Currents conserved

The second term represents the conserved current and the third is a momentum term. Noting that dL/dq = P, the field momentum bscomes... 

Fischer C, Kakoulli I (2006) Multispectral and hyperspectral imaging technologies in conservation current research and potential applications. Rew Conserv 7 3-16 and references therein. 

This type of transformation is not dependent on spacetime and is purely internal [46] in Noether s theorem. Under a global gauge transformation, Noether s theorem gives the conserved current... 

By construction, and taking into account Eqs. (43), the density of electromagnetic helicity is a conserved current for any Maxwell field in vacuum (with the above indicated behavior at infinity) ... 

In this context, environmental monitoring and inventory study are important as the basis for formulating and development of NIP. Several scientists of the region may have been dealing with POPs monitoring, however not much of the results could be accessed widely. Comprehensive monitoring researches since 1990s to date from the Department of Environment Conservation (currently Center for Marine Environmental... 

A mathematical description of an electrochemical system should take into account species fluxes, material conservation, current flow, electroneutrality, hydrodynamic conditions, and electrode kinetics. While rigorous equations governing the system can frequently be identified, the simultaneous solution of all the equations is not generally feasible. To obtain a solution to the governing equations, we must make a number of approximations. In the previous section we considered the mathematical description of electrode kinetics. In this section we shall assume that the system is mass-transport limited and that electrode kinetics can be ignored. 

The conserved current density determined by Noether s theorem is... 

An advantage of the Lagrangian formulation is that it permits the identification of conserved quantities by studying the invariances of the action S. The fundamental result behind this statement is Noether s theorem, which identifies conserved currents associated with invariance of S (and hence L) under infinitesimal continuous transformation. 

This is Noether s theorem. The quantity jf in the parlance of modern physics is known as a conserved current. The theorem (13) states that if the... 

The analysis of mass transfer in electrochemical cells requires the use of equations that describe the condition of electroneutrality (which applies for the entire elecnolyte outside the double layer at an electrode), species fluxes, mass conservation, current density, and fluid hydrodynamics. Often, mass transport events are rate limiting, as compared to kinetics processes at the electrode surface, in which case the overall electrode reaction rate is solely dependent on species mass transfer (e.g., during high-rate electroplating of some metals and for those elecnochemical reactions where the concentration of reactant in solution is low). 

This confirms that = —ecy/y yr is a conserved current, as in (39). Other conserved quantities, such as the energy-momentum tensor can be derived in similar fashion. 

The weak interactions that cause radioactive decay involve charge changing currents. These interactions affect only the nucleus, and play no other role in stable atoms. However, it was soon realized that there is no reason in principle to exclude charge conserving currents in the weak interjictions, though such interactions were not observed until the 1970 s. Such interactions can affect atoms, since the nucleus remains intact. The first discussion... 

Conservative current flux resulting from the electroneutrality of the conducting media ... 

In Subsections 5.1, 5.2, a form of Noether s Theorem has been applied in order to derive the associated weak statements of the conserved currents. This implementation led to Equations 14, 16, which correspond to the conservation of energy and momentum, respectively. These equations express in a clear manner the participation of each primary variable in the statements of conserved currents, a task that proves to be not trivial. To be more specific, in the case of linear and angular momentum-conservation statement 16, only the weak velocities and not, as someone may expect, the momentum type variables enter. Moreover, in the case of energy conservation, it is shown in (Eq. 14) that the weak velocities and not the strong time derivatives of displacement determine the kinetic energy. 

The symbol ( ) denotes parameters of the spatial discretization. Moreover, matrices M, M, M, M contain the space integral of products of the selected spatial interpolation functions. It is worth noting that (Eqs. 20, 22) coincide with those presented in Simo Tarnow (1992), if we adopt the same spatial interpolation for the whole set of variables and their variations. However, in the present paper, they have been explicitly derived in a rational manner. Furthermore, (Eq. 21), which provides the basis for the consistent computation of the linear momentum of the discrete system, does not appear in the aforementioned publication. Note also that examination of the conserved currents is consistent with the weak form of Noether s Theorem as proposed in Section 5 and justifies the procedures employed by Simo Tarnow (1992). Moreover, in case of... 

The electromagnetic current is a conserved current. Mathematically this is expressed by the vanishing of the four-dimensional divergence of... 

It is now postulated that (1.2.12) holds for the entire current, not just for the nucleon piece of it, and therefore V" is a conserved current. But precisely because it is conserved its matrix elements can be shown to be uninfluenced by the strong interactions, as is explained in some detail in Appendix 3, so that... 

The fact that an invariance of leads to a conserved current is known as Noether s theorem ind the currents are often called Noether currents (see, for example, Ramond, 1981). 

This unwelcome discovery is potentially catastrophic for our unified weak and electromagnetic gauge theory. There we have lots of gauge invariance, many conserved currents, both vector and axial-vector, and hence many Ward identities. Moreover the Ward identities play a vital role in proving that the theory is renormalizable. It is the subtle interrelation of matrix elements that allows certain infinities to cancel out and render the theory finite. Thus we cannot tolerate a breakdown of the Ward identities, and we have to ensure that in our theory these triangle anomalies do not appear. 

It may seem odd that a spin 1 particle like the W can transform directly into a spin 0 tt in Fig. 14.6(c). The reason for this is rather subtle (Leader, 1968). In a Feynman diagram the propagator of a vector meson off mass shell is really a mixture of spin 1 and spin 0 pieces. Only on mass shell does it reduce to pure spin 1. However, if the vector meson couples to a conserved current (as does the photon) the spin 0 part gives no contribution. The axial part of the current to which the W couples is, of course, not conserved, which explains its ability to transform into the pseudo-scalar ir. On the other hand, according to CVC (see Section 1.2), the W couples to a conserved vector current and thus it cannot transmute into a spin 0 scalar particle. 

It is these relations together with the exact results for the hadronic matrix elements of conserved currents discussed in Appendix 3 that provide the real basis for the sum rules quoted in eqns (16.4.2) to (16.4.4). 

---