Gauge fields

Note that external and ag fields gauges not only the kinetic term of the QCD low-energy effective action but also its interaction term in Eq. (35). The reason is obvious It is the nonlocal... 

Sheng LM, Liu P. et al., A saddle-field gauge with carbon nanotube field emitters. Diamond and Related Materials, 2005. 14(10) 1695-1699. 

Reservoir pressure is measured in selected wells using either permanent or nonpermanent bottom hole pressure gauges or wireline tools in new wells (RFT, MDT, see Section 5.3.5) to determine the profile of the pressure depletion in the reservoir. The pressures indicate the continuity of the reservoir, and the connectivity of sand layers and are used in material balance calculations and in the reservoir simulation model to confirm the volume of the fluids in the reservoir and the natural influx of water from the aquifer. The following example shows an RFT pressure plot from a development well in a field which has been producing for some time. 

Pure versus Tensorial Gauge Fields The Curl Condition ... 

The quantum phase factor is the exponential of an imaginary quantity (i times the phase), which multiplies into a wave function. Historically, a natural extension of this was proposed in the fonn of a gauge transformation, which both multiplies into and admixes different components of a multicomponent wave function [103]. The resulting gauge theories have become an essential tool of quantum field theories and provide (as already noted in the discussion of the YM field) the modem rationale of basic forces between elementary particles [67-70]. It has already been noted that gauge theories have also made notable impact on molecular properties, especially under conditions that the electronic... 

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... 

Thus, the existence of a (matrix-type) phase g represents the pure-gauge case and the nonexistence of g represents the nonpure YM field case, which cannot be tiansformed away by a gauge. 

Now, we recall the remarkable result of [72] that if the adiabatic electronic set in Eq. (90) is complete (N = oo), then the curl condition is satisfied and the YM field is zero, except at points of singularity of the vector potential. (An algebraic proof can be found in Appendix 1 in [72]. An alternative derivation, as well as an extension, is given below.) Suppose now that we have a (pure) gauge g(R), that satisfies the following two conditions ... 

Then, two things (that are actually interdependent) happen (1) The field intensity F = 0, (2) There exists a unique gauge g(R) and, since F = 0, any apparent field in the Hamiltonian can be transformed away by introducing a new gauge. If, however, condition (1) does not hold, that is, the electronic Hilbert space is truncated, then F is in general not zero within the tmncated set. In this event, the fields A and F cannot be nullified by a new gauge and the resulting YM field is true and irremovable. 

The vanishing of the YM field intensity tensor can be shown to follow from the gauge transformation properties of the potential and the field. It is well known (e.g., Section II in [67]) that under a unitary transfoiination described by the matrix... 

Now the Lagrangean associated with the nuclear motion is not invariant under a local gauge transformation. Eor this to be the case, the Lagrangean needs to include also an interaction field. This field can be represented either as a vector field (actually a four-vector, familiar from electromagnetism), or as a tensorial, YM type field. Whatever the form of the field, there are always two parts to it. First, the field induced by the nuclear motion itself and second, an externally induced field, actually produced by some other particles E, R, which are not part of the original formalism. (At our convenience, we could include these and then these would be part of the extended coordinates r, R. The procedure would then result in the appearance of a potential interaction, but not having the field. ) At a first glance, the field (whether induced internally... 

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

However, this procedure depends on the existence of the matrix G(R) (or of any pure gauge) that predicates the expansion in Eq. (90) for a full electronic set. Operationally, this means the preselection of a full electionic set in Eq. (129). When the preselection is only to a partial, truncated electronic set, then the relaxation to the truncated nuclear set in Eq. (128) will not be complete. Instead, the now tmncated set in Eq. (128) will be subject to a YM force F. It is not our concern to fully describe the dynamics of the truncated set under a YM field, except to say (as we have already done above) that it is the expression of the residual interaction of the electronic system on the nuclear motion. 

As shown in Eq. (92), the gauge field aJ is simply related to the non-adiabatic coupling elements For an infinite set of electtonic adiabatic states [A = 00 in Eq. (90)], Ftc = 0. This important results seems to have been first established... 

In Chapter IV, Englman and Yahalom summarize studies of the last 15 years related to the Yang-Mills (YM) field that represents the interaction between a set of nuclear states in a molecular system as have been discussed in a series of articles and reviews by theoretical chemists and particle physicists. They then take as their starting point the theorem that when the electronic set is complete so that the Yang-Mills field intensity tensor vanishes and the field is a pure gauge, and extend it to obtain some new results. These studies throw light on the nature of the Yang-Mills fields in the molecular and other contexts, and on the interplay between diabatic and adiabatic representations. 

Fig. 21. Magnetostrictive level sensors measure the intersection of two magnetic fields one in the float, the other in the gauge.

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