Four-current density

In the relativistic version of DFT, the ground state energy of the system is a unique functional of the fermion four-current density, / (/-) ... 

In this equation, A, is a given external four-potential and J (x) is the four-current density. As the treatment is for stationary states, the external four-potential and the fermion four-current are independent of time. 

The energy (3.32) is, via the (rather involved) dependence of the RKS-orbitals on the four current density, a functional of this quantity,... 

In Section V it will be shown that the quaternion structure of the fields that correspond to the electromagnetic field tensor and its current density source, implies a very important consequence for electromagnetism. It is that the local limit of the time component of the four-current density yields a derived normalization. The latter is the condition that was imposed (originally by Max Bom) to interpret quantum mechanics as a probability calculus. Here, it is a derived result that is an asymptotic feature (in the flat spacetime limit) of a field theory that may not generally be interpreted in terms of probabilities. Thus, the derivation of the electromagnetic field equations in general relativity reveals, as a bonus, a natural normalization condition that is conventionally imposed in quantum mechanics. 

My investigation proceeded along the following lines [12]. The conventional coupling of an external electromagnetic potential field A to an electrical four-current density of matter is in terms of the scalar interaction Lagrangian... 

We see here that the real and imaginary parts of the complex invariant I correspond to the two invariants of the standard form of electromagnetic theory, namely, the scalar and the pseudoscalar terms. They appear together here in a single complex function because of the reflection-nonsymmetric feature of this theory. The invariant h corresponds to the real-valued modulus of the four-current density of the standard theory. 

In this expression for the electromagnetic field intensity tensor, Q is a constant of proportionality with the dimension of charge, inserted on both sides of Eq. (48) (negative). The four-current density is... 

Integration of the local limit of Eq. (50) for the four-current density source of Maxwell s equations, together with the boundary condition that the... 

Thus, the four-current density jy of this expression of the Maxwell theory has the following general form in a curved spacetime ... 

We have seen in this section that the factorization of Einstein s symmetric, second-rank tensor field equations (10 relations) to a quaternion form (16 relations) not only yields the gravitational and electromagnetic manifestations of matter in a unified field theory but also reveals a feature of quantum mechanics. In particular, it was found that in the flat-space approximation to the curved-space representation in general relativity, the time component of the electromagnetic four-current density corresponds in a one-to-one way with the probability density of quantum mechanics. Its integration over all of space in this limit is found to be unity. 

Figure 3. Changes in resistance with time for four current densities and a PR waveform.

Figure 5. Changes in stress with time for four current densities and a PR waveform. C denotes an initially compressive film and T denotes an initially tensile film.

A BRIEF INTRODUCTION TO FOUR-CURRENT DENSITY FUNCTIONAL THEORY... 

In the non-relativistic theory, the kinetic energy is T = -EVf/2 >, and T < -t-oo implies n E L T ) for the density n via Sobolev s inequality [3]. The relativistic kinetic energy is (in the Schrodinger representation for ) Tr = ( I X) V j I ), and one is tempted to suppose G (T ). If for instance a bispinor orbital behaves like (f) r , then G implies s < 1 for which indeed Tr < +00. In the above exploited duality theory of Legendre transforms the variational space of admitted potentials is the dual of the variational space of admitted four-current densities, hence A G L T ) must be demanded. For a potential A this implies again s < 1 and hence excludes Coulomb potentials although it permits to treat them as a limiting case. 

The four-current density functional H[J] is split into an orbital variation part K and a density integral L ... 

A semiempirical model that considered only the effect of toluene on kinetics was constructed to describe cell voltage as a fimction of contamination time and current density. The parameters independent of the contamination process, i.e., open circuit voltage (E°), cell resistance (RJ, and Tafel slope (fc), were first estimated based on experimental data in the absence of toluene (giving a baseline). Then these parameters were used to empirically obtain the expressions of two other parameters, and Kcbc/ which accoimted for the effect of toluene contamination on transient and steady-state cell performance, using experimental data at various levels of toluene concentration imder four current densities. The model was validated by comparing the contamination testing results with model-predicted results. Several other definitions were also presented, based on the model, such as the threshold toluene concentration and the degradation rate. 

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