(l) invariant theory of the Aharonov-Bohm effect is... 

In the 0(3) invariant theory of the Aharonov-Bohm effect, the holonomy consists of parallel transport using 0(3) covariant derivatives and the internal gauge space is a physical space of three dimensions represented in the basis ((1),(2),(3)). Therefore, a rotation in the internal gauge space is a physical rotation, and causes a gauge transformation. The core of the 0(3) invariant explanation of the Aharonov-Bohm effect is that the Jacobi identity of covariant derivatives [46]... 

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