Gauge transformation density

The electromagnetic field may now formally be interpreted as the gauge field which must be introduced to ensure invariance under local U( 1) gauge transformation. In the most general case the field variables are introduced in terms of the Lagrangian density of the field, which itself is gauge invariant. In the case of the electromagnetic field, as before,... 

Therefore the Lehnert equation (253) correctly conserves action under a local U(l) gauge transformation in the vacuum. Such a transformation leads to a vacuum charge current density as the result of gauge theory itself, because U(l) gauge theory has a scalar internal space that supports A and A. These must be complex in order to define the globally conserved charge ... 

So it becomes clear that the description of the vacuum in gauge theory can be developed systematically by recognizing that, in general, A is an -dimensional vector. On the U(l) level, it is one-dimensional on the 0(3) level, it is three-dimensional and so on. The internal gauge space in this development is a physical space that can be subjected to a local gauge transform to produce physical vacuum charge current densities. 

It has been demonstrated already that local gauge transformation on this Lagrangian leads to Eq. (153), which contains new charge current density terms due to the Higgs mechanism. For our present purposes, however, it is clearer to use the locally invariant Lagrangian obtained from Eq. (325), specifically... 

In classical electrodynamics, the field equations for the Maxwell field A/( depend only on the antisymmetric tensor which is invariant under a gauge transformation A/l A/l + ticduxix), where x is an arbitrary scalar field in space-time. Thus the vector field A/( is not completely determined by the theory. It is customary to impose an auxiliary gauge condition, such as 9/x/Fx = 0, in order to simplify the field equations. In the presence of an externally determined electric current density 4-vector j11, the Maxwell Lagrangian density is... 

This defines the fermion contribution to an isovector gauge current density. Although the Euler-Lagrange equation is gauge covariant by construction, this fermion gauge current is not invariant, because the matrix r does not commute with the 5(7(2) unitary transformation matrices. It will be shown below that the... 

Chemical behaviour depends on chemical potential and electromagnetic interaction. Both of these factors depend on the local curvature of space-time, commonly identified with the vacuum. Any chemical or phase transformation is caused by an interaction that changes the symmetry of the gauge field. It is convenient to describe such events in terms of a Lagrangian density which is invariant under gauge transformation and reveals the details of the interaction as a function of the symmetry. The chemically important examples of crystal nucleation and the generation of entropy by time flow will be discussed next. The important conclusion is that in all cases, the gauge field arises from a symmetry of space-time and the nature of chemical matter and interaction reduces to a function of space-time structure. 

Since the current density is gauge invariant the proof of the theorem can be carried out with an arbitrary representative of the gauge class of v, A) and an arbitrary representative of the gauge class of (r. A ). As representatives we choose those four-potentials having a vanishing electric potential, i.e., for v(r, t) we make a gauge transformation (110) satisfying... 

Solvent-induced effects on NMR shielding of 1,2,4,5-tetrazine and two isomeric tetrazoles are calculated using density functional theory combined with the polarizable continuum model and using the continuous set gauge transformation/ Direct and indirect solvent effects on shielding are also calculated. 

The relativistic generalisation of the Hohenberg-Kohn theorem states that the external four-potential is - except for a gauge transformation - determined by the four-current of the system. The first component e7° is the charge density while the spatial components, J are associated both with orbital currents and the spin density. In the non-relativistic limit, the coupling of electron spin to an external magnetic field is automatically retrieved. 

The magnetic fieid B points toward the viewer. The structures were optimized at the B3LYP/6-31G ievei of density function theory. The current density was caicuiated using the continuous set of gauge transformation (CSGT) method [54]. 

Exchange-correlation functionals using current density are, therefore, also not invariant for the gauge transformation. To ensure gauge invariance, the gauge-invariant vorticity. 

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