4-vector current

E and B are the fundamental force vectors, while P and H are derived vectors associated with the state of matter. J is the vector current density. The Maxwell equations in terms of E and B are... 

We see that the modes near the Fermi surface contributes only some parts of the axial anomaly. As in the vector current, the rest should come from modes in the deep Fermi sea and from anti-particles. To recover the full axial anomaly we add a counter term (See Fig. 7), which is two thirds of the axial anomaly plus a Chern-Simons term ... 

To examine the long-distance behavior of the vector current, we note that the correlator of the vector current for a given gauge field A can be written as... 

This is analogous to the current algebra for the weak and electromagnetic interactions between fermions. We have the two vector current operators [16]... 

One occurrence is a violation of the conservation of the axial vector current. We have that the 1 and 2 currents are conserved and invariant. On the high-energy vacuum we expect that currents should obey... 

Here, the additional term acts like vector current and by making a fit of geologic data, Schrodinger arrived at a finite rest mass. However, he pointed out that this effect might be produced by positive or negative particles revolving around the earth at same distances in the equatorial plane. ... 

The conservation of the matrix current along the connector, I = const, following from Eq. (4) and the boundary condition in Eq. (36), results in conservation of the vector current, I = IN = IB = const. This implies that for all elements of the connector, the unit vectors p coincide, therefore the... 

Because W/tv is an antisymmetric Lorentz tensor, the total iso vector current density must satisfy dvJvw = 0. 

Vector analysis. The author learnt vector analysis in a 1950 postgraduate course, based on the German book of 1932, Classical Electricity and Magnetism, by M. Abraham and R. Becker, published by Blackie and Sons, Glasgow. The book was written so that its definitions of scalar flux (flow in all directions) and of vector current (flow in one direction) fitted both hydrodynamics and electromagnetism. The same definitions were carried over unmodified into the scalar neutron fluxes and vector neutron currents of the nuclear power reactor. In electrochemistry, however, the term exchange current attaches to what is more properly described (see above) as a local, somewhat anisotropic, scalar flux. 

Fick s law of diffusion, J = —D9C/9x, applies when there is a source and a sink, with a vector current between then proportional to the concentration gradient. An equilibrium electrochemical reaction on open circuit is not a sink for reactants nor a source of product. It is a zero-flow blockage, albeit with balanced exchange currents. The flow of ions in the cell is a balance the drive V is balanced by an opposing concentration difference. 

Most of the expression vectors currently in use are constructed and propagated in E. coli bacterium. Large amounts of the vector DNAs are extracted from E. coli cultures and introduced into the chosen host cell. (A notable exception to this generalization are vectors derived from mammalian and insect viruses, but even in these cases at least some of the components of the expression vector are assembled in E. coli.) Thus, vectors assembled by the manipulation of DNA by restriction nucleases and ligase are introduced in the naked-DNA form into E. coli rendered competent to accept extraneous DNA. This process is termed Transformation Once the vector enters the E. coli cell, it has to be replicated and maintained in order to avoid its loss due to the dilution that accompanies rounds of cell division. The presence of a DNA sequence on the vector which functions as the origin of replication permits the vector to replicate in the host cell. 

An operator with the property exhibited in eqn (5.5) is said to be Hermitian if it satisfies this equation for all functions P defined in the function space in which the operator is defined. The mathematical requirement for Hermiticity of H expressed in eqn (5.5) places a corresponding physical requirement on the system—that there be a zero flux in the vector current through the surface S bounding the system To illustrate this and other properties of the total system we shall assume, without loss of generality, a form for H corresponding to a single particle moving under the influence of a scalar potential F(r)... 

The left-hand side of eqns (5.22) and (5.24) may be explicitly evaluated using Schrodinger s equations (5.2) for 4 and and expressed in terms of the quantum vector current density, eqn (5.11). For the time derivative of one has... 

Equation (5.93) is a statement of a physical principle as it can be restated in terms of the flux in the infinitesimal change in vector current density through the surface bounding the atom. In analogy with the definition of the charge density, the single-particle vector current density, j(r), of a many-particle... 

In terms of this vector current the variation in the atomic energy functional is given by... 

Consider an element of surface area dS and define n to be the outwardly directed unit vector normal to dS. Then j n is the component of the vector current density normal to the surface element dS and the current through this element is j n dS. The total current out of some region 1 bounded by a surface S( 2) is obtained by summing the contributions from all its surface elements. The resulting surface integral is called the flux j through the surface S( l) , that is... 

The non-vanishing of the subsystem average of the commutator implies a fluctuation in the value of the observable G over the subsystem as measured by the flux of its vector current density through the surface of the subsystem. Thus one anticipates and finds non-vanishing fluctuations in subsystem expectation values for observables which do not commute with H. 

The opening section of this chapter stressed the importance of the presence of the surface integral in the hypervirial theorem for an open system, eqn (6.2). Unlike the theorem for a total system, eqn (6.4), in which case the average of the commutator of any observable G with the Hamiltonian H vanishes, the corresponding result for an atom in a molecule is proportional to the flux in the effective single-particle vector current density of the property G through the atomic surface. As a result, the hypervirial theorem plays an important role in determining the properties of an atom in a molecule. It also enables one to relate an atomic property to a sum of bond contributions, as is now demonstrated. 

Thus the change in the value of the property A that occurs as a result of an atom entering into chemical combination is given by the flux in the vector current density of its associated generator G through the atomic surface. 

Equation (8.175) is a generalization of Ehrenfest s theorem (Ehrenfest 1927). This theorem relates the forces acting on a subsystem or atom in a molecule to the forces exerted on its surface and to the time derivative of the momentum density mJ(r). It constitutes the quantum analogue of Newton s equation of motion in classical mechanics expressed in terms of a vector current density and a stress tensor, both defined in real space. 

The quantity jg is the vector current density for a particle in the presence of an electromagnetic field and is defined as... 

A parallel development of the atomic contribution to the magnetic susceptibility of a molecule is readily obtained, the charge density being replaced by the vector current density. The magnetic dipole moment m is defined in terms... 

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