Dirac 4-current

There are correspronding results for Qy,Qz and again the L and S parts differ by two orders of magnitude, the large parts reproducing the densities obtained in 2-component Pauli approximation (cf. (9) et seq.). When the curl of the corresponding magnetization density is added to (43), for the case = 1, we obtain the Dirac current density which already includes the spin term. 

This process cannot be described classically, because positrons are the result of the Dirac equation, but it illustrates the fact that a vacuum current (of photons) is made up of the interaction of two Dirac currents, one for the electron, one for the positron, and these are both matter currents. Therefore, there is a transmutation of matter current to vacuum current. On the classical level, this can be described in the scalar internal gauge space as... 

The Dirac current j e M associated with a Dirac spinor is given (Appendix A.3) by the equivalence... 

The vectors e3, u, v or n, w, v are present in the expressions of the E3 space component of space-time vectors, as the Dirac current corresponding to a state or the probability transition current between two states. But, with the use of STA these vectors are also present in the expression of the wave function ip and allow for a simple and clear construction of these currents. 

Both 4> r) corresponds to the Darwin solution for the state whose energy is E, and (14.4) is acceptable in average by means of an integration on the E3 space of a formula in which the Dirac current... 

If ip is associated with a Galilean frame eM, all the properties of invariance met in the Dirac theory are immediately deduced, in particular the one of the Dirac current... 

So, three fundamental properties in the theory of the electron, the energy, the spin, and the gauge, are directly related to a plane orthogonal to the Dirac current. 

Another relatively recent technique, in its own way as strange as Mossbauer spectrometry, is positron annihilation spectrometry. Positrons are positive electrons (antimatter), spectacularly predicted by the theoretical physicist Dirac in the 1920s and discovered in cloud chambers some years later. Some currently available radioisotopes emit positrons, so these particles arc now routine tools. High-energy positrons are injected into a crystal and very quickly become thermalised by... 

All solutions of this Hamiltonian are thereby electronic, whether they are of positive or negative energy and contrary to what is often stated in the literature. Positronic solutions are obtained by charge conjugation. From the expectation value of the Dirac Hamiltonian (23) and from consideration of the interaction Lagrangian (16) relativistic charge and current density are readily identified as... 

Due to its zero-gap electronic stmcture, large graphene sheets are not suitable for FET applications [185, 193]. It has been shown that graphene transistors conduct current even at the point of expected isolator behavior (i.e., Dirac point) [288]. Therefore, current modulation cannot be achieved using macroscopic graphene sheets [185]. In order to have a band gap on graphene, the use of narrow ribbons of... 

The mass dependence of the correction of order a Za) beyond the reduced mass factor is properly described by the expression in (3.7) as was proved in [11, 12]. In the same way as for the case of the leading relativistic correction in (3.4), the result in (3.7) is exact in the small mass ratio m/M, since in the framework of the effective Dirac equation all corrections of order Za) are generated by the kernels with one-photon exchange. In some earlier papers the reduced mass factors in (3.7) were expanded up to first order in the small mass ratio m/M. Nowadays it is important to preserve an exact mass dependence in (3.7) because current experiments may be able to detect quadratic mass corrections (about 2 kHz for the IS level in hydrogen) to the leading nonrecoil Lamb shift contribution. 

The quantum mechanical expression for the charge-weighted current density is obtained from Eq. (26) when we replace the classical velocity r (f) by the Dirac velocity operator caL and evaluate its expectation value (21),... 

According to the Dirac [36] electron theory, the relativistic wavefunction has four components in spin-space. With the Hermitian adjoint wave function , the quantum mechanical forms of the charge and current densities become [31,40]... 

This result, as well as the form of expressions (23) and (24), shows that the charge and current density relations (3), (4), and (8) of the present extended theory become consistent with and related to the Dirac theory. It also implies that this extended theory can be developed in harmony with the basis of quantum electrodynamics. 

The introduced current density j = So(divE)C is thus consistent with the corresponding formulation in the Dirac theory of the electron, but this introduction also applies to electromagnetic field phenomena in a wider sense. 

Close-looping an overunity EM system has very special Dirac sea hole current phenomena involved and special techniques are required. Bedini and the present author have filed a patent application on the major process required, and the details will be released a year from that filing. 

We state, however, that at about COP a 2.0 Special Dirac sea hole current phenomena are encountered in close-looping, as a new kind of decay mechanism from the disequilibrium state back to the Lorentz equilibrium. Bedini and Bearden have filed a patent application for energy transduction processes to overcome this effect and allow close-looping. 

Replacing the Fermi—Dirac distribution by the Boltzman distribution (for U below the Fermi level) and integrating eqn. (151), a current maximum close to the Fermi level of the electrode is predicted with the major contribution within kB T around UF. 

Relativistic quantum chemistry is currently an active area of research (see, for example, the review volume edited by Wilson [102]), although most of the work is beyond the scope of this course. Much of the effort is based on Dirac s relativistic formulation of the Schrodinger equation this results in wave functions that have four components rather than the single component we conventionally think of. As a consequence the mathematical and computational complications are substantial. Nevertheless, it is very useful to have programs for Dirac-Fock (the relativistic analogue of Hartree-Fock) calculations available, as they can provide calibration comparisons for more approximate treatments. We have developed such a program and used it for this purpose [103]. 

These laws are useful but represent cause without effect, that is, fields propagating without sources, and the Maxwell displacement current is an empirical construct, one that happens to be very useful. These two laws can be classified as U(l) invariant because they are derived from a locally invariant U(l) Lagrangian as discussed already. Majorana [114] put these two laws into the form of a Dirac-Weyl equation (Dirac equation without mass)... 

From the symmetric set, an extended set of Maxwell equations was exhibited in Section V.E. This set contains currents and sources for both fields E, B. The old conjecture of Dirac s is vindicated, but the origin of charge density is always electric (i.e., no magnetic monopole). Standard Maxwell s equations are a limiting case in far field. 

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