12-vector identity, 0 electrodynamics

We must next consider more precisely the connection between the description of bodily identical states by the two observers (the requirements of Postulate 1). Quite in general, in fact, a physical theory, and quantum electrodynamics in particular, is fully defined only if the connection between the description of bodily identical states by (equivalent) observers is known for every state of the system and for every pair of observers. Since the observers are equivalent every state which can be described by 0 can also be described by O. Given a bodily state of the same system, observer 0 will ascribe to it a state vector Y0> in his Hilbert space and observer O will attribute to it a state vector T0.) in his Hilbert space. The above formulation of invariance means that there exists a one-to-one correspondence between the vectors Y0> and Y0.) used by observers 0 and O to describe bodily the same state.3 This correspondence guarantees that the two Hilbert spaces are in fact isomorphic. It is, therefore, possible for the two observers to agree to describe states of the system by vectors in the same Hilbert space. A similar statement can be made for the observables there exists a one-to-one correspondence between the operators Q0 and Q0>, which observers 0 and O attribute to observables. The consistency of the theory (Postulate 2) demands, however, that the two observers make the same prediction as the outcome of the same experiment performed on bodily the same system. This requires the relation... 

In this final section, it is shown that the three magnetic field components of electromagnetic radiation in 0(3) electrodynamics are Beltrami vector fields, illustrating the fact that conventional Maxwell-Heaviside electrodynamics are incomplete. Therefore Beltrami electrodynamics can be regarded as foundational, structuring the vacuum fields of nature, and extending the point of view of Heaviside, who reduced the original Maxwell equations to their presently accepted textbook form. In this section, transverse plane waves are shown to be solenoidal, complex lamellar, and Beltrami, and to obey the Beltrami equation, of which B is an identically nonzero solution. In the Beltrami electrodynamics, therefore, the existence of the transverse 1 = implies that of , as in 0(3) electrodynamics. 

All these dipoles are conservative ones with respect to the entity number, the basic quantity in this case of capacitive dipoles. The interesting feature is that the separability is linked to the symmetry between energies-per-entity (here efforts) When the dipole is inseparable, both efforts are equal in magnitude but opposed in direction (for vectors) or value (for scalars). The converse is not true such symmetry may be found in peculiar cases for separable dipoles. For instance, the two potential values Vi and V2 of a capacitor may be equal in magnitude and opposite however, this happens only in case of equal pole capacitances. This is a frequent case in electrodynamics when the two capacitor plates are strictly identical, as in the case of planar capacitor, but this is not general as nonidentical shapes or geometries can also be found. Note that, in physical chemistry, this never happens, because it would correspond to identical partners in a chemical reaction of to identical phases in an interface ... 

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